Advances in Mathematical Physics
Volume 2018 (2018), Article ID 1642139, 18 pages
https://doi.org/10.1155/2018/1642139

Research Article

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

Pierre Gaillard

Université de Bourgogne, Institut de Mathématiques de Bourgogne, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France

Correspondence should be addressed to Pierre Gaillard; pierre.gaillard@u-bourgogne.fr

Received 10 November 2017; Accepted 8 May 2018; Published 1 August 2018

Academic Editor: Giampaolo Cristadoro

Copyright © 2018 Pierre Gaillard. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order giving solutions of order depending on parameters. We obtain order rational solutions that can be written as a quotient of two polynomials of degree in , and in depending on parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order with parameters are constructed and studied in detail by means of their modulus in the plane in function of time and parameters , , , and .

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 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 depending on parameters and then by means of Wronskians of order with parameters. So we construct an infinite hierarchy of solutions to the Johnson equation, depending on real parameters.

New rational solutions depending a priori on parameters at order are constructed, when one parameter tends to .

We obtain families depending on parameters for the th order as a ratio of two polynomials of degree in , and of degree 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 and time .

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

Some notations are given. We define first real numbers such that , ; they depend on a parameter and can be written as

Then, we define , and ; they are functions of , and are defined by the following formulas:

are defined by

, are defined by

As usual is the unit matrix and is the matrix defined by the following:

Then we get the following theorem.

Theorem 1. The function defined by

with

and is the matrix

is a solution to (1), depending on parameters and .

Proof. The solution to the KPI equation can be written as follows by using [12]:

where

and is the matrix

where , , , , , , and 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 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.

3. Solutions to the Johnson Equation by Means of Wronskians

We use the following notations:

with

is the Wronskian of the functions defined by

We consider the matrix defined in (17).

Then we have the following result.

Theorem 2.

where

Proof. First, we remove the factor in each row in the Wronskian for .

Then

with

The determinant can be written as

where , , and ;

, , and .

We have to calculate . So, we use the following lemma.

Lemma 3. Let , let , and let be the matrix formed by replacing in the jth row of by the ith row of . Then

Proof. Let be the transposed matrix in cofactors of . Then .
So .
Then the general term of the product can be expressed by

We obtain

So .

We use the notations and .

Using the preceding lemma, we get

where 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 to evaluate the determinant . To do that, we study two cases.

(1) For , the matrix is a Vandermonde matrix, where the th row of in is replaced by the th row of . Then we have

with being the determinant defined by for and . Thus we get

To compute , we have to simplify the quotient :

is equal to defined by , because .

Thus can be written as

with the notations given in (17).

(2) We can do the same estimations for are made; is first as follows:

with being the determinant defined by for and . Then we get

Then can be expressed as

is replaced by defined by , for the same reason as previously exposed.

Then can be written as

with notations given in (17).

is replaced by . Then can be rewritten as

We compute the two members of the last relation (42) in . 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 to the Johnson equation can be rewritten as

With (24), the following link between Fredholm determinants and Wronskians is obtained:

and

As contains terms and terms , we have the relation , and we get the following theorem.

Theorem 4. The function defined by

is a solution of the Johnson equation which depends on real parameters , , and , with defined in (21).

where , , , , and are defined in (6), (5), and (7).

4. Study of the Limit Case When Tends to

4.1. Rational Solutions of Order Depending on Parameters

An infinite hierarchy of rational solutions to the Johnson equation depending on parameters is obtained. For this, we take the limit when the parameter tends to .

We get the following statement.

Theorem 5. The function

is a rational solution to the Johnson equation. It is a quotient of two polynomials and depending on real parameters and , of degrees in , and in .

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 of coordinates in functions of parameters , , 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 , 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.

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 degree in , and in depending on 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 order depending on real parameters is given. Another expression by means of Wronskians of order depending on real parameters is also constructed. Also rational solutions to the Johnson equation depending on 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.

Figure 1: Solution of order 3 to (1), on the left for , , , , and ; in the center for , , , , and ; and on the right for , , , , and .

Figure 2: Solution of order 3 to (1), on the left for , , , , and ; in the center for , , , , and ; and on the right for , , , , and .

Figure 3: Solution of order 3 to (1), on the left for , , , , and ; in the center for , , , , and ; and on the right for , , , , and .

Figure 4: Solution of order 3 to (1), on the left for , , , , and ; in the center for , , , , and ; and on the right for , , , , and .

Figure 5: Solution of order 3 to (1), on the left for , , , , and ; in the center for , , , , and ; and on the right for , , , , and .

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 real parameters. We will prove that these solutions can be written as a quotient of polynomials of degree in , and in .

Appendix

The solutions to the Johnson equation can be written as

with

Conflicts of Interest

The author declares that there are no conflicts of interest.

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