Multiple soliton solutions for the sixth-order Ramani equation and a coupled Ramani equation

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Abstract

In this work, the completely integrable sixth-order nonlinear Ramani equation and a coupled Ramani equation are studied. Multiple soliton solutions and multiple singular soliton solutions are formally derived for these two equations. The Hirota’s bilinear method is used to determine the two distinct structures of solutions. The resonance relations for the three cases are investigated.

Introduction

In [1], [2], [3], [4], [5], the Painlevé analysis and other methods were used to test the complete integrability of the sixth-order nonlinear Ramani equation, or the KdV6 equation, given byuxxxxxx+15uxuxxxx+15uxxuxxx+45ux2uxx-5(uxxxt+3uxuxt+3utuxx)-5utt=0.

In [2], it was indicated that there are four distinct cases for Eq. (1) to pass the Painlevé test.

The authors in [1], [2], [3], [4], [5] examined Eq. (1) for complete integrability by using the Bäcklund transformation and Lax pairs. In [2], the method of truncated singular expansion was used to derive Lax pair and Bäcklund self-transformation for Eq. (1). In [3], Lax pairs and Bäcklund transformations were used to handle (1).

However, a specific case of a coupled Ramani equation given byuxxxxxx+15uxuxxxx+15uxxuxxx+45ux2uxx-5(uxxxt+3uxuxt+3utuxx)-5utt+18wx=0,wt-wxxx-3wxux-3wuxx=0,was investigated in [3], [4] and by others.

For completely integrable evolution equations, three powerful methods namely, the inverse scattering method, the Bäcklund transformation method [1], [2], [3], [4], [5], [6], and the Hirota bilinear method [7], [8], [9], [10], [11], [12], [13] were thoroughly used to derive the multiple soliton solutions of these equations. The Hirota’s bilinear method is rather heuristic and possesses significant features that make it practical for the determination of multiple soliton solutions [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] for a wide class of nonlinear evolution equations in a direct method. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.

In this work, we aim to study the Ramani equation (1) and a generalized form of a coupled Ramani equation of the formuxxxxxx+15uxuxxxx+15uxxuxxx+45ux2uxx-5(uxxxt+3uxuxt+3utuxx)-5utt+bwx=0,wt-wxxx-3wxux-3wuxx=0.

We will test the last equation for complete integrability, and to show that the coupled Ramani equation given (2), (3) is just a specific case among many others where b will take other values other than 18. The Hirota’s bilinear method [7], [8], [9], [10], [11], [12], [13] will be used to derive multiple regular soliton solutions and multiple singular soliton solutions for the two models given by (1), (4), (5).

Section snippets

The Ramani equation

We will first study the sixth-order Ramani equationuxxxxxx+15uxuxxxx+15uxxuxxx+45ux2uxx-5(uxxxt+3uxuxt+3utuxx)-5utt=0.

Unlike the KdV equation, the Ramani equation (6) includes a second derivative with respect to time t. In this section we will apply the Hirota’s bilinear sense to determine multiple soliton solutions and multiple singular soliton solutions for this equation.

A coupled Ramani equation

In this section we will study a generalized form of a coupled Ramani equation given byuxxxxxx+15uxuxxxx+15uxxuxxx+45ux2uxx-5(uxxxt+3uxuxt+3utuxx)-5utt+bwx=0,wt-wxxx-3wxux-3wuxx=0.

Notice that the coefficient of wx is not fixed by 18 as in [3], [4]. To achieve our goal, we introduce an auxiliary variable z.

Discussion

The Ramani or the KdV6 equation was treated to confirm its integrability. Moreover, the integrability of a coupled Ramani equation was tested and also confirmed. The Hirota’s bilinear method was employed to achieve our goal for this work. We showed that the coupled Ramani equation is not unique, instead other forms were established provided that ab = 18. The resonance relation does not exist for the two problems.

References (30)

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