Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case
In this paper, we study the spectral theory of soliton condensates - a special limit of soliton gases - for the focusing NLS (fNLS). In particular, we analyze the kinetic equation for the fNLS circular condensate, which represents the first example of an explicitly solvable fNLS condensate with nont...
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| description | In this paper, we study the spectral theory of soliton condensates - a special limit of soliton gases - for the focusing NLS (fNLS). In particular, we analyze the kinetic equation for the fNLS circular condensate, which represents the first example of an explicitly solvable fNLS condensate with nontrivial large-scale space-time dynamics. Solution of the kinetic equation was obtained by reducing it to Whitham-type equations for the endpoints of spectral arcs. We also study the rarefaction and dispersive shock waves for circular condensates, as well as calculate the corresponding average conserved quantities and the kurtosis. We want to note that one of the main objects of the spectral theory - the nonlinear dispersion relations - is introduced in the paper as some special large genus (thermodynamic) limit of the Riemann bilinear identities that involve the quasimomentum and the quasienergy meromorphic differentials.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 070, 26 pages
Soliton Condensates for the Focusing Nonlinear
Schrödinger Equation: a Non-Bound State Case
Alexander TOVBIS a and Fudong WANG bc
a) University of Central Florida, Orlando FL, USA
E-mail: alexander.tovbis@ucf.edu
b) School of Sciences, Great Bay University, Dongguan, P.R. China
c) Great Bay Institute for Advanced Study, Dongguan, P.R. China
E-mail: fudong@gbu.edu.cn
Received December 30, 2023, in final form July 16, 2024; Published online July 31, 2024
https://doi.org/10.3842/SIGMA.2024.070
Abstract. In this paper, we study the spectral theory of soliton condensates – a special
limit of soliton gases – for the focusing NLS (fNLS). In particular, we analyze the kinetic
equation for the fNLS circular condensate, which represents the first example of an explicitly
solvable fNLS condensate with nontrivial large scale space-time dynamics. Solution of the
kinetic equation was obtained by reducing it to Whitham type equations for the endpoints
of spectral arcs. We also study the rarefaction and dispersive shock waves for circular
condensates, as well as calculate the corresponding average conserved quantities and the
kurtosis. We want to note that one of the main objects of the spectral theory – the nonlinear
dispersion relations – is introduced in the paper as some special large genus (thermodynamic)
limit the Riemann bilinear identities that involve the quasimomentum and the quasienergy
meromorphic differentials.
Key words: soliton condensate; focusing nonlinear Schrödinger equation; kurtosis; nonlinear
dispersion relations; dispersive shock wave
2020 Mathematics Subject Classification: 37K40; 35P30; 37K10
1 Introduction
1.1 Soliton gases for integrable equations
Many nonlinear integrable equations have special solutions representing solitary waves – localized
traveling wave solutions also known as solitons. Solitons have some very peculiar and well studied
properties. One of the most celebrated of them is the property that two solitons with different
velocities retain their shapes and velocities after the interaction, so that the only result of this
interaction is a phase change (e.g., a shift in the position of the center) of the above solitons.
There are also more complicated n-soliton solutions which can be considered as ensembles of n
interacting solitons.
The spectral theory of soliton gases (see, for example, [9] or [8, Section 3]) is based on the
natural idea to interpret solitons as particles of some gas with elastic two-particle interactions.
Assuming that a given soliton (a particle) undergoes consistent interactions (phase shifts) with
other solitons (particles) in the gas, it is clear that its actual velocity will be affected by the
persistent interactions, namely, by the density and the characteristics of other solitons in the
gas interacting with the given one. This idea goes back to the paper of V.E. Zakharov [18],
This paper is a contribution to the Special Issue on Evolution Equations, Exactly Solvable Mod-
els and Random Matrices in honor of Alexander Its’ 70th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Its.html
mailto:alexander.tovbis@ucf.edu
mailto:fudong@gbu.edu.cn
https://doi.org/10.3842/SIGMA.2024.070
https://www.emis.de/journals/SIGMA/Its.html
2 A. Tovbis and F. Wang
where the formula for the effective velocity of a soliton in a rarefied Korteweg–de Vries (KdV)
soliton gas was first presented. However, the case of a dense soliton gas required a different
approach, which was suggested by G. El in [7] for KdV soliton gases and later by G. El and
A. Tovbis in [9] for the focusing nonlinear Schrödinger equation (fNLS) soliton gases. This
approach is based on considering a certain large n limit of special n-phase nonlinear wave (finite
gap) solutions to an integrable system, called the thermodynamic limit. The subject of our
interest in this paper will be not the large n limit of finite gap solutions to integrable equations,
but rather the scaled continuum limit of wavenumbers kj and of frequencies ωj , associated with
these solutions. These limits will be interpreted as the density of states (DOS) u(z) and the
density of fluxes (DOF) v(z) respectively of the corresponding soliton gas, where z denotes the
spectral parameter. The main results of the paper will be described in Section 1.3 below, while
some brief background information on spectral theory of soliton gases and soliton condensates
will be given in Section 1.2. More detailed information on the subject can be found in the review
paper [8]. We now turn to a brief discussion of the thermodynamic limit – the key concept of
the spectral theory of soliton gases.
Since finite gap solutions can be represented in terms of the Riemann theta functions, it is
convenient for us to describe soliton gases starting from the corresponding Riemann surfaces.
In particular, genus n finite gap solutions to the fNLS, considered in this paper, can be defined
in terms of a genus n Schwarz symmetrical hyperelliptic Riemann surface Rn. Additional in-
formation in the form of n initial phases is required to define a particular finite gap solution.
However, the wavenumbers kj and frequencies ωj of any finite gap solution to the fNLS, associ-
ated with Rn, are defined in terms of Rn only. In particular, let dpn, dqn be the second kind
real normalized meromorphic differentials on Rn with poles only at infinity (both sheets) and
the principal parts ±1 for dpn and ±2z for dqn there respectively, where the spectral parameter
z ∈ Rn. Here real normalized mean that all the periods of dpn, dqn on Rn are real. Note that
the above conditions uniquely define dpn, dqn on Rn. Then the vectors k⃗, ω⃗ of the (real) periods
of dpn, dqn with respect to a fixed homology basis (A and B cycles) of Rn are vectors of the
wavenumbers and frequencies of finite gap solutions on Rn, respectively, i.e.,
kj =
∮
Aj
dpn, k̃j =
∮
Bj
dpn, ωj =
∮
Aj
dqn, ω̃j =
∮
Bj
dqn,
where k⃗ = (k1, . . . , kn, k̃1, . . . , k̃n), ω⃗ = (ω1, . . . , ωn, ω̃1, . . . , ω̃n). The differentials dpn, dqn are
known as quasimomentum and quasienergy differentials, respectively, [10, 11].
Denote by wj,n = wj the j-th normalized holomorphic differential on Rn, j = 1, . . . , n, that
are defined by the condition
∫
Ak
wj = δk,j , k = 1, . . . , n, where δk,j is the Kronecker delta. The
well known Riemann bilinear relations (RBR)∑
j
[∮
Aj
wm
∮
Bj
dpn −
∮
Aj
dpn
∮
Bj
wm
]
= 2πi
∑
Res
(∫
wmdpn
)
, (1.1)
∑
j
[∮
Aj
wm
∮
Bj
dqn −
∮
Aj
dqn
∮
Bj
wm
]
= 2πi
∑
Res
(∫
wmdqn
)
, (1.2)
m = 1, . . . , n, form systems of linear equations for k⃗, ω⃗ respectively, where the summation is
taking over the only poles z = ∞± (infinities on both sheets of Rn) of dpn, dqn. Indeed, taking
real and imaginary parts of (1.1), one gets∑
j
kj Im
∮
Bj
wm = −2πRe
(∑
Res
(∫
wmdpn
))
, (1.3)
k̃m −
∑
j
kj Re
∮
Bj
wm = −2π Im
(∑
Res
(∫
wmdpn
))
, m = 1, . . . , n. (1.4)
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 3
The matrix of the n×n system of linear equations is positive definite, since it is the imaginary
part Im τ of the Riemann period matrix τ =
∮
Bj
wm. Once kj are known, the values of k̃j can
be calculated from (1.4). Thus, the systems (1.3)–(1.4) always have a unique solution. Similar
results are true for (1.2). So, the wavenumbers and frequencies vectors k⃗, ω⃗ are connected via
the underlying Riemann surface Rn. Thus, the Riemann bilinear relations for the quasimo-
mentum and quasienergy differentials dpn, dqn, connecting the wavenumbers and frequencies,
form the nonlinear dispersion relations (NDR) for the finite gap solutions to the fNLS, defined
by Rn.
One of the main subjects of the spectral theory of soliton gases is the thermodynamic limit
of scaled vectors k⃗, ω⃗, i.e., the thermodynamic limit of Riemann bilinear relations (1.1)–(1.2),
which leads to the continuum version of the NDR established in [9] in the form of two Fredholm
integral equations (1.5)–(1.6).
For simplicity of the exposition, let us assume that N is even and only one (Schwarz sym-
metrical) band γ0 intersects R. Before considering the thermodynamic limit, we want to address
the choice of the homology basis. Namely, the A cycles are chosen as counterclockwise loops
around each of the bands in C+ (on the main sheet) and clockwise loops around each of the
bands in C− (on the main sheet). Each such loop contains only one band (branch cut) inside.
That leaves one exceptional band γ0 not surrounded by a loop.
