Adjoint solutions and superposition principle for linearizable Krichever-Novikov equation
Existence of an operator equality for equations connected by nonlocal transformations allowed us to propose a method of finding of a new solution of the initial equation adjoint to its known solution. This approach is used for construction of exact solutions for the linearizable Krichever-Novikov eq...
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| author | Tychynin, V. Тичинін, В. |
| author_facet | Tychynin, V. Тичинін, В. |
| author_institution_txt_mv | [
{
"author": "V. Tychynin",
"institution": "Prydniprovs’ka State Academy of Civil Engineering and Architecture"
}
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| author_sort | Tychynin, V. |
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| description | Existence of an operator equality for equations connected by nonlocal transformations allowed us to propose a method of finding of a new solution of the initial equation adjoint to its known solution. This approach is used for construction of exact solutions for the linearizable Krichever-Novikov equation and for the corresponding linear equation. The formula of nonlinear nonlocal superposition of solutions for this nonlinear equation is derived and applied to generation of its solutions. |
| first_indexed | 2026-08-04T01:06:20Z |
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Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 181�192
ÓÄÊ 517.912:512.816
Adjoint solutions and superposition
principle for linearizable
Krichever�Novikov equation
V.A. Tychynin
Prydniprovs'ka State Academy of Civil Engineering and Architecture,
Dnipro, Ukraine
E-mail: tychynin@ukr.net
Íàÿâíiñòü îïåðàòîðíî¨ ðiâíîñòi äëÿ ðiâíÿíü, ïîâ'ÿçàíèõ íåëîêàëüíèìè
ïåðåòâîðåííÿìè, äîçâîëèëà çàïðîïîíóâàòè ìåòîä çíàõîäæåííÿ iíøî-
ãî ðîçâ'ÿçêó âèõiäíîãî ðiâíÿííÿ, ÿêèé ïðè¹äíàíèé äî âiäîìîãî éîãî
ðîçâ'ÿçêó. Öåé ïiäõiä çàñòîñîâàíî äëÿ ïîáóäîâè òî÷íèõ ðîçâ'ÿçêiâ ëi-
íåàðèçîâàíîãî ðiâíÿííÿ Êðè÷åâåðà�Íîâiêîâà òà âiäïîâiäíîãî ëiíiéíî-
ãî ðiâíÿííÿ. Âèâåäåíî ôîðìóëó íåëiíiéíî¨ íåëîêàëüíî¨ ñóïåðïîçèöi¨
ðîçâ'ÿçêiâ, ÿêó âèêîðèñòàíî äëÿ ðîçìíîæåííÿ òî÷íèõ ðîçâ'ÿçêiâ öüîãî
íåëiíiéíîãî ðiâíÿííÿ.
Existence of an operator equality for equations connected by nonlocal
transformations allowed us to propose a method of �nding of a new solu-
tion of the initial equation adjoint to its known solution. This approach
is used for construction of exact solutions for the linearizable Krichever�
Novikov equation and for the corresponding linear equation. The formula
of nonlinear nonlocal superposition of solutions for this nonlinear equation
is derived and applied to generation of its solutions.
1. Introduction. A wide range of efficient methods for study of
nonlinear partial differential equations are being developed at the mo-
ment. A considerable part of them are based on a fundamental idea
of symmetry and, in particular, on the group-theoretical method sug-
gested by Lie [6, 14, 16]. The most important generalizations of the ba-
sic symmetry group approach are realized in the concepts of conditional
(nonclassical) symmetries, weak symmetries [5, 9, 15] and nonlocal sym-
metries of differential equations [1, 2, 3, 4, 7, 10, 12, 13, 17, 18, 21, 23, 24].
Therefore, development of other approaches to seek for new symmetries
182 V.A. Tychynin
and methods for investigation of these equations is of importance and
stays relevant.
Finite nonlocal transformations are efficiently used to study and solve
nonlinear partial differential equations for a long time [8, 10, 21, 23, 24].
In particular, a number of interesting results for nonlinear equations
connected among themselves by means of the nonlocal transformations
of variables were obtained and formulae generating solutions or nonlocal
nonlinear superposition were derived [11, 19, 22, 23].
