# The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three

Authors: Pierre Gaillard
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x,y) plane in function of time t and parameters a1, a2, b1, and b2.


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## Body

## 1. Introduction

The Johnson equation was introduced in 1980 by Johnson [1] to describe waves surfaces in shallow incompressible fluids [2, 3]. This equation was derived for internal waves in a stratified medium [4]. The Johnson equation is dissipative; it is well known that there is no solution with a linear front localized along straight lines in the (x,y) plane. This Johnson equation is, for example, able to explain the existence of the horseshoe-like solitons and multisoliton solutions quite naturally.We consider the Johnson equation (J) in the following normalization:(1)ut+6uux+uxxx+u2tx-3uyyt2=0,where as usual subscripts x, y, and t mean partial derivatives.The first solutions were constructed in 1980 by Johnson [1]. Other types of solutions were found in [5]. A new approach to solve this equation was given in 1986 [6] by giving a link between solutions of the Kadomtsev-Petviashvili (KP) [7] and solutions of the Johnson equation. In 2007, other types of solutions were obtained by using the Darboux transformation [8]. More recently, in 2013, other extensions have been considered as the elliptic case [9].Here, we consider the famous Kadomtsev-Petviashvili (KPI), which can be written in the following form:(2)4ut-6uux+uxxxx-3uyy=0.The KPI equation first appeared in 1970 [7] in a paper written by Kadomtsev and Petviashvili. This equation is considered as a model for surface and internal water waves [10] and in nonlinear optics [11].In the following, we will use the KPI equation to construct solutions to the Johnson equation but in another way different from this used in [6]. Indeed, these last authors consider another representation of KPI equation given by(3)ut+6uux+uxxxx-3uyy=0,and so the transformations between solutions of (3) and (1) are different from those we use to transform solutions to (2) in solutions to (1).In fact, to obtain solutions to (1) from solutions to (2), we use the following transformation:(4)x;y;t⟼x1=-ix-iy2t12;y1=yt;t1=4itIn this paper, we give solutions by means of Fredholm determinants of order 2N depending on 2N-1 parameters and then by means of Wronskians of order 2N with 2N-1 parameters. So we construct an infinite hierarchy of solutions to the Johnson equation, depending on 2N-1 real parameters.New rational solutions depending a priori on2N-2 parameters at order N are constructed, when one parameter tends to 0.We obtain families depending on2N-2 parameters for the Nth order as a ratio of two polynomials of degree 2N(N+1) in x, t and of degree 4N(N+1) in y.In this paper, we construct only rational solutions of order 3, depending on 4 real parameters; we construct the representations of their modulus in the plane of the coordinates(x,y) according to the four real parameters ai and bi for 1≤i≤2 and time t.


