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### Preprint submitted on 3 Jan 2020

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**Families of solutions to the CKP equation with** **multi-parameters**

### Pierre Gaillard

**To cite this version:**

### Pierre Gaillard. Families of solutions to the CKP equation with multi-parameters. 2020. �hal-

### 02427660�

## Families of solutions to the CKP equation with multi-parameters

### Pierre Gaillard

### Universit´e de Bourgogne,

### Institut de math´ematiques de Bourgogne, 9 avenue Alain Savary BP 47870

### 21078 Dijon Cedex, France :

### E-mail: Pierre.Gaillard@u-bourgogne.fr

### Abstract. We construct solutions to the CKP (cylindrical Kadomtsev-Petviashvili)) equation in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions are called solutions of order N; they depend on 2N − 1 parameters.

### They can be written as a quotient of 2 polynomials of degree 2N (N + 1) in x, t and 4N (N + 1) in y depending on 2N − 2 parameters.

### We explicitly construct the expressions up to order 5 and we study the patterns of their modulus in plane (x, y) and their evolution according to time and parameters.

### Keywords : Fredholm determinants, wronskians, rational solutions, CKP equation.

### PACS numbers :

### 33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd

### 1. Introduction

### We consider the CKP equation which can be written in the form (u _{t} + 6uu _{x} + u _{xxx} + u

### 2t ) _{x} − 3 u _{yy}

### t ^{2} = 0, (1)

### subscripts x, y and t denoting partial derivatives.

### Johnson [1] first proposed this equation in 1980. That equation is considered to describe wave surfaces in shallow incompressible fluids [2, 3]. This equation was later derived for internal waves in a stratified medium [4]. The CKP equation is a dissipative equation. There is no soliton-like solution with a linear front localized along straight lines in the (x, y) plane.

### The first solutions were constructed by Johnson in 1980 [1]. Other types of solutions were found by Golinko, Dryuma, and Stepanyants in 1984 [5]. This equation was solved with a new ap- proach in 1986 [6] by giving a connection between solutions of the Kadomtsev-Petviashvili (KP) [7] and solutions of the CKP equation. The Darboux transformation [8] gave another types of solutions. In 2013, the extension to the elliptic case was considered [9].

### In the following, we give the results of the author about the representations of solutions to

### the CKP equation. We have expressed the solutions in terms of Fredholm determinants of order

### 2N depending on 2N − 1 parameters. We have also given another representation in terms of wronskians of order 2N with 2N − 1 parameters. These representations allow to obtain an infinite hierarchy of solutions to the CKP equation, depending on 2N − 1 real parameters.

### We have used these results to build rational solutions to the CKP equation, when a parameter tends to 0.

### Rational solutions of order N depending on 2N − 2 parameters without the presence of a limit have been constructed. These families depending on 2N − 2 parameters for the N -th order can be written as a ratio of two polynomials of degree 2N (N + 1) in x, t and 4N (N + 1) in y depending on 2N − 2 parameters.

### That provides an effective method to build an infinite hierarchy of rational solutions of order N depending on 2N − 2 real parameters. We present here the representations of their modulus in the plane of the coordinates (x, y) and their evolution according to time and the 2N − 2 real parameters a _{i} and b _{i} and time t for N an integer such that 1 ≤ N ≤ 5.

### 2. Solutions to the CKP equation expressed in terms of Fredholm determinants We need to give some notation in the following. We define first real numbers λ _{j} such that

### −1 < λ _{ν} < 1, ν = 1, . . . , 2N ; they depend on a parameter ǫ and can be written as

### λ _{j} = 1 − 2ǫ ^{2} j ^{2} , λ _{N} _{+j} = −λ _{j} , 1 ≤ j ≤ N, (2) Then, we define κ _{ν} , δ _{ν} , γ _{ν} and x _{r,ν} ; they are functions of λ _{ν} , 1 ≤ ν ≤ 2N and are defined by the following formulas :

### κ _{j} = 2 q

### 1 − λ ^{2} _{j} , δ _{j} = κ _{j} λ _{j} , γ _{j} =

### s 1 − λ _{j} 1 + λ _{j} ,;

### x _{r,j} = (r − 1) ln γ _{j} − i

### γ _{j} + i , r = 1, 3, τ _{j} = −12iλ ^{2} _{j} q

### 1 − λ ^{2} _{j} − 4i(1 − λ ^{2} _{j} ) q 1 − λ ^{2} _{j} , κ _{N+j} = κ _{j} , δ _{N} _{+j} = −δ _{j} , γ _{N+j} = γ _{j} ^{−1} ,