Let us remind here the main concepts and terminology of the nonlocal
transformations method. Assume that a given nonlocal transformation
T : xi = hi(y, v(k)), uK = HK(y, v(k)),
i = 1, . . . , n, K = 1, . . . ,m, (1)
maps an initial (source) equation
F0(x, u(n)) = 0 (2)
into an equation Φ(y, v(q)) = 0 of order q = n+ k that admits factoriza-
tion to another equation which we call a target equation
F1(y, v(s)) = 0, (3)
i.e.,
Φ(y, v(q)) = λF1(y, v(s)). (4)
Here λ is a differential operator of order n + k − s. This results in
algorithms for finding solutions of (2) via known solutions of (3). Exis-
tence of factorization equation (4) gives rise to a technique of finding of
a special solution to the initial equation (2) from a known solution of
the equation Φ(y, v(q)) = 0. The symbol u(r) denotes the tuple of partial
derivatives of the function u from order zero up to order r. In the case of
two independent variables, we use the special notation of the variables:
x1 = x, x2 = t and thus ut = ∂u/∂t = ∂tu, ux = ∂u/∂x = ∂xu.
The paper is organized as follows. In the next section we begin with
some preliminary remarks on the concept of adjoint solution of the initial
equation. Then we apply it to the linearisable Krichever–Novikov equa-
tion derived from the linear one via the known nonlocal transformation.
In Section 3 this concept is applied to the case of the nonlocal invariance
Adjoint solutions for linearizable Krichever–Novikov equation 183
of the linearizable Krichever–Novikov equation. Examples of adjoint so-
lutions are constructed.
2. Adjoint solution of the initial equation. This section is
devoted to construction of solutions of the initial equation generated
from known solutions of the appropriate inhomogeneous target equation.
Existence of a factorization equation (4) gives rise to a technique [20] of
construction of the special solution to the initial equation (2). Further
we call it an adjoint solution.
We assume that a given function v = f(y) is not a solution of equa-
tion (3), that is, substituting this function into (3), we get another equa-
tion with discrepancy w(y)
F1(y, v(s)) = w(y). (5)
Suppose, nevertheless, that equation (4) holds and the equation
λ(y, v(s))F1(y, v(s)) = λ(y, v(s))w(y) = 0 (6)
appears to be true. Here w(y) runs through the set of solutions of a linear
equation λ(y, v(s))w(y) = 0 with variable coefficients of spatial form.
Solving (6) with respect to the unknown function w(y, v(k)(y)), one can
find its solution as a function depending on y, v(k)(y)
w = W (y, v(k)). (7)
After substitution of (7) into the equation (5) we obtain an inhomo-
geneous equation for the dependent variable v:
F1(y, v(s)) = W (y, v(k)). (8)
Hence the result of transformation (1) takes a form
Φ(y, v(q)) = λF2(y, v(s)) = λ(F1(y, v(s))−W (y, v(k)). (9)
The function v(y) determined by (9) satisfies the equation T F0(x, u(n))
= Φ(y, v(q)). Therefore, substituting v(y) obtained in this way into the
formulae of nonlocal transformation T , one can find an appropriate so-
lution of the given equation (2). Moreover, having the information on
symmetries of the inhomogeneous equation (8), one can construct a r-
parametrical family of solutions for it and, consequently, find the corre-
sponding parametrical sets of solutions to the initial equation (2). If we
184 V.A. Tychynin
let in (8) W (y, v(k)) = 0 the equality (9) returns us to the connection
of solutions of the initial equation and (3). That is why further we call
such a special solution an adjoint. In what follows, we illustrate this
approach by some examples.
3. Adjoint solutions constructed via the linearization. In this
section we use the nonlocal transformation that connects the linear equa-
tion and the Krichever–Novikov equation of a special form and illustrate
the proposed approach by some examples.