## 2. Solutions to Johnson Equation Expressed by Means of Fredholm Determinants

Some notations are given. We define first real numbersλj such that -1<λν<1, ν=1,…,2N; they depend on a parameter ϵ and can be written as(5)λj=1-2ϵ2j2,λN+j=-λj,1≤j≤N.Then, we define κν,δν,γν, and xr,ν; they are functions of λν,1≤ν≤2N, and are defined by the following formulas:(6)κj=21-λj2,δj=κjλj,γj=1-λj1+λj,;xr,j=r-1ln⁡γj-iγj+i,r=1,3,τj=-12iλj21-λj2-4i1-λj21-λj2,κN+j=κj,δN+j=-δj,γN+j=γj-1,xr,N+j=-xr,j,τN+j=τjj=1,…,N.eν1≤ν≤2N are defined by(7)ej=2i∑k=11/2M-1akje2k+1-i∑k=11/2M-1bKje2k+1,eN+j=2i∑k=11/2M-1akje2k+1+i∑k=11/2M-1bkje2k+1,1≤j≤N,ak,bk∈R,1≤k≤N.ϵν, 1≤ν≤2N are defined by(8)ϵj=1,ϵN+j=01≤j≤N.As usual I is the unit matrix and Dr=(djk)1≤j,k≤2N is the matrix defined by the following:(9)dνμ=-1ϵν∏η≠μγη+γνγη-γμexp⁡κνx+κνy12-2δνyt+4iτνt+xr,ν+eν.Then we get the following theorem.Theorem 1.
The functionv defined by(10)vx,y,t=-2nx,y,t2dx,y,t2with(11)nx,y,t=det⁡I+D3x,y,t,(12)dx,y,t=det⁡I+D1x,y,t,and Dr=(djk)1≤j,k≤2N is the matrix(13)dνμ=-1ϵν∏η≠μγη+γνγη-γμexp⁡κνx+κνy12-2δνyt+4iτνt+xr,ν+eνis a solution to (1), depending on 2N-1 parameters ak,bk,1≤k≤N-1 and ϵ.Proof.
The solutionv to the KPI equation can be written as follows by using [12]:(14)vx,y,t=-2nx,y,t2dx,y,t2where(15)nx,y,t=det⁡I+D3x,y,t,(16)dx,y,t=det⁡I+D1x,y,t,and Dr=(djk)1≤j,k≤2N is the matrix (17)dνμ=-1ϵν∏η≠μγη+γνγη-γμexp⁡iκνx-2δνy+τνt+xr,ν+eν.where κν, δν, xr,ν, γν, τν, eν, and ϵν are defined in (6), (5), (7), and (8).
The connection between the solutions to the Johnson equation and these to the KPI equation was already explained in [6] but with another expression of the KPI equation (3).
Here, the knowledge of a solutionu to the KPI equation (2) gives a solution to the Johnson equation (1). Let us consider u(x,y,t) a solution of the KPI equation (2), then the function(18)u~x1,y1,t1 for (19)x1=-ix-iy2t12,y1=yt,t1=4it is a solution to the KPI equation (2). Using this crucial transformation, the solution to the Johnson equation takes the form (20)vx,y,t=-2det⁡I+D3x,y,t2det⁡I+D1x,y,t2 with the matrix Dr defined in (17).
So we get the solutions to (14) by means of Fredholm determinants.


