Hirota bilinear equations with linear subspaces of hyperbolic and trigonometric function solutions

doi:10.1016/j.amc.2013.06.019

Applied Mathematics and Computation

Volume 220, 1 September 2013, Pages 226-234

https://doi.org/10.1016/j.amc.2013.06.019 Get rights and content

Abstract

Linear superposition principles of hyperbolic and trigonometric function solutions are analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of N-soliton solutions formulated by linear combinations of hyperbolic and trigonometric functions. An algorithm using weights is discussed and a few illustrative application examples are presented.

Introduction

It is significantly important in mathematical physics to search for exact solutions to nonlinear differential equations. Exact solutions play a vital role in understanding various qualitative and quantitative features of nonlinear phenomena. There are diverse classes of interesting exact solutions, such as traveling wave solutions and soliton solutions, but it often needs specific mathematical techniques to construct exact solutions due to the nonlinearity present in dynamics [1], [2]. Typical integrable equations, such as the KdV equation, the Boussinesq equation and the KP equation, possess multi-soliton solutions, generated from combinations of multiple exponential waves on the basis of their Hirota bilinear forms [3]. Various equations of mathematical physics can be written as Hirota bilinear forms through dependent variable transformations [3], [4]. Wronskian solutions, including solitons, positons and complexitons [5], [6], [7], [8], and quasi-periodic solutions [9], [10], [11] can be presented systematically based on Hirota bilinear forms. Recently, Hirota bilinear operators are generalized and their applications are presented in [12].

Besides soliton solutions, another class of interesting multiple exponential wave solutions is linear combinations of exponential waves, which implies the existence of linear subspaces of solutions. Hirota bilinear equations which possess linear subspaces of exponential traveling wave solutions are discussed and it is shown that a kind of nonlinear equations can possess such a linear superposition principle under some conditions [13], [14].

In this paper, we would like to explore when Hirota bilinear equations possess linear subspaces of hyperbolic and trigonometric function solutions, aiming to construct a specific sub-class of N-soliton solutions formulated by linear combinations of hyperbolic and trigonometric functions.

Based on the Hirota bilinear formulation, we will present sufficient and necessary conditions, with an algorithm, to guarantee the applicability of the linear superposition principles to hyperbolic and trigonometric function solutions. A few illustrative examples will be computed.

Section snippets

Linear superposition principles

We begin with a Hirota bilinear equation $P (D_{x_{1}}, D_{x_{2}}, \dots, D_{x_{M}}) f \cdot f = 0,$ where P is a polynomial in the indicated variables, satisfying $P (0, 0, \dots, 0) = 0,$ and $D_{x_{i}}, 1 ⩽ i ⩽ M,$ are Hirota’s differential operators defined by $D_{y}^{p} f (y) g (y) = {(\partial_{y} - \partial_{y^{'}})}^{p} {f (y) g (y^{'})|}_{y^{'} = y} = \partial_{y^{'}}^{p} {f (y + y^{'}) g (y - y^{'})|}_{y^{'} = 0}, p ⩾ 1 .$

Note that a term of odd degree in P produces zero in the resulting Hirota bilinear equation, and so we assume that P is an even polynomial, i.e., $P (- x_{1}, - x_{2}, \dots, - x_{M}) = P (x_{1}, x_{2}, \dots, x_{M}) .$

Let us now introduce N wave variables: $η_{i} = k_{i} \cdot x = k_{1, i} x_{1} + k_{2, i} x_{2} + \dots k_{M, i} x$

Applications

We would like to propose an opposite procedure for conversely constructing Hirota bilinear equations that possess N-wave solutions formulated by linear combinations of hyperbolic or trigonometric functions. This is an opposite question on applying the linear superposition principles in Theorem 1, Theorem 2. The problem can be reduced to how to construct an even multivariate polynomial P(x₁, x₂, … , x_M) satisfying the system (2.10) or (2.15). Based on the idea in [12], [13], [14], an algorithm using

Conclusions

In this paper, we analyzed when Hirota bilinear equations possess the linear superposition principle of hyperbolic or trigonometric function solutions and discussed how to construct multivariable polynomials which generate such Hirota bilinear equations. An algorithm using weights and a few illustrative examples are given.

Future research questions [12], [13], [14] include how to achieve other parameterizations of wave numbers and frequencies by using several parameters and how to create

Acknowledgements

This work is supported by the National Nature Science Foundation of China (61070233).

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Hirota bilinear equations with linear subspaces of hyperbolic and trigonometric function solutions

Abstract

Introduction

Section snippets

Linear superposition principles

Applications

Conclusions

Acknowledgements

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Nonlinear Anal.

Comput. Math. Appl.

Appl. Math. Comput.

The Direct Method in Soliton Theory