It is well-known [8] that the Krichever–Novikov equation
ut + 3
4u
−1
x u2
xx − uxxx = 0 (10)
can be obtained by applying the nonlocal transformation
w =
√
ux (11)
to the homogeneous linear third-order partial differential equation
wt − wxxx = 0. (12)
The operator equation (4) connecting these two equations has the form
−4u2
x∂x
(
ut + 3
4u
−1
x u2
xx − uxxx
)
= 0. (13)
Suppose a function S(x, t) (discrepancy) is defined such that the inhomo-
geneous equation
ut + 3
4u
−1
x u2
xx − uxxx = S(x, t)
is satisfied. Then the condition ∂xS(x, t) = 0 follows from (13). In
particular, if we let S(x, t) be a linear function of time, i.e., S(x, t) = ht,
the corresponding equation takes the form
ut + 3
4u
−1
x u2
xx − uxxx = ht. (14)
This equation admits the Lie algebra spanned by the following operators
X1 = ∂x, X2 = ∂t + ht∂u, X3 = ∂u,
X4 = (u− ht)∂u, X5 = 1
3x∂x + t∂t + ht2∂u. (15)
1) We obtain a simple group-invariant solution of equation (14) sol-
ving the characteristic equation generated by the sum of the first two
Adjoint solutions for linearizable Krichever–Novikov equation 185
operators of the algebra (15), ux + ut − ht = 0. The corresponding Lie
ansatz is
u(x, t) = − 1
2hx
2 + hxt+ f(t− x).
Substituting this expression into (14), we find the reduced ordinary dif-
ferential equation
4f ′′′(hω − f ′) + 3f ′′2 − 6hf ′′ − 4f ′2 + 8hωf ′ − 4h2ω2 + 3h2 = 0,
where ω = t−x. The general solution of this equation allows us to write
down the required solution of (14)
u(x, t) = 1
2ht
2 + 1
16c2 sin 2(t− x+ c1)
+ 1
2c2 sin(t− x+ c1) + 3
8c2(t− x+ c1) + c3,
where c1, c2, c3 are arbitrary constants.
Applying the nonlocal transformation (11) to the obtained solution,
we get the corresponding solution of the linear equation:
w(x, t) = 1
4
√
−2c2 cos 2(t− x+ c1)− 8c2 cos(t− x+ c1)− 6c2.
One can compare the above solution with another solution of (12) being
obtained in the form w(x, t) = f(ct− x) determining a wave of unchan-
ging profile moving at the constant velocity c:
w(x, t) = c̄1 + c̄2 sin(ct− x) + c̄3 cos(ct− x).
2) Another group-invariant solution of the equation (14) correspon-
ding to the operator X5 of the Lie algebra (15) has an implicit form
u(x, t) = 1
2ht
2 +
∫ t/x3
expQ(k) dk + c3,
Q(k) = 2
9
√
3c1B1 − 16
3
√
3B2 + 4
3B3 + 2
9
√
3B4
− 16
3
√
3c1B5 + 4
3c1B6 + c2, (16)
where
B1 =
∫ k Y 2
3
(
1
9
√
3√
b
)
b3/2
(
c1Y− 1
3
(
1
9
√
3√
b
)
+ J− 1
3
(
1
9
√
3√
b
)) db,
186 V.A. Tychynin
B2 =
∫ k J 2
3
(
1
9
√
3√
b
)
b1/2
(
c1Y− 1
3
(
1
9
√
3√
b
)
+ J− 1
3
(
1
9
√
3√
b
)) db,
B3 =
∫ k 4
√
3bJ 2
3
(
1
9
√
3√
b
)
− J− 1
3
(
1
9
√
3√
b
)
b
(
c1Y− 1
3
(
1
9
√
3√
b
)
+ J− 1
3
(
1
9
√
3√
b
)) db,
B4 =
∫ k J 2
3
(
1
9
√
3√
b
)
b3/2
(
c1Y− 1
3
(
1
9
√
3√
b
)
+ J− 1
3
(
1
9
√
3√
b
)) db,
B5 =
∫ k Y 2
3
(
1
9
√
3√
b
)
b1/2
(
c1Y− 1
3
(
1
9
√
3√
b
)
+ J− 1
3
(
1
9
√
3√
b
)) db,
B6 =
∫ k 4
√
3bY 2
3
(
1
9
√
3√
b
)
− Y− 1
3
(
1
9
√
3√
b
)
b
(
c1Y− 1
3
(
1
9
√
3√
b
)
+ J− 1
3
(
1
9
√
3√
b
)) db,
where c1, c2, c3 are arbitrary constants, Jα(x) and Yα(x) are Bessel
functions of the first and the second kinds respectivelly. Applying the
formula (11) to this solution, we get such nonstationary solution of the
linear equation (12):
w(x, t) = ± 1
x2
√
3te
1
9
√
3c1B̄1− 8
3
√
3B̄2+ 2
3 B̄3+ 1
9
√
3B̄4− 8
3 c1B̄5+ 2
3 c1B̄6+
c2
2 .