## 3. Solutions to the Johnson Equation by Means of Wronskians

We use the following notations:(21)ϕr,ν=sin⁡Θr,ν,1≤ν≤N,ϕr,ν=cos⁡Θr,ν,N+1≤ν≤2N,r=1,3, with(22)Θr,ν=-iκνx2+i-κνy24+δνyt-ixr,ν2+2τνt+γνw-ieν2,1≤ν≤2N.Wr(w) is the Wronskian of the functions ϕr,1,…,ϕr,2N defined by (23)Wrw=det⁡∂wμ-1ϕr,νν,μ∈1,…,2N.We consider the matrix Dr=(dνμ)ν,μ∈[1,…,2N] defined in (17).Then we have the following result.Theorem 2.
(24) det ⁡ I + D r = k r 0 × W r ϕ r , 1 , … , ϕ r , 2 N 0 ,where (25)kry=22Nexp⁡i∑ν=12NΘr,ν∏ν=22N∏μ=1v-1γν-γμ.Proof. First, we remove the factor 2i-1eiΘr,ν in each row ν in the Wronskian Wr(w) for 1≤ν≤2N.Then(26)Wr=∏ν=12NeiΘr,ν2i-N2-N×W~r,with (27)W~r=1-e-2iΘr,1iγ11+e-2iΘr,1…iγ12N-11+-12Ne-2iΘr,11-e-2iΘr,2iγ21+e-2iΘr,2…iγ22N-11+-12Ne-2iΘr,2⋮⋮⋮⋮1-e-2iθr,2Niγ2N1+e-2iΘr,2N…iγ2N2N-11+-12ne-2iΘr,2N The determinant W~r can be written as (28)W~r=det⁡αjkej+βjk, where αjk=-1kiγjk-1, ej=e-2iΘr,j, and βjk=iγjk-1,1≤j≤N,1≤k≤2N;α j k = - 1 k - 1 i γ j k - 1, ej=e-2iΘr,j, and βjk=(iγj)k-1,N+1≤j≤2N,1≤k≤2N.We have to calculateW~r. So, we use the following lemma.Lemma 3.
LetA=(aij)i,j∈[1,…,N], let B=(bij)i,j∈[1,…,N], and let (Hij)i,j∈[1,…,N] be the matrix formed by replacing in A thejth row of A by theith row of B. Then(29)det⁡aijxi+bij=det⁡aij×det⁡δijxi+det⁡Hijdet⁡aijProof.
LetA~=(a~ji)i,j∈[1,…,N] be the transposed matrix in cofactors of A. Then A×A~=det⁡A×I.
Sodet⁡(A~)=det⁡AN-1.
Then the general term of the product(cij)i,j∈[1,..,N]=(aijxi+bij)i,j∈[1,..,N]×(a~ji)i,j∈[1,..,N] can be expressed by (30)cij=∑s=1Naisxi+bis×a~js=xi∑s=1naisa~js+∑s=1nbisa~js=δijdet⁡Axi+det⁡Hij.We obtain (31)det⁡cij=det⁡aijxi+bij×det⁡AN-1=det⁡AN×det⁡δijxi+det⁡Hijdet⁡A.So det⁡(aijxi+bij)=det⁡(A)×det⁡(δijxi+det⁡(Hij)/det⁡(A)).We use the notationsU=(αij)i,j∈[1,…,2N] and V=(βij)i,j∈[1,…,2N].Using the preceding lemma, we get(32)W~r=det⁡αijei+βij=det⁡αij×det⁡δijei+det⁡Hijdet⁡αij=det⁡U×det⁡δijei+det⁡Hijdet⁡U,where Hiji,j∈[1,…,N] is the matrix obtained by replacing in U thejth row of U by theith row of V defined previously.U is the classical Vandermonde determinant that is equal to(33)det⁡U=iN2N-1∏2N≥l>m≥1γl-γm.We have to compute det⁡(Hij) to evaluate the determinant W~r. To do that, we study two cases.(1) For1≤j≤N, the matrix Hij is a Vandermonde matrix, where the jth row of U in U is replaced by the ith row of V. Then we have(34)det⁡Hij=-1N2N+1+N-1iN2N-1×M,with M=M(m1,…,m2N) being the determinant defined by mk=γk for k≠j and mj=-γi. Thus we get(35)det⁡Hij=-iN2N-1×∏2N≥l>k≥1,ml-mk=-iN2N-1×∏2N≥l>m≥1,l≠j,m≠jγl-γm×∏l<j-γi-γl×∏l>jγl+γi,=-1jiN2N-1×∏2N≥l>m≥1,l≠j,m≠jγl-γm×∏l≠jγl+γi. To compute W~r, we have to simplify the quotient qij≔det⁡(Hij)/det⁡(U):(36)qij=-1jiN2N-1×∏2N≥l>m≥1,l≠j,m≠jγl-γm×∏l≠jγl+γiiN2N-1∏2N≥l>m≥1γl-γm=-1j∏l≠jγl+γi∏l<jγj-γl∏l>jγl-γj=-1j∏l≠jγl+γi-1j-1∏l≠jγl-γj=-∏l≠jγl+γi∏l≠jγl-γj.qij is equal to rij defined by -∏l≠j(γl+γi)/∏l≠i(γl-γi), because det⁡(δijxi+det⁡(qij)/det⁡(A))=det⁡(δijxi+det⁡(rij)/det⁡(A)).Thusrij can be written as(37)rij=-1ϵi∏l≠jγl+γiγl-γj=cije-2iΘr,i0, with the notations given in (17).(2) We can do the same estimations forN+1≤j≤2N are made; det⁡Hij is first as follows:(38)det⁡Hij=-1N2N+1+N-1iN2N-1×M,with M=M(m1,…,m2N) being the determinant defined by mk=γk for k≠j and mj=-γi. Then we get(39)det⁡Hij=iN2N-1×∏2N≥l>k≥1,ml-mk=iN2N-1×∏2N≥l>m≥1,l≠j,m≠jγl-γm×∏l<j-γi-γl×∏l>jγl+γi,=-1j-1iN2N-1×∏2N≥l>m≥1,l≠j,m≠jγl-γm×∏l≠jγl+γi. Then qij≔det⁡(Hij)/det⁡(U) can be expressed as(40)qij=-1j-1iN2N-1×∏2N≥l>m≥1,l≠j,m≠jγl-γm×∏l≠jγl+γiiN2N-1∏2N≥l>m≥1γl-γm=-1j-1∏l≠jγl+γi∏l<jγj-γl∏l>jγl-γj=-1j-1∏l≠jγl+γi-1j-1∏l≠jγl-γj=∏l≠jγl+γi∏l≠jγl-γj.qij is replaced by rij defined by ∏l≠j(γl+γi)/∏l≠i(γl-γi), for the same reason as previously exposed.Thenrij can be written as(41)rij=-1ϵi∏l≠jγl+γiγl-γi=-1ϵi∏l≠jγl+γi∏l≠iγl-γi=cije-2iΘr,i0, with notations given in (17).e i is replaced by e-2iΘr,i. Then det⁡W~r can be rewritten as(42)det⁡W~r=det⁡U×det⁡δijei+det⁡Hijdet⁡U=det⁡U×det⁡δijei+rij=det⁡U∏i=12Ne-2iΘidet⁡δij+-1ϵi∏l≠iγl+γiγl-γie2iΘr,i. We compute the two members of the last relation (42) in y=0. Using (33), we get(43)det⁡W~r0=iN2N-1∏2N≥l>m≥1γl-γm∏i=12Ne-2iΘr,i0×det⁡δij+-1ϵi∏l≠iγl+γiγl-γie2iΘr,i0=iN2N-1∏j=22N∏i=1j-1γj-γie-2i∑i=12NΘr,i0det⁡δij+cij=iN2N-1∏j=22N∏i=1j-1γj-γie-2i∑i=12NΘr,i0det⁡I+Cr=iN2N-1∏j=22N∏i=1j-1γj-γie-2i∑i=12NΘr,i0det⁡I+Dr. Thus, the Wronskian Wr given by (26) can be rewritten as(44)Wrϕr,1,…,ϕr,2N0=∏j=12NeiΘr,j02-2Ni-N×W~r=∏j=12NeiΘr,j02-2Ni-NiN2N-1∏j=22N∏i=1j-1γj-γie-2i∑i=12NΘr,i0det⁡I+Dr=2-2N∏j=22N∏i=1j-1γj-γie-i∑i=12NΘr,i0det⁡I+Dr. Then(45)det⁡I+Dr=kr0Wrϕ1,…,ϕ2N0.This finishes the proof of Theorem 2.Then the solutionv to the Johnson equation can be rewritten as(46)vx,y,t=-2det⁡I+D3x,y,t2det⁡I+D1x,y,t2. With (24), the following link between Fredholm determinants and Wronskians is obtained: (47)det⁡I+D3=k30×W3ϕr,1,…,ϕr,2N0 and (48)det⁡I+D1=k10×W1ϕr,1,…,ϕr,2N0. As Θ3,j(0) contains N terms x3,j1≤j≤N and N terms -x3,j1≤j≤N, we have the relation k3(0)=k1(0), and we get the following theorem.Theorem 4.
The functionv defined by (49)vx,y,t=-2W3ϕ3,1,…,ϕ3,2N02W1ϕ1,1,…,ϕ1,2N02 is a solution of the Johnson equation which depends on 2N-1 real parameters ak, bk, and ϵ, with ϕνr defined in (21).(50)ϕr,ν=sin⁡-iκνx2+i-κνy24+δνyt-ixr,ν2+2τνt+γνw-ieν2,1≤ν≤N,ϕr,ν=cos⁡-iκνx2+i-κνy24+δνyt-ixr,ν2+2τνt+γνw-ieν2,N+1≤ν≤2N,r=1,3, where κν, δν, xr,ν, γν, and eν are defined in (6), (5), and (7).