### x _{r,N+j} = −x _{r,j} , , τ _{N+j} = τ _{j} j = 1, . . . , N.

### (3)

### e _{ν} 1 ≤ ν ≤ 2N are defined by : e _{j} = 2i

### P 1/2 M−1

### k=1 a _{k} (je) ^{2} ^{k+1} − i P 1/2 M −1

### k=1 b _{K} (je) ^{2} ^{k+1} , e _{N} _{+j} = 2i

### P 1/2 M−1

### k=1 a _{k} (je) ^{2} ^{k+1} + i P 1/2 M −1

### k=1 b _{k} (je) ^{2} ^{k+1}

### , 1 ≤ j ≤ N, a _{k} , b _{k} ∈ R, 1 ≤ k ≤ N.

### (4)

### ǫ _{ν} , 1 ≤ ν ≤ 2N are defined by :

### ǫ _{j} = 1, ǫ _{N+j} = 0 1 ≤ j ≤ N. (5)

### As usual I is the unit matrix and D _{r} = (d _{jk} ) _{1≤j,k≤2N} the matrix defined by : d _{νµ} = (−1) ^{ǫ}

^{ν}

### Y

### η6=µ

### γ _{η} + γ _{ν} γ _{η} − γ _{µ}

### exp(κ _{ν} x + ( κ _{ν} y

### 12 − 2δ _{ν} )yt + 4iτ _{ν} t + x _{r,ν} + e _{ν} ). (6) Then we get :

### Theorem 2.1 The function v defined by

### v(x, y, t) = −2 |n(x, y, t)| ^{2}

### d(x, y, t) ^{2} (7)

### where

### n(x, y, t) = det(I + D _{3} (x, y, t)), (8)

### d(x, y, t) = det(I + D _{1} (x, y, t)), (9)

### and D _{r} = (d _{jk} ) _{1≤j,k≤2N} the matrix d _{νµ} = (−1) ^{ǫ}

^{ν}

### Y

### η6=µ

### γ _{η} + γ _{ν} γ _{η} − γ _{µ}

### exp(κ _{ν} x + ( κ _{ν} y

### 12 − 2δ _{ν} )yt + 4iτ _{ν} t + x _{r,ν} + e _{ν} ). (10) is a solution to (1), depending on 2N − 1 parameters a _{k} , b _{k} , 1 ≤ k ≤ N − 1 and ǫ.

### A proof of this result is a consequence of previous works of the author [18, 20, 36, 43].

### 3. Wronskian representation of the solutions to the CKP equation We define here the following notations :

### φ _{r,ν} = sin Θ _{r,ν} , 1 ≤ ν ≤ N, φ _{r,ν} = cos Θ _{r,ν} , N + 1 ≤ ν ≤ 2N, r = 1, 3, (11) with

### Θ r,ν = −iκ ν x

### 2 + i( −κ ν y

### 24 + δ _{ν} )yt − i x _{r,ν}

### 2 + 2τ ν t + γ _{ν} w − i e _{ν}

### 2 , 1 ≤ ν ≤ 2N. (12) W _{r} (w) is the wronskian of the functions φ _{r,1} , . . . , φ _{r,2N} defined by

### W _{r} (w) = det[(∂ _{w} ^{µ−1} φ _{r,ν} ) ν, µ∈[1,...,2N] ]. (13) We consider the matrix D _{r} = (d _{νµ} ) ν, µ∈[1,...,2N ] defined in (10).

### Then we have the following result Theorem 3.1

### det(I + D _{r} ) = k _{r} (0) × W _{r} (φ _{r,1} , . . . , φ _{r,2N} )(0), (14) where

### k _{r} (y) = 2 ^{2N} exp(i P 2N ν=1 Θ r,ν ) Q 2N

### ν=2

### Q ν−1

### µ=1 (γ ν − γ _{µ} ) .

### It is also a consequence of previous works of the author [18, 20, 36, 43].

### 4. Rational solutions to the CKP equation

### From those two preceding results, we can construct rational solutions to the CKP equation as a quotient of two determinants.

### We use the following notations : X _{ν} = −iκ _{ν} x

### 2 + i( −κ _{ν} y

### 24 + δ _{ν} )yt − i x _{3,ν}

### 2 + 2τ _{ν} t + γ _{ν} w − i e _{ν} 2 , Y _{ν} = −iκ _{ν} x

### 2 + i( −κ _{ν} y

### 24 + δ _{ν} )yt − i x _{1,ν}

### 2 + 2τ _{ν} t + γ _{ν} w − i e _{ν}

### 2 ,

### for 1 ≤ ν ≤ 2N , with κ _{ν} , δ _{ν} , x _{r,ν} defined in (3) and parameters e _{ν} defined by (4).

### We define the following functions :

### ϕ _{4j+1,k} = γ _{k} ^{4j−1} sin X _{k} , ϕ _{4j+2,k} = γ _{k} ^{4j} cos X _{k} ,

### ϕ _{4j+3,k} = −γ _{k} ^{4j+1} sin X _{k} , ϕ _{4j+4,k} = −γ _{k} ^{4j+2} cos X _{k} , (15) for 1 ≤ k ≤ N , and

### ϕ _{4j+1,N+k} = γ _{k} ^{2N−4j−2} cos X _{N+k} , ϕ _{4j+2,N+k} = −γ _{k} ^{2N} ^{−4j−3} sin X _{N+k} ,

### ϕ _{4j+3,N+k} = −γ _{k} ^{2N} ^{−4j−4} cos X _{N+k} , ϕ _{4j+4,N+k} = γ _{k} ^{2N} ^{−4j−5} sin X _{N+k} , (16) for 1 ≤ k ≤ N .