B̄i, i = 1, . . . , 6, are the same as introduced above with k = t
x3 .
4. Adjoint solutions found via the nonlocal invariance. Beside
a nonlocal linearization, the Krichever–Novikov equation (10) admits the
auto-Bäcklund transformation [8]:
ux = v−1
x v2
xx, (17)
ut = 2v−1
x vxxvxxxx − 2v−2
x v2
xxvxxx + 5
4v
−3
x v4
xx − v−1
x v2
xxx,
where v(x, t) is another solution of the same equation
vt + 3
4v
−1
x v2
xx − vxxx = 0.
In other words, the equation (10) stays invariant under the nonlocal
transformation (17). This connection is realized by means of the operator
equality(
−4vxxv
2
x∂x + 8v3
x∂xx
)
·
(
vt + 3
4v
−1
x v2
xx − vxxx
)
= 0. (18)
Adjoint solutions for linearizable Krichever–Novikov equation 187
We assume existence of the function v(x, t) that is a solution of the
inhomogeneous equation
vt + 3
4v
−1
x v2
xx − vxxx = W (x, t). (19)
Solving the partial differential equation generated by (18)
−4vxxv
2
xWx(x, t) + 8v3
xWxx(x, t) = 0
with respect to W (x, t, v, vx, vxx), we obtain the general solution
W (x, t) = f1(t) + f2(t)
∫ √
vx dx. (20)
To exclude an integral term in (20), we differentiate (19) with respect
to x and set for simplicity f1(t) = 0, f2(t) = K in (20), where K is
a constant. So, instead of (19) we consider the equation
∂x
(
vt + 3
4v
−1
x v2
xx − vxxx
)
−K
√
vx = 0. (21)
This inhomogeneous equation admits an infinite-dimensional Lie algebra
spanned by the following operators
X1 = ∂t + F1(t)∂v, X2 = ∂x + F2(t)∂v,
X3 = 1
3∂x + t∂t +
(
7
3v + F3(t)
)
∂v. (22)
Here Fi(t), i = 1, 2, 3, are arbitrary functions of the time variable. This
algebra allows us to get a wide range of group-invariant solutions of the
equation (21). We choose v(x, t) in the traveling wave solution form
v(x, t) = G(ω), ω = x − ht, where h is a fixed constant. Substituting
this expression into (21), we get the reduced equation
4G′(ω)2G′′′′(ω)− 6G′(ω)G′′(ω)G′′′(ω)
+ 3G′′(ω)3 + 4hG′(ω)2G′′(ω) + 4KG′(ω)
5/2
= 0, (23)
which admits a solution
v(x, t) = G(ω) =
∫
Y (ω) dω + c4,
where Y (ω) = Z(ω) + ω + c3 and Z(ω) is the function determined by
the equation∫ Z(ω)
H−1 df = 0. (24)
188 V.A. Tychynin
H(Z(ω), f) is an implicit solution of any of two equations
∓
(
2
√
f − c2 ∓ Ph
)
h3/2 +Kh arctan
(
Hh+K
√
f√
hfP
)
= 0, (25)
P =
√
−hH
2 + 2
√
fKH − c1f
f
.
To use the formula (17) and to verify a new solution, we need an expres-
sion for ux to be a function of ω = x − ht. First we find a solution of
the equation (10) differentiated with respect to x. We set u(x, t) = L(ω)
and substitute that into the equation
∂x
(
ut + 3
4u
−1
x u2
xx − uxxx
)
= 0. (26)
Implementation of the reduction procedure leads to the ordinary dif-
ferential equation
4L′(ω)2L′′′′(ω)− 6L′(ω)L′′(ω)L′′′(ω)
+ 3L′′(ω)3 + 4hL′(ω)2L′′(ω) = 0. (27)
An implicit solution of this equation is determined by the integral equa-
tion ∫ L(ω)
Q(a)
−1
da− ω − c̄4 = 0,
where function Q(a) is defined by the equations
∓
∫ Q(a) hf√
hf(c̄1 − 4fh2 + 4
√
fh2c̄2 − h2c̄22)
df + a+ c̄3 = 0,
where c̄1, c̄2, c̄3 are arbitrary constants.