## 4. Study of the Limit Case Whenϵ Tends to 0

### 4.1. Rational Solutions of OrderN Depending on 2N-2 Parameters

An infinite hierarchy of rational solutions to the Johnson equation depending on2N-2 parameters is obtained. For this, we take the limit when the parameter ϵ tends to 0.We get the following statement.Theorem 5.
The functionv(51)vx,y,t=limϵ→0⁡-2W3x,y,t2W1x,y,t2is a rational solution to the Johnson equation. It is a quotient of two polynomials n(x,y,t) and d(x,y,t) depending on 2N-2 real parameters a~j and b~j,1≤j≤N-1, of degrees 2N(N+1) in x, t and 4N(N+1) in y.


### 4.2. Families of Rational Solutions of Order 3 Depending on 4 Parameters

Here we construct families of rational solutions to the Johnson equation of order 3 explicitly; they depend on 4 parameters.We only give the expression without parameters and we give it in the appendix because of the length of the solutions.We construct the patterns of the modulus of the solutions in the plane(x,y) of coordinates in functions of parameters ai, b,1≤i≤2, and time t.The role of the parametersai and bi for the same integer i is the same one; one will be interested primarily only in parameters ai.The study of these configurations makes it possible to give the following conclusions. The variation of the configuration of the module of the solutions is very fast according to timet. When time t grows from 0 to 0,01, one passes from a rectilinear structure with a height of 98 to a horseshoe structure with a maximum height equal to 4. The role played by the parameters ai and bi is the same for same index i. When variables x, y, and time tend towards infinity, the modulus of the solutions tends towards 2 in accordance with the structure of the polynomials which will be studied in a forthcoming article.