A B cycle Bj is represented by a loop going through the branch cut encircled by Aj and
through the exceptional branch cut γ0, see Figure 1. If one starts shrinking just one branch
cut of Rn, corresponding to Aj (and, generically, its Schwarz symmetrical) to a point while
leaving the other branch cuts unchanged, the system (1.1)–(1.2) implies that the corresponding
kj , ωj → 0, which can be associated with a solitonic limit. Indeed, an fNLS soliton is spectrally
represented by a pair of Schwarz symmetrical points and a complex norming constant, the
latter can be viewed as an analog of initial phases. For example, an elliptic solution to the fNLS
ψm = eit(2−m) dn(x,m), where dn denotes the corresponding Jacobi elliptic function, in the limit
m→ 1 goes into a soliton solution ψ1 = eit sech(x). That observation illustrates the connection
between finite gap and multi-soliton solutions.
Re z
γ0
Bj
Aj
Figure 1. The hyperelliptic Riemann surface Rn, indicating the orientations of the canonical homology
basis.
Considering the thermodynamic limit, we assume (see [8, 9]) that the number n + 1 of
the branch cuts of Rn is growing so that the centers of branch cut (spectral band) are accu-
mulating on some (Schwarz symmetrical) compact Γ in C with a normalized continuous den-
sity function ϕ(z) > 0 on Γ+ = Γ ∩ C+
(
i.e.,
∫
Γ+ ϕ(w)dλ(w) = 1 for some reference mea-
4 A. Tovbis and F. Wang
sure λ(w), see Section 1.2 below
)
. Simultaneously, all the bandwidth are shrinking at the order
e−ν(z)n, where ν(z) is a continuous non-negative function on Γ+, in such a way that the dis-
tance between any two bands should be of the order at least O(1/n). The wavenumbers kj
and frequencies ωj are called solitonic wavenumbers and frequencies respectively, because in
the thermodynamic limit they go to zero. The function σ(z) = 2ν(z)
ϕ(z) is called spectral scal-
ing function. In the thermodynamic limit, the system of linear equations (1.3) for the soli-
tonic wavenumbers kj turns into the integral equation (1.5) for the scaled continuum limit u(z)
of kj , see details in [9]. Similarly, the imaginary part of (1.2) for the solitonic frequencies ωj
turns into the integral equation (1.6) for the scaled continuum limit v(z) of ωj . Thus, the
NDR (1.5)–(1.6), discussed in Section 1.2 below, which is one of the main subjects of this
paper, represent the thermodynamic limit of the systems of linear equations for kj , ωj , i.e.,
the thermodynamic limit of imaginary parts of the RBR (1.1)–(1.2). So far, the derivation
of the NDR (1.5)–(1.6) was made rigorous [16] in the case when Γ+ is a curve and σ > 0
on Γ+. In the case σ(z) ≡ 0 on Γ+, i.e., in the case of sub-exponential decay of the band-
width, a soliton gas was called soliton condensate [9]. Soliton condensates have a natural
extremal property: if Γ+ is fixed but σ ≥ 0, σ ∈ C(Γ+), is allowed to vary, the maximal
average intensity of the fNLS soliton gas is attained at σ ≡ 0, i.e., for the soliton condensate,
see [13].
1.2 fNLS bound state soliton condensate
In this paper we consider soliton gases for the fNLS
iψt + ψxx + 2|ψ|2ψ = 0,
where x, t ∈ R are the space-time variables and ψ : R2 → C is the unknown complex-valued
function.
The nonlinear dispersion relations (NDR) for the fNLS soliton gas are integral equations (see,
for example, [9, 16]):∫
Γ+
log
∣∣∣∣z − w̄
z − w
∣∣∣∣u(w) dλ(w) + σ(z)u(z) = Im z, z ∈ Γ+, (1.5)∫
Γ+
log
∣∣∣∣z − w̄
z − w
∣∣∣∣v(w)|dλ(w) + σ(z)v(z) = −4Re z Im z, z ∈ Γ+, (1.6)
where Γ+ ⊂ C+ is a compact, λ(w) is a reference measure (for example, the arclength if Γ+
is a curve) and σ(z) : Γ+ 7→ [0,∞) is a continuous function. As it was mentioned above, the
uniqueness and existence of solutions to the discrete NDR ((1.3) and its analog for dqn) follow
from the positive-definiteness of the imaginary part of the Riemann period matrix. Similar
results for the continuous NDR (1.5)–(1.6) were obtained in [13], where each of the equations
was considered as a variational equation for the minimizer of the Green’s energy functional.
Definition 1.1. Let S be a subset of C and let z0 ∈ C. Then S is thick (or non-thin) at z0 if
z0 ∈ S \ {z0} and if, for every superharmonic function u defined on a neighborhood of z0,
lim inf
z→z0
z∈S\{z0}
u(z) = u(z0).
Otherwise, S is thin at z0.
A connected set with more than one point (for example, a contour) is thick at all of its
points. On the other hand, a countable set is thin at every point. We consider Γ+ to be a finite
collection of arc or closed regions in C. Here is one of the results from [13].
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 5
Theorem 1.2. Let σ be continuous on Γ+, and S0 =
{
z ∈ Γ+ | σ(z) = 0
}
. Suppose S0 is either
empty or thick at each z0 ∈ S0 (see Definition 1.1). Then the solution u(z) to (1.5) exists and
is unique; moreover, u(z) ≥ 0 on Γ+.
Similar results can be proven for (1.6) with the exception that, in general, v(z) can take both
positive and negative values on Γ+. In particular, the statement of Theorem 1.2 is valid in the
case σ ≡ 0, i.e., in the case of a soliton condensate. The ratio
s(z) =
v(z)
u(z)
(1.7)
represents the effective velocity of a tracer soliton (elements of the gas with the spectral char-
acteristic z) in the soliton gas.
Soliton gases (and condensates) discussed so far were equilibrium soliton gases, that is, the
spectral properties of these gases were uniform all over the physical (x, t)-plane. One can also
consider non-equilibrium soliton gases, where the parameters of the NDR (1.5)–(1.6) are slowly
varying with (x, t) on large (x, t) scales. That is, we assume that σ = σ(z;x, t), Γ+ = Γ+(x, t)
and so, the DOS u = u(z;x, t) and DOF v = v(z;x, t). In this case, the NDR (1.5)–(1.6) should
be supplemented by the “continuity equation”
∂tu(z;x, t) + ∂xv(z;x, t) = 0, (1.8)
thus forming the system known as kinetic equation see [9] (to be precise, often the NDR (1.5)–
(1.6) in the kinetic equation are replaced by the equation of state, which represents an integral
equation for the unknown s(z) in terms of Γ+ and u(z)).
Equation (1.8) can be used to illustrate 3 different scales naturally appearing in our approach
to soliton gases: the large number of micro-scale soliton-soliton interactions allows one to derive
the NDR describing meso-scale DOS u(z) and DOF v(z), assuming that at these relatively large
space-time scales the compact Γ+ and the spectral scaling function σ(z), determining u, v, are
virtually independent on x, t. Finally, equation (1.8) describes the macro-scale dynamics of
u(z;x, t) and v(z;x, t) for non-equilibrium soliton gases, where the large scale x, t dependence
of Γ+ and σ has to be taken into account.
Even though the existence and uniqueness of the solutions to the NDR (1.5)–(1.6) were
established in Theorem 1.2, the explicit form of such solutions is known in a very few special
cases, such as, for example, periodic gases, see [16], or bound state soliton condensates, where
σ ≡ 0 and Γ+ ⊂ a + iR+ for some a ∈ R. fNLS soliton gases with such Γ+ are called bound
state gases. This name reflects the fact that, as can be easily seen from (1.5)–(1.6), the effective
speed (1.7) for bound state gases is s = −4a for all z ∈ Γ+, i.e., all the elements (tracers) of the
gas have the same effective velocity. It was observed in [13] that, for a bound state condensate,
the DOS u(z) is proportional to the dp
dz , where dp is the quasimomentum differential on the
(limiting) hyperelliptic Riemann surfaceR defined by Γ. Similar results are valid for KdV soliton
condensates with u(z), v(z) proportional to dp
dz ,
dq
dz , where dq is the quasienergy differential on R.
For non-equilibrium KdV soliton condensates, it was proved in [3] that the integro-differential
kinetic equation for the DOS u = u(z;x, t) and DOF v = v(z;x, t) reduces to the multi-phase
KdV-Whitham modulation equations for the endpoints of Γ derived by Flaschka, Forest and
McLaughlin [10] and Lax and Levermore [14]. Riemann problems for soliton condensates and
explicit solutions for the kinetic equation describing generalized rarefaction and dispersive shock
waves were recently considered in [3].
Extension of these results of [3] to the fNLS bound states condensate is trivial since, as it was
mentioned earlier, s(z;x, t) is constant. In the present paper we provide the first known explicit
solutions of the NDR (1.5)–(1.6) for a class of non bound state fNLS condensates (i.e., σ = 0),
6 A. Tovbis and F. Wang
namely, for circular condensates, where the contour Γ+ is a collection of arcs (bands) located on
an upper semicircle |z| = ρ, where ρ is a real positive constant.