### We define the functions ψ _{j,k} for 1 ≤ j ≤ 2N , 1 ≤ k ≤ 2N in the same way, the term X _{k} is only replaced by Y _{k} .

### ψ _{4j+1,k} = γ _{k} ^{4j−1} sin Y _{k} , ψ _{4j+2,k} = γ _{k} ^{4j} cos Y _{k} ,

### ψ _{4j+3,k} = −γ _{k} ^{4j+1} sin Y _{k} , ψ _{4j+4,k} = −γ _{k} ^{4j+2} cos Y _{k} , (17) for 1 ≤ k ≤ N , and

### ψ _{4j+1,N+k} = γ ^{2N−4j−2} _{k} cos Y _{N+k} , ψ _{4j+2,N+k} = −γ _{k} ^{2N} ^{−4j−3} sin Y _{N+k} ,

### ψ _{4j+3,N+k} = −γ _{k} ^{2N−4j−4} cos Y _{N+k} , ψ _{4j+4,N+k} = γ _{k} ^{2N} ^{−4j−5} sin Y _{N+k} , (18) for 1 ≤ k ≤ N .

### The following ratio

### q(x, t) := W _{3} (0) W _{1} (0) can be written as

### q(x, t) = ∆ _{3}

### ∆ _{1} = det(ϕ _{j,k} ) _{j, k∈[1,2N} _{]} det(ψ _{j,k} ) j, k∈[1,2N]

### . (19)

### The terms λ _{j} depending on ǫ are defined by λ _{j} = 1 − 2jǫ ^{2} . All the functions ϕ _{j,k} and ψ _{j,k} and their derivatives depend on ǫ. They can all be prolonged by continuity when ǫ = 0.

### We use the following expansions

### ϕ _{j,k} (x, y, t, ǫ) =

### N−1

### X

### l=0

### 1

### (2l)! ϕ _{j,1} [l]k ^{2l} ǫ ^{2l} + O(ǫ ^{2N} ), ϕ _{j,1} [l] = ∂ ^{2l} ϕ _{j,1}

### ∂ǫ ^{2l} (x, y, t, 0), ϕ _{j,1} [0] = ϕ _{j,1} (x, y, t, 0), 1 ≤ j ≤ 2N, 1 ≤ k ≤ N, 1 ≤ l ≤ N − 1, ϕ _{j,N+k} (x, y, t, ǫ) =

### N −1

### X

### l=0

### 1

### (2l)! ϕ _{j,N+1} [l]k ^{2l} ǫ ^{2l} + O(ǫ ^{2N} ), ϕ _{j,N+1} [l] = ∂ ^{2l} ϕ _{j,N+1}

### ∂ǫ ^{2l} (x, y, t, 0), ϕ _{j,N+1} [0] = ϕ _{j,N+1} (x, y, t, 0), 1 ≤ j ≤ 2N, 1 ≤ k ≤ N, 1 ≤ l ≤ N − 1.

### We have the same expansions for the functions ψ _{j,k} . ψ _{j,k} (x, y, t, ǫ) =

### N−1

### X

### l=0

### 1

### (2l)! ψ _{j,1} [l]k ^{2l} ǫ ^{2l} + O(ǫ ^{2N} ), ψ _{j,1} [l] = ∂ ^{2l} ψ _{j,1}

### ∂ǫ ^{2l} (x, y, t, 0),

### ψ _{j,1} [0] = ψ _{j,1} (x, y, t, 0), 1 ≤ j ≤ 2N, 1 ≤ k ≤ N, 1 ≤ l ≤ N − 1,

### ψ _{j,N+k} (x, t, ǫ) =

### N −1

### X

### l=0

### 1

### (2l)! ψ _{j,N+1} [l]k ^{2l} ǫ ^{2l} + O(ǫ ^{2N} ), ψ _{j,N+1} [l] = ∂ ^{2l} ψ _{j,N+1}

### ∂ǫ ^{2l} (x, y, t, 0), ψ _{j,N+1} [0] = ψ _{j,N+1} (x, t, 0), 1 ≤ j ≤ 2N, 1 ≤ k ≤ N, N + 1 ≤ k ≤ 2N..

### Then we get the following result :

### Theorem 4.1 The function v defined by

### v(x, y, t) = −2 | det((n _{jk)}

_{j,k∈[1,2N]}

### )| ^{2}

### det((d _{jk)}

_{j,k∈[1,2N]}