## 4.1. Rational Solutions of OrderN Depending on 2N-2 Parameters

An infinite hierarchy of rational solutions to the Johnson equation depending on2N-2 parameters is obtained. For this, we take the limit when the parameter ϵ tends to 0.We get the following statement.Theorem 5.
The functionv(51)vx,y,t=limϵ→0⁡-2W3x,y,t2W1x,y,t2is a rational solution to the Johnson equation. It is a quotient of two polynomials n(x,y,t) and d(x,y,t) depending on 2N-2 real parameters a~j and b~j,1≤j≤N-1, of degrees 2N(N+1) in x, t and 4N(N+1) in y.


## 4.2. Families of Rational Solutions of Order 3 Depending on 4 Parameters

Here we construct families of rational solutions to the Johnson equation of order 3 explicitly; they depend on 4 parameters.We only give the expression without parameters and we give it in the appendix because of the length of the solutions.We construct the patterns of the modulus of the solutions in the plane(x,y) of coordinates in functions of parameters ai, b,1≤i≤2, and time t.The role of the parametersai and bi for the same integer i is the same one; one will be interested primarily only in parameters ai.The study of these configurations makes it possible to give the following conclusions. The variation of the configuration of the module of the solutions is very fast according to timet. When time t grows from 0 to 0,01, one passes from a rectilinear structure with a height of 98 to a horseshoe structure with a maximum height equal to 4. The role played by the parameters ai and bi is the same for same index i. When variables x, y, and time tend towards infinity, the modulus of the solutions tends towards 2 in accordance with the structure of the polynomials which will be studied in a forthcoming article.


## 5. Conclusion

We have constructed solutions to the Johnson equation, starting from the solutions of the KPI equation, what makes it possible to obtain rational solutions. These solutions are expressed by means of quotients of two polynomials of degree2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters.Here we have given a new method to construct solutions to the Johnson equation related to previous results [12–14].We have given two types of representations of the solutions to the Johnson equation. An expression by means of Fredholm determinants of order2N depending on 2N-1 real parameters is given. Another expression by means of Wronskians of order 2N depending on 2N-1 real parameters is also constructed. Also rational solutions to the Johnson equation depending on 2N-2 real parameters are obtained when one of parameters (ϵ) tends to zero.The patterns of the modulus of the solutions in the plane(x,y) and their evolution according to time and parameters have been studied in Figures 1, 2, 3, 4, and 5.Figure 1
Solution of order 3 to (1), on the left for t=0, a1=0, b1=0, a2=0, and b2=0; in the center for t=0, a1=103, b1=0, a2=0, and b2=0; and on the right for t=0, a1=0, b1=103, a2=0, and b2=0.Figure 2
Solution of order 3 to (1), on the left for t=0, a1=103, b1=0, a2=0, and b2=0; in the center for t=0, a1=0, b1=0, a2=0, and b2=106; and on the right for t=0,01, a1=0, b1=103, a2=0, and b2=0.Figure 3
Solution of order 3 to (1), on the left for t=0,01, a1=0, b1=0, a2=106, and b2=0; in the center for t=0,1, a1=103, b1=0, a2=0, and b2=0; and on the right for t=1, a1=103, b1=0, a2=0, and b2=0.Figure 4
Solution of order 3 to (1), on the left for t=0,1, a1=0, b1=0, a2=106, and b2=0; in the center for t=1, a1=0, b1=0, a2=106, and b2=0; and on the right for t=10, a1=103, b1=0, a2=0, and b2=0.Figure 5
Solution of order 3 to (1), on the left for t=10, a1=0, b1=0, a2=106, and b2=0; in the center for t=100, a1=106, b1=0, a2=0, and b2=0; and on the right for t=103, a1=105, b1=103, a2=0, and b2=0.In another study, we will give a more general representation of rational solutions to the Johnson equation. It can be written without limit at orderN depending on 2N-2 real parameters. We will prove that these solutions can be written as a quotient of polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y.



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