We then obtain the Whitham equations describing the x, t dynamics of the endpoints of
the arcs of Γ+, as well as some explicit solutions for the kinetic equation describing generalized
rarefaction and dispersive shock waves. We also analyze the kurtosis κ for equilibrium and
non-equilibrium circular fNLS gases. Here the kurtosis is the fourth normalized moment
κ =
⟨|ψ|4⟩
⟨|ψ|2⟩2
of the probability density function (PDF) of the random wave amplitude ψ. The fNLS evolution
of the so-called partially coherent waves, whose amplitude is given by a slowly varying random
function with a given (e.g., Gaussian) statistics, was studied in [4]. In particular, it was shown
there that long time fNLS evolution of partially coherent waves leads to the doubling of the
initial κ, so that the initial κ0 = 2, corresponding to the Gaussian distribution, eventually
becomes κ∞ = 4, indicating “fat tail” distribution and, thus, potential presence of rogue waves.
In Section 5, we calculate the kurtosis κ for genus one and zero circular condensates and show
that κ = 2 for a rarefaction wave and κ > 2 for a dispersive shock wave. Considering special
families of genus one condensates in the limit of diminishing bands, we show that any value
κ > 2 can be approached along certain trajectories in the corresponding parameter space.
The rest of the paper is organized as follows: in Section 1.3, we present the main results of
the paper (Theorems 1.3 and 1.4). The proof of Theorem 1.3 is given in Section 2. In Section 3,
we apply Theorem 1.3 to study the circular condensates in genus 0 and 1 cases, where the
explicit formulae for the DOS and DOF are presented. In Section 4, we study the modulational
dynamics of circular condensates. In Section 5, we analyze the kurtosis of the genus 0 and 1
circular condensates.
1.3 Main results
In this paper, we study the fNLS soliton condensate supported on a compact Γ+ ⊂ S+, where
S+ ⊂ C+ is the centered at z = 0 semicircle of the radius ρ > 0.
We consider Γ+ =
⋃N
k=1{ρeiθ : θ ∈ [α2k−2, α2k−1]}, where 0 ≤ α0 < α1 < · · · < α2N−1≤ π,
a collection of N ∈ N closed non-degenerate arcs (bands) of S+. We call zj = ρeiαj , j =
0, . . . , 2N − 1, endpoints of the bands. The bands are interlaced with N + 1 gaps lying on S+,
where the arcs from z = ±ρ to the nearest band are also considered as gaps, see Figure 2. Any
of the latter gaps are considered to be collapsed if the corresponding ±ρ ∈ Γ+. We also take the
reference measure λ(w) in (1.5)–(1.6) to be simply the arclength. The conformal map
p(z) =
1
2
(
z
ρ
+
ρ
z
)
(1.9)
maps C+ onto C with two branch cuts from ±1 to ±∞ respectively, where p(S+) = [−1, 1].
Let RN denotes the hyperelliptic Riemann surface with the branch cuts on (−∞,−1], [1,∞)
and on
Γ̂+ = p
(
Γ+
)
⊂ [−1, 1],
where p(Γ+) =
⋃N
k=1[a2k−2, a2k−1] and aj = p(zj) = p(ρ exp(iαj)).
Denote by û(p), v̂(p) solutions of singular integral equations
πH[û](p) :=
∫
Γ̂+
û(q)dq
q − p
= −p, πH[v̂](p) = 4ρ
(
2p2 − 1
)
on Γ̂+; (1.10)
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 7
R
z0
z1
z2N−1
z2N−2
0−ρ ρ
ρ
p(z) = 1
2
(
z
ρ + ρ
z
)
a0
a1
a2N−2
a2N−1 1−1−∞ +∞
C+ z C\(−∞,−1] ∪ [1,∞) p
Figure 2. Illustration of the conformal map p(z) from the upper half plane with n circular slits (namely,⋃N
k=1{ρeiθ : θ ∈ [α2k−2, α2k−1]}) to C\(−∞,−1]∪ (1,∞] with N slits (namely,
⋃N
k=1[a2k−2, a2k−1], where
aj = p(ρ exp(iαj))) within the interval [−1, 1]. In the left figure, zj = ρeiαj indicate the branch points.
The genus of RN , see the right plane, is N .
here H denotes the finite Hilbert transform (FHT) on Γ̂+. It is straightforward to show (see,
for example, [13]) that
û(p) =
P (p)
R(p)
, v̂(p) =
Q(p)
R(p)
, (1.11)
where P , Q are polynomials of degree N +1 and N +2 respectively and R(p) =
∏
j(p− p(zj))
1
2
taken over all the endpoints of Γ̂+ and normalized by R(p) ∼ pN as p → ∞ in C. However,
P , Q are not uniquely defined by (1.10), since the FHT H has an N -dimensional kernel. The
following theorem establishes solutions of the NDR (1.5)–(1.6) in terms of û, v̂.
Theorem 1.3. The NDR (1.5)–(1.6) for fNLS circular soliton condensate (i.e., with σ ≡ 0)
have solutions
u(z) = û(p(z)) =
P (p(z))
R(p(z))
, where πH[û] = −p on Γ̂+ = p(Γ+),
v(z) = v̂(p(z)) =
Q(p(z))
R(p(z))
, where πH[v̂] = 4ρ(2p2 − 1) on Γ̂+ = p(Γ+). (1.12)
Here P (p), Q(p) are polynomials of degrees N +1, N +2 respectively, p ∈ R, û(p)dp√
1−p2
, v̂(p)dp√
1−p2
are
second kind meromorphic differentials on RN and u, v satisfy the conditions∫
cj
u(z) d arg z = 0,
∫
cj
v(z) d arg z = 0 (1.13)
for all j = 0, . . . , N , where cj = {ρeiθ : θ ∈ (α2j−1, α2j)}, α−1 = 0, and α2N = π, denote
the gaps on S+ \ Γ+ where S+ = {|z| = ρ, 0 ≤ arg z ≤ π} (see Figure 2). In the case when
an endpoint ±ρ ∈ Γ+ and so the corresponding gap cj collapses, the corresponding integral
conditions in (1.13) should be replaced by u(±ρ) = 0, v(±ρ) = 0. Also, u(z), v(z) ∈ R on Γ+
and u > 0 on Γ+ \ R.
According to Theorem 1.2, a solution to NDR is unique. We now state the second main theo-
rem about non-uniform circular soliton condensate for the fNLS, where the endpoints zj = ρeiαj
of the bands on S+ can depend on x, t. It is governed by the equation (1.8) and reflect large
scale changes of u, v.
Let α⃗(x, t) = (α0(x, t), . . . , α2N−1(x, t)) denote the vector of x, t dependent endpoints of Γ+.
Assuming smoothness of u(z;x, t), v(z;x, t) and α⃗(x, t), and using Theorem 1.3 , we follow the
approach of [10] to show that the continuity equation (1.8) can be written as a Whitham type
equations on the evolution of α⃗(x, t). A similar result for the KdV soliton condensates was
obtained in [3].
8 A. Tovbis and F. Wang
Theorem 1.4. If the fNLS circular soliton condensate that described in Theorem 1.3, with
Γ+ ∩ R = ∅, is non-equilibrium, then (1.8) is equivalent to the system of modulation (Whitham)
equations given by
∂tαj + Vj(α⃗)∂xαj = 0, j = 0, . . . , 2N − 1, (1.14)
where zj = ρeiαj and
Vj(α⃗) =
Q(cosαj)
P (cosαj)
(1.15)
are bounded velocities, with P , Q as in Theorem 1.3.
Proof. Let us first obtain (1.14) from (1.8). According to Theorem 1.3, (1.8) can be written as(
P (p)
R(p)
)
t
+
(
Q(p)
R(p)
)
x
= 0, (1.16)
which should be valid for all p ∈ Γ̂+. Denoting T = R2, equation (1.16) can be written as
2T (Pt +Qx) = PTt +QTx or 2(Pt +Qx) = P (log T )t +Q(log T )x. (1.17)
Since
(log T )r = −
2N−1∑
j=0
(aj)r
p− aj
,
where r is either x or t, the second equation (1.17) yields
P (aj)(aj)t +Q(aj)(aj)x = 0, j = 0, 1, . . . , 2N − 1,
if we take limit p→ aj . Given aj = cosαj , the latter equation implies (1.14)–(1.15). Note that
all the zeros of P (p) are on the gaps and, thus, the velocities Vj defined by (1.15) are bounded.
Assume now that the evolution of the endpoints satisfies (1.14) or, equivalently,
∂taj + Vj (⃗a)∂xaj = 0, j = 0, . . . , 2N − 1,
where a⃗ = (a0, . . . , a2N−1). Then, following [10], we observe that the only poles of the differential
Ω = ∂tûdp + ∂xv̂dp on RN are the second-order poles at each aj . But the modulation equa-
tions (1.14) show that the principal parts of Ω at each p = aj is zero, that is, Ω is a holomorphic
differential. Since all the gap integrals (and, thus, the B-periods) of a holomorphic differential Ω
are zeros, we obtain the kinetic equation ∂tû+ ∂xv̂ = 0, which is equivalent to (1.8). ■
The modulation equations (1.14)–(1.15) form a strictly hyperbolic system of first-order quasi-
linear PDEs provided that all the branchpoints are distinct (otherwise it will be just hyperbolic).
This system is in the diagonal (Riemann) form with all the coefficients (velocities) being real.
The Cauchy data for this system consists of α⃗(x, 0). The system has a unique local (classic)
real solution provided α⃗(x, 0) is of C1 class, see [5, Theorem 7.8.1], and real and the veloci-
ties Vj(α⃗) are smooth and real. Thus, the fNLS circular gas is (at least locally) preserved under
the evolution described by the kinetic equation.