Now we apply the formula (17) to the obtained solution of the equa-
tion (23) and find the corresponding expression for ux(x, t):
ux(x, t) = L̂′(ω) = G′(ω)−1G′′(ω)2.
After simplification we get
L̂′(ω) =
B2
Y
, Y (ω) = Z(ω) + ω + c3.
Adjoint solutions for linearizable Krichever–Novikov equation 189
Here B(ω, Y ) are the implicit functions determined by the equations
∓
(
2
√
Y − c2
)
h5/2 + Th3/2 +Kh arctan
(
Bh+K
√
Y√
hY T
)
= 0,
T =
√
−hB
2 + 2
√
Y KB − c1Y
Y
.
The function Z(ω) was implicitly determined above by equations (24),
(25). Substitution of this solution into the equation (27) takes it to zero.
Knowing the Lie algebra (22) of the inhomogeneous equation (21) we
can construct a wide family of group-invariant solutions, and, therefore,
obtain various solutions of the equation (10). The new solution of the
equation (26) constructed above obviously can be generated via the in-
variance algebra admitted by this equation or by means of its any other
symmetry. The symmetry solutions of the special inhomogeneous target
equation allow us to generate different solutions for the initial equation.
What type of the symmetry of initial equations do we have in this case?
As a target equation is broken by a discrepancy appearance, it seems
naturally to call it a forced symmetry.
5. The superposition formula and generation of solutions.
We return to the homogeneous linear third-order differential equation
(12) and the nonlocal transformation (11), connecting this equation
with (10). We choose a linear superposition principle for (12) setting
wIII(x, t) ≡ w(x, t) = wI(x, t) + wII(x, t).
Here wI(x, t), wII(x, t) are some known solutions of the linear equation.
As equations are connected by the nonlocal transformation (11), the
corresponding principle of nonlinear nonlocal superposition of solutions
for equation (10) can be constructed.
Theorem 1. The nonlinear nonlocal superposition formula of solutions
for equation (10) has the form
u(x, t) = uI(x, t) + uII(x, t)
+ 2
∫ √
uI(x, t)
√
uII(x, t) dx+ s(t), (28)
where the arbitrary function s(t) is defined by the equation
ut = 2
(√
uI(x, t) +
√
uII(x, t)
)
∂2
x
(√
uI(x, t) +
√
uII(x, t)
)
190 V.A. Tychynin
−
(
∂x
(√
uI(x, t) +
√
uII(x, t)
))2
. (29)
Given solutions uI and uII, the new solution of (10) is found integra-
ting the third term of (28). We get the specialization of the function s(t)
substituting the expression (28) into (29) and solving the equations ob-
tained with respect to s.
We illustrate utilization of the proposed superposition formula for
generation of solutions of the equation (10).
1) It can be easily verified that
uI = 1
1280c1
(
x5 + 5c2x
4 + 10c22x
3 + 10c32x
2 + 5c42x+ c52
)
+ c3,
uII = k1x
5
are time-independent solutions of the equation (10). Applying the formu-
la (28) adduced in Theorem 1 we find a time-dependent solution
uIII = 1
3840c13/2
(
a1x
5 + a2x
4 + a3x
3 + 30
√
c1c
3
2x
2
+ 15
√
c1c
4
2x+ 960c1c
2
2
√
5k1t+ κ
)
,
a1 = 3
√
c1 + 96
√
5k1c1 + 3840k1c1
3/2,
a2 = 15c2
√
c1 + 240
√
5k1c1c2,
a3 = 160
√
5k1c1c
2
2 + 30
√
c1c
2
2,
κ = 3
√
c1c2
5 + 380c
3/2
1 (c3 + c4).
