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

We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N 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.


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 (, ) 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: where as usual subscripts , , and  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: 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 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: In this paper, we give solutions by means of Fredholm determinants of order 2 depending on 2 − 1 parameters 2 Advances in Mathematical Physics and then by means of Wronskians of order 2 with 2 − 1 parameters.So we construct an infinite hierarchy of solutions to the Johnson equation, depending on 2 − 1 real parameters.
New rational solutions depending a priori on 2 − 2 parameters at order  are constructed, when one parameter tends to 0.
We obtain families depending on 2 − 2 parameters for the th order as a ratio of two polynomials of degree 2(+ 1) in ,  and of degree 4( + 1) in .
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 (, ) according to the four real parameters   and   for 1 ≤  ≤ 2 and time .

Solutions to Johnson Equation Expressed by Means of Fredholm Determinants
Some notations are given.We define first real numbers   such that −1 <  ] < 1, ] = 1, . . ., 2; they depend on a parameter  and can be written as Then, we define  ] ,  ] ,  ] , and  ,] ; they are functions of  ] , 1 ≤ ] ≤ 2, and are defined by the following formulas: ] 1 ≤ ] ≤ 2 are defined by ] , 1 ≤ ] ≤ 2 are defined by As usual  is the unit matrix and   = (  ) 1≤,≤2 is the matrix defined by the following: Then we get the following theorem.
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 solution  to the KPI equation (2) gives a solution to the Johnson equation (1).Let us consider (, , ) a solution of the KPI equation ( 2), then the function for is a solution to the KPI equation (2).Using this crucial transformation, the solution to the Johnson equation takes the form with the matrix   defined in (17).So we get the solutions to ( 14) by means of Fredholm determinants.

Solutions to the Johnson Equation by Means of Wronskians
We use the following notations: with () is the Wronskian of the functions  ,1 , . . .,  ,2 defined by We consider the matrix Then we have the following result.

det (𝐼 + 𝐷
where Proof.First, we remove the factor (2 with
Using the preceding lemma, we get where (  ) ,∈[1,...,] is the matrix obtained by replacing in  the jth row of  by the ith row of  defined previously. is the classical Vandermonde determinant that is equal to We have to compute det(  ) to evaluate the determinant W .
To do that, we study two cases.
(1) For 1 ≤  ≤ , the matrix   is a Vandermonde matrix, where the th row of  in  is replaced by the th row of .Then we have det with  = ( 1 , . . .,  2 ) being the determinant defined by   =   for  ̸ =  and   = −  .Thus we get To compute W , we have to simplify the quotient   fl det(  )/ det(): Thus   can be written as with the notations given in (17).
Then   can be written as with notations given in (17).  is replaced by  −2Θ , .Then det W can be rewritten as We compute the two members of the last relation (42) in  = 0. Using (33), we get Thus, the Wronskian   given by (26) can be rewritten as Then This finishes the proof of Theorem 2.
Then the solution V to the Johnson equation can be rewritten as where  ] ,  ] ,  ,] ,  ] , and  ] are defined in ( 6), (5), and (7).For this, we take the limit when the parameter  tends to 0. We get the following statement.

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 (, ) of coordinates in functions of parameters   , , 1 ≤  ≤ 2, and time .
The role of the parameters  and   for the same integer  is the same one; one will be interested primarily only in parameters   .
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 time .When time  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   and   is the same for same index .When variables , , 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.

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 degree 2( + 1) in ,  and 4( + 1) in  depending on 2 − 2 parameters.
Here we have given a new method to construct solutions to the Johnson equation related to previous results [12][13][14].
We have given two types of representations of the solutions to the Johnson equation.An expression by means of Fredholm determinants of order 2 depending on 2 − 1 real parameters is given.Another expression by means of Wronskians of order 2 depending on 2−1 real parameters is also constructed.Also rational solutions to the Johnson equation depending on 2 − 2 real parameters are obtained when one of parameters () tends to zero.
The patterns of the modulus of the solutions in the plane (, ) and their evolution according to time and parameters have been studied in Figures 1, 2, 3, 4, and 5.
In another study, we will give a more general representation of rational solutions to the Johnson equation.It can be written without limit at order  depending on 2 − 2 real parameters.We will prove that these solutions can be written as a quotient of polynomials of degree 2( + 1) in ,  and 4( + 1) in .

𝜖 Tends to 0 4 . 1 .
Rational Solutions of Order  Depending on 2 − 2 Parameters.An infinite hierarchy of rational solutions to the Johnson equation depending on 2 − 2 parameters is obtained.