Remark 1.5. As it is well known, systems of hyperbolic equations may develop singularities in
the x, t plane, which, in the case of modulation equations (1.14)–(1.15), lead to collapse of a band
or a gap, or to appearance of a new “double point” that will open into a band or gap. In any case,
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 9
at a point of singularity (also known as a breaking point), two or more endpoints from a⃗(x, t)
collide or a new pair(s) of collapsed double points appear, so that the Riemann surface RN
develop a singularity. In this paper we do not intend to discuss details of transition of the circular
condensate between regions of different genera while passing through a breaking point. However,
we would like to mention that since the differentials û(p)dp, v̂(p)dp are imaginary normalized
differentials, they undergo a continuous transition through breaking points, see [1]. Thus, fNLS
circular condensate is preserved under the kinetic equation evolution through breaking points
(change of genus). Some examples of such evolution can be found in Section 4.
2 Proof of Theorem 1.3
In the particular case of a genus zero circular condensate where the point z = ρ is on the band,
the NDR where solved in [9]. The proof of Theorem 1.3 presented below in a sense resembles
the proof of the solution to the NDR for a bound state fNLS soliton condensate from [13].
The conformal map (1.9) has the inverse z = ρ
(
p +
√
p2 − 1
)
or z = ρeiξ, where p = cos ξ.
In the variables ξ, θ, where w = ρeiθ, each NDR equation in (1.5) with σ ≡ 0 can be written as
−ρ
∫
Γ̃+
log
∣∣∣∣∣sin ξ−θ
2
sin ξ+θ
2
∣∣∣∣∣ψj(θ) dθ = ϕj(ξ), j = 1, 2, (2.1)
where ψ1(ξ) = u
(
ρeiξ
)
, ψ2(ξ) = v
(
ρeiξ
)
, Γ̃+ is the preimage of Γ+ under the map z = ρeiξ,
ϕ1(ξ) = ρ sin ξ, ϕ2(ξ) = −4ρ2 sin ξ cos ξ,
and the integration in Γ̂+ ⊂ R goes in the negative direction. Here and henceforth, we always
assume that the integration over Γ̂+ goes in the positive direction, and, therefore, change the
sign in the left-hand side of (2.1). Since
d
dξ
log
∣∣∣∣∣sin ξ−θ
2
sin ξ+θ
2
∣∣∣∣∣ = 1
2
[
cot
ξ − θ
2
− cot
ξ + θ
2
]
=
sin θ
cos θ − cos ξ
,
differentiation in ξ of (2.1) yields
ρ
∫
Γ̃+
ψj(θ) sin θ
cos θ − cos ξ
dθ = − d
dξ
ϕj(ξ), j = 1, 2,
or
πH[û](p) = −p, πH[v̂](p) = 4ρ
(
2p2 − 1
)
, (2.2)
where û(q) = ψ1(θ), v̂(q) = ψ2(θ) and q = cos θ.
Inversion of the first FHT in (2.2) has the form (1.11), where the degreeN+1 polynomial P (p)
is defined up to a kernel of H acting on Γ̂+. As it is well known (and can be easily verified),
this kernel is an N -dimensional space that consists of functions K(p)
R(p) , where K(p) is an arbitrary
polynomial of degree N − 1.
We now prove that the N unknown coefficients of K(p) are uniquely defined by condi-
tions (1.13) for û. We start with proving that û(p)dp√
1−p2
has zero residue at p = ∞, i.e., it is a second
kind meromorphic differential onRN . Indeed, substituting û(p) = P (p)
R(p) into (2.2) and calculating
the H[û] through the residue at p = ∞, we obtain
a = −i, b =
i
2
2N−1∑
j=0
aj , (2.3)
10 A. Tovbis and F. Wang
where P (p) = 1
π
(
apN+1 + bpN + · · ·
)
and aj = cosαj are the endpoints of Γ̂+. Thus
û(p)dp√
1− p2
=
1−
∑
j aj
2p + · · ·√
1− p−2
∏
j
(
1− aj
p
) 1
2
dp =
(
1 +O
(
p−2
))
dp,
which complete the argument.
Now equations (1.13) for u can be written as∫ a2j
a2j−1
û(p)dp√
1− p2
= 0, j = 0, . . . , N, (2.4)
with a−1 = 1, a2N = −1 and the corresponding integral in (2.4) should be replaced by û(±1) = 0
if±1 ∈ Γ̂+ respectively. The fact that Ω1 =
û(p)dp√
1−p2
is a second kind differential implies that one of
the equations (2.4) is a tautology and so the remaining n conditions simply define a normalization
of Ω1. In particular, if all except one gap are A-cycles, then (2.4) implies that Ω1 is an A-
normalized meromorphic differential. Thus, equations (2.4) always have a unique solution. The
cases ±1 ∈ Γ̂+ can be treated as limits of small closing gaps from ±1 to the nearest endpoint.
To complete our arguments for u(z), we need to show that conditions (1.13) for u must be
satisfied. In this proof we follow the arguments of [13, Theorem 6.1], where similar conditions
were derived for the bound state fNLS soliton condensate, i.e., when all the bands were situated
on the imaginary axis. The idea of the proof is related to the fact that the solution u(z) to (1.5)
is the density of the equilibrium measure for the corresponding Green’s energy [13]. If u is such
an equilibrium density then the Green’s potential G[u] :=
∫
Γ+ log
∣∣ z−w̄
z−w
∣∣u(w)|dw| of u should be
continuous at every regular point of Γ+, see [15]. In our case, all points of Γ+ are regular and
so G[u] must be continuous in C.
Equations (2.4) imply that there exists at least one zero in each gaps (intervals). Since there
are n+ 1 gaps and the degree of P (p) is N + 1, we conclude that each gap has exactly one root
of P (p) and, so, all the roots of the polynomial P (p) are real. Thus, according to (2.3), P (p) is
purely imaginary on R and, so, u > 0 on Γ+.
Similar arguments hold for the solution v(z) of the second NDR (1.12). For example, repre-
senting Q(p) = 1
π
(
apN+2+bpN+1+cpN+· · ·
)
, it follows from (1.10) that a = −8iρ, b = −a
2
∑
j aj
and c = 4iρ+ iρ
(∑
j a
2
j − 6
∑
j<k ajak
)
. So, a, b, c ∈ iR. Equations (2.3) imply that N + 1 roots
are real. Thus it follows that all the roots of Q are real.
3 Genus 0 and genus 1 circular condensates
In this section, the results of Theorem 1.3 are applied to two simple cases: the genus 0 and 1
circular soliton condensate. We remind that by genus we mean the genus of the Riemann
surface RN , N = 1, 2, 3, which, generically, is equal to the number of bands N (or the number
of gaps minus one), see Figure 2. However, since in this section we always have α0 = 0 and
α2N−1 = π, two gaps are collapsed and the genus of the Riemann surface is reduced to N = 0, 1.
As before, we denote
aj = p
(
ρeiαj
)
= cos(αj), j = 1, 2, 3, 4.
The support Γ+ for the genus one circular condensate is illustrated by Figure 3. In what
follows, we will not mention α0 and α2N+3 since they are always fixed. In both cases, the exact
solutions to the NDR equations can be explicitly represented with the help of complete elliptic
integrals. For higher genus situation, the solution to the NDR equations can be represented
using hyperelliptic integrals.
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 11
O
ρ
z0
z1
z2
z3
z4
z5
Figure 3. The plot of Γ+ showing bands and gaps of the fNLS circular condensate, where {zj = ρeiαj}5j=0
are the endpoints of the bands respectively and ρ is the radius; α0 = arg z0 = 0, α5 = arg z5 = π are
fixed. If any of the gaps, shown on the plot, is closed, the genus becomes zero.
Corollary 3.1. Given the contour Γ+ as shown in Figure 3, the solution to the NDR (1.5)–(1.6)
is given by
u(z) = u1(p; a1, a2, a3, a4) =
−
√
1− p2
π
(
−p2 + 1
2 l1p+A
R(p)
)
, (3.1)
v(z) = v1(p; a1, a2, a3, a4) =
ρ
√
1− p2
π
(
8p3 − 4l1p
2 +
(
4l2 − l21
)
p+B
R(p)
)
, (3.2)
where the subindex in u1, v1 indicates the genus, p = p(z) is given by (1.9),
R(p) =
√√√√ 4∏
j=1
(p− aj), l1 =
4∑
j=1
aj , l2 =
∑
1≤i<j≤4
aiaj ,
A =
E(m)
2K(m)
(a2 − a4)(a1 − a3)−
1
2
(a1a2 + a3a4),
B = −E(m)
K(m)
(a2 − a4)(a1 − a3)l1 + (a1a2 − a3a4)(a1 + a2 − a3 − a4),
m =
(a1 − a2)(a3 − a4)
(a1 − a3)(a2 − a4)
,
and, as in Theorem 1.3, the branch of R(p) is chosen so that R(p) ∼ p2 as p→ ∞. The effective
velocity is then given by
s(z) = s1(p; a1, a2, a3, a4) = −
(
8p3 − 4l1p
2 +
(
4l2 − l21
)
p+B
p2 − 1
2 l1p−A
)
ρ. (3.3)
Proof. Based on Theorem 1.3, the general solution to the first NDR reads
u(z) = u1(p; a1, a2, a3, a4) =
1
π
−
√
1− p2
(
−p2 + c1p+ c0
)√
(p− a1)(p− a2)(p− a3)(p− a4)
,
where c1 =
1
2 l1, c0 = A are determined by the (gap-vanishing) normalization conditions:∫ a3
a4
u(p)
dp√
1− p2
= 0,
∫ a1
a2
u(p)
dp√
1− p2
= 0.