2) Choosing c1 = 0 in (16), we obtain a simpler solution
uI =
∫ y
x3
exp
(
− 16
3
√
3B̃1 + 4
3 B̃2 + 2
9
√
3B̃3 + c2
)
dk + c3,
B̃1 =
∫ k
J 2
3
(
1
9
√
3√
b
)
b−1/2J− 1
3
(
1
9
√
3√
b
)−1
db,
B̃2 =
∫ k 4
√
3bJ 2
3
(
1
9
√
3√
b
)
− J− 1
3
(
1
9
√
3√
b
)
b
(
J− 1
3
(
1
9
√
3√
b
)) db,
B̃3 =
∫ k
J 2
3
(
1
9
√
3√
b
)
b−3/2J− 1
3
(
1
9
√
3√
b
)−1
db. (30)
Adjoint solutions for linearizable Krichever–Novikov equation 191
Let the second solution be
uII = k1x
5.
Then
uIII
x = −3tx−4 exp
(
− 16
3
√
3B̂1 + 4
3 B̂2 + 2
9
√
3B̂3 + c2
)
+ 5k1x
4
+ 2
√
5
√
k1x4
√
−3tx−4 exp
(
− 16
3
√
3B̂1 + 4
3 B̂2 + 2
9
√
3B̂3 + c2
)
,
where B̂i, i = 1, 2, 3, are the same as those introduced in (30) but k = t
x3 .
One can easily verify that obtained expression satisfies (26). More solu-
tions may be constructed by utilization of the previous theorem and
application of the Lie symmetry transformations or any other formula
generating solutions.
6. Conclusion. The concept of an adjoint solution of the initial
equation was developed in this paper, and used for construction of new
solutions of linearizable Krichever–Novikov equation and for the connec-
ted linear one. Some of them were obtained in an explicit form, while
others have a parametrical representations with functional parameters
given in implicit form. The Lie symmetry solutions of the special inho-
mogeneous target equation allowed us to generate appropriate solutions
for the given initial equation. The superposition formula was derived in
the present paper and applied for the generation of solutions to the equa-
tion (10). All the found solutions can be naturally extended by means of
the Lie symmetry transformations or any other formula generating new
solutions. The results obtained for the equation (10) can be extended to
similar classes of equations.
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|
| id | oai:trim.imath.kiev.ua:article-375 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:20Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/70/8d6366cba0f38fa5b9f1ed3120d09f70.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3752020-08-13T08:52:53Z Adjoint solutions and superposition principle for linearizable Krichever-Novikov equation Приєднані розв'язки та принцип суперпозиції для лінеаризованого рівняння Крічевера-Новікова Tychynin, V. Тичинін, В. Existence of an operator equality for equations connected by nonlocal transformations allowed us to propose a method of finding of a new solution of the initial equation adjoint to its known solution. This approach is used for construction of exact solutions for the linearizable Krichever-Novikov equation and for the corresponding linear equation. The formula of nonlinear nonlocal superposition of solutions for this nonlinear equation is derived and applied to generation of its solutions. Наявність операторної рівності для рівнянь, пов'язаних нелокальними перетвореннями, дозволила запропонувати метод знаходження іншого розв'язку вихідного рівняння, який приєднаний до відомого його розв'язку. Цей підхід застосовано для побудови точних розв'язків лінеаризованого рівняння Кричевера-Новікова та відповідного лінійного рівняння. Виведено формулу нелінійноїнелокальної суперпозиції розв'язків, яку використано для розмноження точних розв'язків цього нелінійного рівняння. Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/375 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 181-192 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 181-192 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 181-192 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/375/372 Авторське право (c) 2019 В. Тичинін |
| spellingShingle | Tychynin, V. Тичинін, В. Adjoint solutions and superposition principle for linearizable Krichever-Novikov equation |
| title | Adjoint solutions and superposition principle for linearizable Krichever-Novikov equation |
| title_alt | Приєднані розв'язки та принцип суперпозиції для лінеаризованого рівняння Крічевера-Новікова |
| title_full | Adjoint solutions and superposition principle for linearizable Krichever-Novikov equation |
| title_fullStr | Adjoint solutions and superposition principle for linearizable Krichever-Novikov equation |
| title_full_unstemmed | Adjoint solutions and superposition principle for linearizable Krichever-Novikov equation |
| title_short | Adjoint solutions and superposition principle for linearizable Krichever-Novikov equation |
| title_sort | adjoint solutions and superposition principle for linearizable krichever-novikov equation |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/375 |
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