The computation of v is similar and thus omitted. The effective velocity can be computed
directly from equation (1.7). ■
In the case of α3 = α4, one of the gaps disappears and we are in the genus 0 situation, defined
only by α1, α2.
12 A. Tovbis and F. Wang
Corollary 3.2. In the case α4 = α3 (genus zero, see Figure 3), the solution to the NDR (1.5)–
(1.6) is given by
u(z) = u0(p; a1, a2) =
1
π
√
1− p2
(
p− 1
2(a2 + a1)
)√
(p− a2)(p− a1)
, (3.4)
v(z) = v0(p; a1, a2) = − 8
π
ρ
√
1− p2
(
p2 − 1
2(a2 + a1)p− 1
8(a1 − a2)
2
)√
(p− a2)(p− a1)
, (3.5)
where p = p(z) is given by (1.9), aj = cos(αj), j = 1, 2, and the square-root function takes the
principal branch.
The effective velocity is then given by
s(z) = s0(p; a1, a2) =
(
−8p+
(a2 − a1)
2
p− (a2 + a1)/2
)
ρ. (3.6)
Proof. Using the conditions of Theorem 1.3 and having in mind that û(±1) = v̂(±1) = 0, we
obtain (3.4) and
v(z) = −8ρ
π
√
1− p2
(p− a1)(p− a2)
(
p2 − a1 + a2
2
p+ c0
)
,
where c0 is the constant that is determined by the gap vanishing condition∫ a1
a2
v(z(p))(1− p2)−1/2dp = 0.
Solving the latter equation, we obtain
c0 = −1
8
(a1 − a2)
2.
Given solutions u, v, we obtain (3.6) for the effective velocity s. ■
Remark 3.3. As it was mentioned in Remark 1.5, imaginary normalized differentials have a con-
tinuous transition through breaking points, i.e., through points of collapse of a band or a gap.
In particular, genus zero solutions (see equations (3.4)–(3.6)) can be directly obtained from the
genus one solutions (see equations (3.1)–(3.3)) by taking the limit a3 → a+4 . That is to say,
u0(p; a1, a2) = lim
a3→a+4
u1(p; a1, a2, a3, a4).
Similar limits work for the solutions to the density of fluxes and the corresponding effective
velocities.
Remark 3.4. As it was shown in the proof of Theorem 1.3, the DOS u > 0 on Γ+; however, the
DOF v may have a zero z0 on Γ+, which also coincides with the zero of the effective velocity s.
In Figure 4, we show different cases of the location of z0, defined by the branch points (α1, α2).
Remark 3.5. If in the conditions of Corollary 3.2 we further assume α2 = π, then the solutions
to the NDR reduce to
u(z) = u0(p; a1,−1) =
1
π
(1− p)
(
p+ 1−a1
2
)√
(p− a1)(1− p)
, (3.7)
v(z) = v0(p; a1,−1) =
1
π
ρ(1− p)
(
−8p2 + 4(a1 − 1)p+ (a1 + 1)2
)√
(1− p)(p− a1)
, (3.8)
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 13
0 :
arg(z)
0
0.5
1
u
0
(z
)
0 :
arg(z)
-10
-5
0
5
10
s 0
(z
)
0 :
arg(z)
0
1
2
3
4
u
0
(z
)
0 :
arg(z)
-10
-5
0
5
10
s 0
(z
)
0 :
arg(z)
0
0.5
1
1.5
u
0
(z
)
0 :
arg(z)
-10
-5
0
5
10
s 0
(z
)
Figure 4. Three plots of the DOS u and the corresponding effective velocity based on the results
of Corollary 3.2. Here ρ = 1. Left: α1 = 0.1π, α2 = 0.25π; the only zero of s(z) lies on the band
{ρeiθ : θ ∈ (α2, π)}; Middle: α1 = 0.2π, α2 = 0.8π; s(z) has no zero on Γ+; Right: α1 = 0.7π, α2 = 0.9π;
the only zero of s(z) lies in the arc {ρeiθ : θ ∈ (0, α1)}. In each plot, the vertical dashed lines indicate α1
and α2.
where p = p(z) is given by (1.9), so that
s(z) =
(
−8p+
(a1 + 1)2
p+ (1− a1)/2
)
ρ. (3.9)
Expressions (3.7) and (3.9) were calculated in [9, equations (72) and (73)].
4 Modulational dynamics for the circular condensate
Solutions to the kinetic equations for the fNLS circular condensate, obtained in Theorems 1.3
and 1.4, look very similar to that for the KdV soliton condensate, obtained in [3]. Following the
ideas of [3], in this section we consider the evolution of step function initial conditions for the
circular gas modulation equations, i.e., the Riemann problem, that, as in the KdV case, produce
rarefaction and dispersive shock wave solutions.
Remark 4.1. As it was pointed out by one of the referees, similar phenomena in the context
of the “generalized hydrodynamics”, namely, the shock waves in the zero-entropy GHD, was
reported in [6].
Consider the step function initial data for the modulation equations (1.14)
a1(x, t = 0) ≡ cos(α1(x, t = 0)) =
{
q−, x < 0,
q+, x > 0,
q+ ̸= q−. (4.1)
that corresponds to the genus zero circular condensate described in Remark 3.5, i.e., the initial
date for the branch point a1 ∈ Γ̂+. That defines (see (3.7)) the DOS:
u(z;x, t = 0) =
{
u0(p(z); q−,−1), x < 0,
u0(p(z); q+,−1), x > 0,
(4.2)
and a similar expression for the DOF v, see (3.8), where q± ∈ (−1, 1).
14 A. Tovbis and F. Wang
In what follows, we will consider self-similar solutions to the modulation system (1.14), i.e.,
we consider x/t = ξ = const. It is well known that the behavior of such solutions, including the
genus of the corresponding hyperelliptic Riemann surface R = R(x, t) (i.e., the Riemann surface
in the p plane, whose branchcuts are intervals on the real line.) for u, v depends on whether
q− > q+ or q− < q+. The first case, the genus of R(x, t) stays zero and the dynamics of the DOS
u(z;x, t) is characterized by the rarefaction wave solution of the modulation equation (1.14). The
latter case, however, implies immediate wave-breaking, which can be regularized by introducing
a genus one surface R1(x, t) and the corresponding dispersive shock wave DOS u(z;x, t) that
connects u = u0(p; q−,−1) for large negative and u = u0(p; q+,−1) for large positive x. Such
regularization is well known for describing the dispersive shock wave modulations (see [12])
of the KdV equation with step initial data. The two types of behavior of the modulational
dynamics of the DOS are shown as in Figure 5. The following theorem summarizes main results
of the section.
Theorem 4.2. Suppose u(z;x, 0) is given by (4.2) with a1(x, 0) given by (4.1). Then for any
(x, t) ∈ R× R+ we have
(i) If q− > q+,
a1(x, t) =
q−, x < V1−t,
−1
6
x
ρt +
1
3
, V1−t < x < V1+t,
q+, x > V1+t,
and V1± = −6ρ
(
q± − 1
3
)
,
so that the DOS is given by
u(z;x, t) = u0(p; a1(x, t),−1) =
1
π
(1− p)
(
p+ 1−a1(x,t)
2
)√
(p− a1(x, t))(1− p)
, (4.3)
where p = p(z) is given by (1.9) and the expression for the DOF is given by
v(z;x, t) = v0(p; a1(x, t),−1)
= − 8
π
ρ
√
1− p2
(
p2 − 1
2(a1(x, t)− 1)p− 1
8(a1(x, t) + 1)2
)√
(p+ 1)(p− a1(x, t))
. (4.4)
(ii) If q− < q+, then a2(x, t) is uniquely and implicitly determined by the following equation:
x
ρt
= −2(q+ + a2 + q− − 1)− 4(a2 − q−)(q+ − a2)
(q+ − q−)µ(m) + q− − a2
, V2−t < x < V2+t, (4.5)
where
m =
(1 + q−)(q+ − a2)
(q+ − q−)(1 + a2)
, µ(m) =
E(m)
K(m)
, (4.6)
V2− =
(
−16a21 + 8a1a3 + 2a23 − 8a1 + 4a3 + 2
2a1 − a3 + 1
)
ρ,
V2+ = (−2a1 − 4a3 + 2)ρ. (4.7)
so that
u(z;x, t) =
u0(p; q−,−1), x < V2−t,
u1(p; q+, a2(x, t), q−,−1), V2−t < x < V2+t,
u0(p; q+,−1), x > V2+t,
(4.8)
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 15
(a) (b)
Figure 5. Solutions to the kinetic equation for the circular condensate with step function initial data
(Riemann problem) (4.2). (a) Rarefaction wave solution when q− > q+; Dashed line: a1(x, t); Colors:
the values of DOS u0(p; a1(x, t),−1); (b) Dispersive shock wave solution when q− < q+. Dashed line:
a2(x, t); Colors: the value of DOS u1(p; q+, a2(x, t), q−,−1).
where p = p(z) is given by (1.9); and the expression for the DOF is given by
v(z;x, t) =
v0(p; q−,−1), x < V2−t,
v1(p; q+, a2(x, t), q−,−1), V2−t < x < V2+t,
v0(p; q+,−1), x > V2+t.
(4.9)
4.1 Proof of Theorem 4.2
Proof of the first part (i). Applying Theorem 1.4, we obtain the modulation equation for
moving the branch point α1:
∂tα1 + V1∂xα1 = 0,
where
V1 = −6ρ
(
cos(α1)−
1
3
)
.
Since a1 = cos(α1), the modulation equation is equivalent to
∂ta1(x, t)− 6ρ
(
a1 −
1
3
)
∂xa1(x, t) = 0.
Considering the initial data (4.1), the self-similar solution to the modulation equation is given by
a1(x, t) =
q−, x < V1−t,
−1
6
x
ρt +
1
3
, V1−t < x < V1+t,
q+, x > V1+t,
where
V1− = V1|a1=q− = −6ρ
(
q− − 1
3
)
, V1+ = V1|a1=q+ = −6ρ
(
q+ − 1
3
)
.
16 A. Tovbis and F. Wang
Clearly, 0 > V1+ > V1−, which generates a rarefaction wave. In this case, the DOS is given by
u(z;x, t) = u0(p; a1(x, t),−1) =
1
π
(1− p)
(
p+ 1−a1(x,t)
2
)√
(p− a1(x, t))(1− p)
.
Proof of the second part (ii): The previously derived solution (the rarefaction wave) is
not well defined for q− < q+ since a wave breaking occurs immediately. To resolve the issue, it
is necessary to introduce higher genus DOS that connects u0(p; q−,−1) and u0(p; q+,−1). This
can be done by using the genus one DOS (see equation (3.1)):
u(z;x, t) = u1(p; a1 = q+, a2(x, t), a3 = q−,−1).
Following Theorem 1.4, the motion of a2(x, t) is governed by the following modulation equa-
tion:
∂ta2 + V2∂xa2 = 0,
where
V2 = V2(a1, a2, a3,−1) = −2ρ(a1 + a2 + a3 − 1)− 4ρ(a2 − a3)(a1 − a2)
(a1 − a3)µ(m) + a3 − a2
(4.10)
with
m =
(1 + a3)(a1 − a2)
(a1 − a3)(1 + a2)
, µ(m) =
E(m)
K(m)
.
By a direct computation, we obtain
V2− = lim
a2→q−
V2 =
(
−16a21 + 8a1a3 + 2a23 − 8a1 + 4a3 + 2
2a1 − a3 + 1
)
ρ,
V2+ = lim
a2→1
V2 = (−2a1 − 4a3 + 2)ρ.
Moreover, since
V2+ − V2− =
2ρ(6a1 − a3 + 5)(a1 − a3)
2a1 − a3 + 1
,
it is evident that
V2+ > V2−.
Then the solution to the modulation equation for a2 is defined implicitly by the following system:
V2(a1 = q+, a2, a3 = q−,−1) =
x
t
, V2−t < x < V2+t. (4.11)
Since the solution a2(x, t) is defined implicitly, we need to prove that the function V2 as
a function of a2 is invertible. In our case, it is equivalent to show that for any fixed a1, a3, the
function V2(a1, a2, a3,−1) is monotonic with respect to a2. So, to complete the proof, we need
to prove the following proposition.
Proposition 4.3. V2(a1, a2, a3,−1), as defined by equation (4.10), is monotonic as a function
of a2 on the interval (a3, a1) for any −1 < a3 < a1 < 1 being fixed.
Before we prove the proposition, we need the following lemma.
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 17
Lemma 4.4. Let µ(m) = E(m)
K(m) , then
1−m < µ(m) < 1− m
2
, m ∈ (0, 1). (4.12)
Proof. It is well known that (see Byrd–Friedman [2, formulas (710.00) and (710.02)])
E′(m) =
E −K
2m
, K ′(m) =
E − (1−m)K
2m(1−m)
, (4.13)
where prime means differentiation with respect to m. Then
E − (1−m)K = 2m(1−m)K ′(m) > 0,
where we have used the fact that K ′(m) > 0 for m ∈ (0, 1). Thus, the first inequality in (4.12)
is proven. Using both equations (4.13), we obtain
d
dm
(
E −
(
1− m
2
)
K
)
= −m
2
K ′(m),
so that
E −
(
1− m
2
)
K < 0.
Thus, µ(m) < 1− m
2 for m ∈ (0, 1). ■
Proof of Proposition 4.3. We prove the statement by contradiction. Suppose there exists
a2 ∈ (a3, a1) such that
∂a2V2 = 0.
This leads to
µ(m) =
(a2 − a3)
(
a21 − a1a3 − 3a22 + 3a2a3 + a1 − 3a2 + 2a3
)
2(a1 − a3)
(
2a1a2 − a1a3 − 3a22 + 2a2a3 + a1 − 2a2 + a3
) .
Using (4.6) to express a2 in terms of a1, a3 and m, we obtain
µ(m) =
1
2
+
(a1 + 1)(a3 + 1)
2(m(a1 − a3) + a3 + 1)(a1 − a3)
+
[
(a1 + 1)2 + (a3 + 1)2
]
m− (a3 + 1)2
2(a1 − a3)
(
(a1 − a3)m2 − 2(a1 + 1)m+ a3 + 1
) . (4.14)
Since a1 > a3 and m ∈ (0, 1), the first two terms has no singularities. The denominator of
the third term processes two zeros
(a1 + 1)±
√
a21 − a1a3 + a23 + a1 + a3 + 1
a1 − a3
,
but since m < 1, we see that the right-hand side of (4.14) has a unique simple pole at
mcr =
a1 + 1−
√
a21 − a1a3 + a23 + a1 + a3 + 1
a1 − a3
.
On the one hand, since
(a1 + 1)2 −
(√
a21 − a1a3 + a23 + a1 + a3 + 1
)2
= (a3 + 1)(a1 − a3) > 0, mcr > 0.
18 A. Tovbis and F. Wang
On the other hand, since
mcr − 1/2 = (a1 − a3)
−1
(
(a1 + a3)/2 + 1−
√
a21 − a1a3 + a23 + a1 + a3 + 1
)
and
((a1 + a3)/2 + 1)2 −
(√
a21 − a1a3 + a23 + a1 + a3 + 1
)2
= −(a1 − a3)
2 < 0,
we obtain mcr < 1/2. Since the singularity mcr ∈ (0, 1/2), we consider two cases: (1) m ∈
(0,mcr); (2) m ∈ (mcr, 1). In each case, we will construct a contradiction using Lemma 4.4. In
the first case, subtract right-hand side of (4.14) by 1−m/2, we get
m2F (m)
2(m(a1 − a3) + a3 + 1)((a1 − a3)m2 − 2(a1 + 1)m+ a3 + 1)
, (4.15)
where
F (m) = (a1 − a3)
2m2 − (a1 − a3)(3a1 − 2a3 + 1)m
+ a23 − (3a1 + 1)a3 + 3a21 + 3a1 + 1.
Since a1 > a3, the denominator of (4.15) is positive for any m ∈ (0,mcr). From the expression
of the quadratic function F (m), we see that the axis of symmetry is
3a1 − 2a3 + 1
2(a1 − a3)
,
which is obviously strictly greater than 1. This implies F (m) is a decreasing function for
m ∈ (0, 1). Thus, F (m) ≥ F (1) = (a1 + 1)2 > 0 and we have shown that
right-hand side of (4.14) > 1− m
2
.
However, due to Lemma 4.4, this contradicts the inequality satisfied by µ(m).
In the second case, subtract right-hand side of (4.14) by 1−m, we get
m(m− 1)G(m)
2(m(a1 − a3) + a3 + 1)((a1 − a3)m2 − 2(a1 + 1)m+ a3 + 1)
, (4.16)
where
G(m) = (a1 − a3)
2m2 − (a1 − a3)(3a1 − a3 + 2)
m
2
− (a3 + 1)2
2
.
In this case, the denominator is negative since m > mcr. As for the quadratic function G(m), it
is easy to check G(0) < 0 and G(1) < 0, together with the fact that the leading coefficient of G
is positive, we have G(m) < 0 for any m ∈ (0, 1). This implies the expression (4.16) is negative
and we have shown
right-hand side of (4.14) < 1−m.
Again, due to Lemma 4.4, this contradicts the inequality satisfied by µ(m).
Hence, we have shown
∂a2V2 ̸= 0, ∀a2 ∈ (a3, a1).
And this means V2(a1, a2, a3) is a monotonic function of a2 for a2 ∈ (a3, a1). ■
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 19
Based on Proposition 4.3, we have shown that the solution a2 defined by the equation (4.11)
is well defined. Now, let’s check the boundary behaviors as x → V2+t and x → V2−t. A direct
calculation shows
lim
x→V2−t
u1(p; a1 = q+, a2(x, t), a3 = q−,−1) = u0(p; q−,−1),
lim
x→V2+t
u1(p; a1 = q+, a2(x, t), a3 = q−,−1) = u0(p; q+,−1).
Thus, the genus one DOS, as given by
u(z;x, t) =
u0(p; q−,−1), x < V2−t,
u1(p; q+, a2(x, t), q−,−1), V2−t < x < V2+t,
u0(p; q+,−1), x > V2+t.
connects two genus zero DOS: u0(p; q−,−1) and u0(p; q+,−1). Similarly, using the expressions
for the DOF (namely, equations (3.2) and (3.5)), we get the modulated DOF as given by equa-
tions (4.9) and (4.4) respectively. This completes the proof for Theorem 4.2.
5 Kurtosis in genus 0 and genus 1 circular condensate
In this section, we will compute the fourth normalized moment κ =
〈
|ψ|4
〉
/
〈
|ψ|2
〉2
of the fNLS
circular condensate |ψ|-the kurtosis. In the genus 0 case, we obtain that the kurtosis for the
condensate is always 2, while in the genus 1 case, the kurtosis is greater than 2 but finite. Below
we will give the explicit formulae for computing kurtosis in genus 0 and genus 1 case. The main
tool is to use the formulae of computing the averaged conserved quantities for the fNLS soliton
gas, which are recently developed by the authors in [16].
It is well known that the fNLS has infinite many conservation laws ((fj)t = (gj)x, j ≥ 1),
where the densities and the currents, fj and gj , can be determined recursively (see, for example,
Wadati’s paper [17]). In order to compute the kurtosis for the circular condensate, we will need
the first few densities and currents:
f1 = |ψ|2, f3 = |ψ|4 + ψψxx, g2 = |ψx|2 − |ψ|4 − ψψxx.
According to [16], the averaged conserved quantities are given by
⟨f1⟩ = 2I1 := 4
∫
Γ+
(Im z)u(z)|dz|,
⟨f3⟩ = −8
3
I3 := −16
3
∫
Γ+
(
Im z3
)
u(z)|dz|,
⟨g2⟩ = −2J2 := −4
∫
Γ+
(
Im z2
)
v(z)|dz|,
where
Ij = 2
∫
Γ+
u(ξ) Im ξj |dξ|, Jj = 2
∫
Γ+
v(ξ) Im ξj |dξ|, (5.1)
and Γ+ is the support of the circular condensates.
Since the total derivatives do not contribute to the average, through integration by parts, the
following identities follow
⟨f3⟩ =
〈
|ψ|4 − |ψx|2
〉
, ⟨g2⟩ =
〈
2|ψx|2 − |ψ4|
〉
.
20 A. Tovbis and F. Wang
Then by the definition of kurtosis, we have
κ =
〈
|ψ|4
〉〈
|ψ|2
〉2 = −
4
3I3 +
1
2J2
I21
. (5.2)
Below, we use the formula (5.2) to derive formulas for computing the kurtosis for the genus 0
and genus 1 condensate. The main ingredient of the computation is computing the averaged
densities
(
Ij
)
and averaged fluxes
(
Jj
)
. The following proposition provides a fairly simple way
to compute these quantities.
Proposition 5.1. Let Γ+ be the contour for the circular condensate associating with the hyper-
elliptic Riemann surface Rn, then the averaged densities
(
Ij
)
and the averaged fluxes
(
Jj
)
can
be computed by the following formulae: for j ∈ Z+,
Ij = 2πiρj+1Res
{
P (z(p))Uj−1(p)
R(z(p))
, p = ∞
}
, (5.3)
Jj = 2πiρj+1Res
{
Q(z(p))Uj−1(p)
R(z(p))
, p = ∞
}
, (5.4)
where P , Q, R are defined in Theorem 1.3 and
Uj(p) =
sin[(j + 1) arccos(p)]
sin[arccos(p)]
. (5.5)
is the j-th Chebyshev polynomial of the second kind.
Proof. Using (5.1) and (5.5) and changes of variables ξ = ρeiθ, p = cos θ, we calculate
Ij = 2
∫
Γ+
u(ξ) Im ξj |dξ| = 2ρj+1
∫
u
(
ρeiθ
)
sin(jθ) dθ
= −2ρj+1
∫
Γ̂+
u(z(p))
sin(jθ)
sin θ
dp = −ρj+1
∮
γ̂
P (z(p))Uj−1(p)
R(z(p))
dp (5.6)
= 2πiρj+1Res
{
P (z(p))Uj−1(p)
R(z(p))
, p = ∞
}
, (5.7)
where Uj−1(p) is the (j − 1)-th Chebyshev polynomial of the second kind. The equality (5.6)
comes from an application of the Cauchy’s theorem on the Riemann surface Rn, the loop γ̂
encloses Γ̂+ counterclockwisely. Then a direct application of the residues theorem leads to
the equality (5.7), which is exact the formula (5.3) for computing the averaged densities. By
replacing u, P by v, Q respectively, one get the formula (5.4) for computing the averaged
fluxes. ■
Based on the above proposition, in order to compute the kurtosis, we just need the first few
averaged quantities, namely, I1, I3 and J2. The following proposition gives an explicit formula
for computing the kurtosis for the genus one circular condensate.
Proposition 5.2. Let Γ+ be the contour on Figure 6, then the kurtosis for the genus one
condensate is given by
κ =
κnum
κden
, (5.8)
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 21
O
ρ
z0
z1
z2
z3
z4
Figure 6. The curve Γ+ for computing the genus 1 kurtosis and its modulational dynamics. Here z4 =
ρeiα4 = −ρ and z0 = ρeiα0 = ρ are fixed. When considering the modulational dynamics, α1 = arg z1 =
arccos(q+), α3 = arg z3 = arccos(q−), where q− < q+ are given by the initial data for the Riemann
problem. At t = 0 we have z2 = z1, but for t > 0 the motion of the point z2 = ρ exp{i arccos(a2(x, t))} is
determined through the equation (4.5).
where
κnum = 6a42 + 24(a1 − a3 + 1)a32 + 4
[
(a1 − a3)
2 − (8a3 − 10)(a1 − a3)− 3
]
a22
+ 8
[
(a1 − a3)
3 + 5(a1 − a3)
2 +
(
2a21 + 2a23 − 7
)
(a1 − a3)− 9
]
a2
+ 2
[
3(a1 − a3)
4 + 4(a1 − a3)
3 − 6(a1 − a3)
2
]
+ 8
(
4a3 + 2a21 + 2a23 − 9
)
(a1 − a3)
+ 54− 16(a1 − a3)(a2 + 1)
(
3
(
a1 + a2 + a3 +
1
3
)2
− 2(a1a2 + a1a3 + a2a3)−
28
3
)
µ,
κden = 3 [(a1 + a2 − a3 + 3)(a1 + a2 − a3 − 1)− 4(a2 + 1)(a1 − a3)µ]
2 ,
µ =
E(m)
K(m)
, m =
(1 + a3)(a1 − a2)
(a1 − a3)(a2 + 1)
.
Proof. First we use Proposition 5.1 to compute I1, I3 and J2, and then substitute them into
the formula (5.2) to compute the kurtosis. After some algebra, we get the kurtosis formula as
stated. ■
Remark 5.3. Notice that the kurtosis for the circular condensate is independent of the radius ρ
of the semicircle S+.
Corollary 5.4. The kurtosis for the genus 0 condensate is always 2 for any α1, α2 ∈ (0, π) and
α1 < α2.
Proof. This case can be degenerated from Proposition 5.2 by taking the limit a3 → −1+. In
fact, we have
lim
a3→−1+
κnum = 6(a1 − a2 + 2)2(a1 − a2 − 2)2,
lim
a3→−1+
κden = 3(a1 − a2 + 2)2(a1 − a2 − 2)2,
which immediately imply
κ = 2. ■
From the last section, we know that the genus one DOS/DOF (see equations (4.8) and (4.9))
actually connects two genus zero DOS/DOF (see equations (4.3) and (4.4)) as a2 → a1 or
a2 → a3. Also, we have already shown that the kurtosis for the genus zero condensate is
always 2, it would be interesting to study the dynamic of the kurtosis as the branch point a2
moving from a1 to a3. A careful analysis to the formula (5.8), we obtain the following theorem.
22 A. Tovbis and F. Wang
Theorem 5.5. Let a1, a3 be fixed and satisfy −1 < a3 < a1 < 1. Then for any a2 ∈ [a3, a1]
the genus 1 kurtosis κ, given by formula (5.8), is greater or equal to 2 and is finite. Moreover,
κ = 2 if and only if a2 = a1 or a2 = a3.
Proof. Using the explicit formula for the genus one kurtosis (5.8), we define a new function
H(µ̃) = κnum − 2κden
by replacing µ with µ̃ and consider µ̃ as a new variable not depending on a1, a2, a3. To show
κ ≥ 2, it suffices to show H(µ̃) is positive for any µ̃ ∈ (1−m, 1−m/2), where m = (1+a3)(a1−a2)
(a1−a3)(a2+1) .
A direct computation shows
H(0) = −32(a1 + 1)(a2 + 1)(a1 − a3)(a2 − a3) < 0,
H(1) = −32(a2 + 1)(a3 + 1)(a1 − a3)(a1 − a2) < 0,
H(1−m) = 32(a1 + 1)(a3 + 1)(a2 − a3)(a1 − a2) > 0,
H(1−m/2) = 8(a3 + 1)2(a1 − a2)
2 > 0.
Note that the functionH(µ̃) is a quadratic function of µ̃. The above observation (as visualized
on Figure 7) shows there is a zero in the interval (0, 1−m) and another zero in (1−m/2, 1), which
implies H(µ̃) > 0 for any µ̃ ∈ (1 −m, 1 −m/2) and for all a2 ∈ (a3, a1), see inequality (4.12).
This proves that κ > 2 for all a2 ∈ (a3, a1). On the one hand, as a2 → a1 or a2 → a3, a direct
computation shows κ = 2 in both cases. On the other hand, if there exists some µ̃ such that
H(µ̃) = 0, then either m = 0 or m = 1 (otherwise we already show H(µ̃) > 0). Since a1 > a3,
m = 0 or m = 1 implies a2 = a1 or a2 = a3 respectively. Thus, we have shown that κ ≥ 2 and
κ = 2 if and only if a2 = a3 or a1 = a2.
0 1−m 1− m
2 01
0 < m < 1 m = 0 m = 1
1 10 1
2
H(µ̃) H(µ̃) H(µ̃)
Figure 7. Three plots of H(µ̃) regarding different values of m.
To show that the kurtosis is finite, it suffices to show that, according to the definition of the
kurtosis, I1 is positive and I3, J2 are finite. Since I1 =
1
2
〈
|ψ|2
〉
, it is obvious positive. Since u, v
are integrable with respect to the arc-length measure, we have
|I3| ≤
∥∥Im z3
∥∥
L∞(|dz|)∥u(z)∥L1(|dz|) <∞,
|J2| ≤
∥∥Im z2
∥∥
L∞(|dz|)∥v(z)∥L1(|dz|) <∞.
Together with definition of the kurtosis, we conclude that κ <∞. The proof is done. ■
5.1 The kurtosis for DSW
In this subsection, we study the modulation dynamics of the kurtosis. As discussed in Section 4,
there are two types of wave phenomenon, the rarefaction wave and the dispersive shock wave.
According to Theorem 5.4, the kurtosis for the rarefaction wave is always 2. As for the dispersive
shock wave, we follow the same setting in Section 4 for the dispersive shock wave. Replacing a2
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 23
(a) (b)
Figure 8. The density plots of the modulated kurtosis κ(a1, a2(x, t), a3), with a1 = 0.9, a3 = −0.4.
In (a), the white line indicates the line x − V2+t = 0 and the red line indicates the line x − V2−t = 0,
where V2± are defined by equations (4.7). (b) shows the zoom-in density plot near the red line.
in equation (5.8) by a2(x, t) as implicitly defined by equation (4.5), we obtain the kurtosis for
the dispersive shock wave. Denote the modulated kurtosis by κmod = κmod(a2(x, t)). Recall for
each fixed a1, a3 such that −1 < a3 < a1 < 1, we have defined
V2− =
−16a21 + 8a1a3 + 2a23 − 8a1 + 4a3 + 2
2a1 − a3 + 1
, V2+ = −2a1 − 4a3 + 2,
which in turn define two rays in the x− t plane
L± := {(x, t) : x− V2±t = 0, t > 0}.
These two rays split the x− t plane with t > 0 into three regions:
D1 := {(x, t) : x− V2+t > 0},
D2 := {(x, t) : x− V2+t < 0, x− V2−t > 0},
D3 := {(x, t) : x− V2−t < 0}.
In regions D1 and D3, the kurtosis is 2 and in the region D2, the kurtosis, according to Theo-
rem 5.5, is strictly great than 2 and finite as long as a1 < 1. As an illustrative example, we take
a1 = 0.9, a3 = −0.4 and plot the modulated kurtosis κmod(a2(x, t)) in the x− t plane as well as
the plot of the kurtosis near the ray L− in Figure 8.
5.2 Scaling limit of the kurtosis of certain genus one circular condensate
In this subsection, we consider certain type of limiting configurations of the circular condensate
and study the corresponding kurtosis. Specifically, we set a3 = −a2 ≤ 0 and study the limit
lim
(a1,a2)→(1−,0+)
κ(a1, a2,−a2) (5.9)
along a certain path L. In the next theorem, we show that for any given s > 2, one can always
find a path Ls such that the limit in (5.9) is s.
24 A. Tovbis and F. Wang
Figure 9. Three curves (see the expression in (5.10)) in the (a1, a2) plane with s = 3, 5, 7. The scaling
limit lim(a1,a2)→(1,0) κ(a1, a2,−a2) along those curves will respectively go to 3, 5, 7. One the line a1 = a2,
the kurtosis is equal to 2.
Theorem 5.6. For any s > 2 the limit
lim
c→0+
κ(a1, a2,−a2)
taken along the curvea1 = 1− c,
a2 = 4 exp
{
8
3(2− s)c2
}
(5.10)
in the parameter plane (a1, a2) is equal s.
Proof. The kurtosis (5.8) can be written in the following form
κ(a1, a2,−a2) =
P1 + P2µ
(Q1 +Q2µ)2
,
where P1, P2, Q1, Q2 are all polynomials of a1, a2. Replacing a1 = 1 − c and we consider the
Taylor approximation of those polynomials near (c, a2) = (0, 0), which are given as follows:
P1 = −128
3
a2 + 32c2 +O
(
a2c, a
2
2
)
, P2 =
64
3
+O(a2, c),
Q1 = −4c+ 8a2 +O
(
a22, c
2, a2c
)
, Q2 = −4 +O(a2, c),
where O
(
{Aj}nj=1
)
means the correction term is bounded by some linear combination of {Aj}nj=1.
Using the asymptotic approximations of complete elliptic integral of the first and the second
kind (see formulas (900.05) and (900.07) in Byrd–Friedman [2]), it is straightforward to show
µ(m) =
1
log 4√
1−m
+O((m− 1) log(1−m)) as m→ 1−.
Notice, as (c, a2) → (0, 0), we have
m = 1− 4a2 +O
(
a22, a2c
)
,
Soliton Condensates for the Focusing Nonlinear Schrödinger Equation 25
which implies
µ(m) =
1
log 2√
a2
+O
(
−a2
log(a2)
)
as (a2, c) → (0, 0).
Then, after some algebraic manipulations, we arrive at the following leading behavior of the
kurtosis
κ =
−128
3 a2 + 32c2 + 64
3
1
log 2√
a2(
4c− 8a2 +
4
log 2√
a2
)2 + o(1).
Since a2 is obviously dominated by −1/ log(a2) as a2 → 0+, we can further simplify the
kurtosis to
κ =
32c2 + 64
3
1
log 2√
a2(
4c+ 4
log 2√
a2
)2 + o(1).
Let us denote S = 1
log 2√
a2
and
κ0 =
2c2 + 4
3S
(c+ S)2
,
then κ = κ0 + o(1) as c→ 0+. Since
κ0 = 2 +
4
3S − 4cS − 2S2
(c+ S)2
,
for c, S are sufficiently close to 0, it is evident that κ0 ≥ 2. Apparently, S will always dominate
cS + S2 as (a2, c) sufficiently close to (0, 0), thus, we have
κ0 = 2 +
4
3S
(c+ S)2
+ o(1).
Note that S = −3
4(2− s)c2, we know, as c → 0+, c dominates S, thus the denominator is then
dominated by c2. And we eventually get
κ = 2 +
4
3
S
c2
+ o(1),
whose limit as c→ 0+ will be 2. This completes the proof. ■
Remark 5.7. Since the kurtosis is a continuous function of a1, a2, a3 as long as a1 > a2 > a3,
Theorem 5.6 also implies that for any given number that is greater than 2, there exists certain
configuration (a genus one circular condensate) such that the kurtosis equals the given number.
Acknowledgements
The authors would also like to thank the anonymous referees for useful suggestions and com-
ments. Fudong Wang is supported by Guangdong Basic and Applied Basic Research Foundation
B24030004J. The work of Alexander Tovbis is supported in part by NSF Grant DMS-2009647.
26 A. Tovbis and F. Wang
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1 Introduction
1.1 Soliton gases for integrable equations
1.2 fNLS bound state soliton condensate
1.3 Main results
2 Proof of Theorem 1.3
3 Genus 0 and genus 1 circular condensates
4 Modulational dynamics for the circular condensate
4.1 Proof of Theorem 4.2
5 Kurtosis in genus 0 and genus 1 circular condensate
5.1 The kurtosis for DSW
5.2 Scaling limit of the kurtosis of certain genus one circular condensate
References
|
| id | nasplib_isofts_kiev_ua-123456789-212350 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T04:01:57Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Tovbis, Alexander Wang, Fudong 2026-02-05T09:55:43Z 2024 Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case. Alexander Tovbis and Fudong Wang. SIGMA 20 (2024), 070, 26 pages 1815-0659 2020 Mathematics Subject Classification: 37K40; 35P30; 37K10 arXiv:2312.16406 https://nasplib.isofts.kiev.ua/handle/123456789/212350 https://doi.org/10.3842/SIGMA.2024.070 In this paper, we study the spectral theory of soliton condensates - a special limit of soliton gases - for the focusing NLS (fNLS). In particular, we analyze the kinetic equation for the fNLS circular condensate, which represents the first example of an explicitly solvable fNLS condensate with nontrivial large-scale space-time dynamics. Solution of the kinetic equation was obtained by reducing it to Whitham-type equations for the endpoints of spectral arcs. We also study the rarefaction and dispersive shock waves for circular condensates, as well as calculate the corresponding average conserved quantities and the kurtosis. We want to note that one of the main objects of the spectral theory - the nonlinear dispersion relations - is introduced in the paper as some special large genus (thermodynamic) limit of the Riemann bilinear identities that involve the quasimomentum and the quasienergy meromorphic differentials. The authors would also like to thank the anonymous referees for their useful suggestions and comments. FudongWang is supported by the Guangdong Basic and Applied Basic Research Foundation B24030004J. The work of Alexander Tovbis is supported in part by NSF Grant DMS-2009647. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case Article published earlier |
| spellingShingle | Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case Tovbis, Alexander Wang, Fudong |
| title | Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case |
| title_full | Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case |
| title_fullStr | Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case |
| title_full_unstemmed | Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case |
| title_short | Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case |
| title_sort | soliton condensates for the focusing nonlinear schrödinger equation: a non-bound state case |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212350 |
| work_keys_str_mv | AT tovbisalexander solitoncondensatesforthefocusingnonlinearschrodingerequationanonboundstatecase AT wangfudong solitoncondensatesforthefocusingnonlinearschrodingerequationanonboundstatecase |