1-Soliton solution of Kadomtsev–Petviashvili equation with power law nonlinearity

doi:10.1016/j.amc.2009.04.001

Applied Mathematics and Computation

Volume 214, Issue 2, 15 August 2009, Pages 645-647

https://doi.org/10.1016/j.amc.2009.04.001 Get rights and content

Abstract

This paper obtains the 1-soliton solution of the Kadomtsev–Petviasvili equation with power law nonlinearity using the solitary wave ansatz. An exact soliton solution is obtained and a couple of conserved quantities are also computed.

Introduction

Nonlinear evolution equations in 1 + 1 as well as 1 + 2 dimensions have been studied for the past few decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. A remarkable amount of progress has been made in the understanding of these equation, especially with respect to their integrability aspects. Many nonlinear evolution equations are now integrable by various newly found modern methods of integrability. These techniques include the Wadati trace method, Adomian decomposition method, F-expansion method, exponential function method and many more. These methods are also applicable to multi-dimensional nonlinear evolution equations as well as vector nonlinear evolution equations.

The Kadomtsev–Petviasvili (KP) equation, also known as the two-dimensional Korteweg-de Vries equation, is one such nonlinear evolution in 1 + 2 dimensions that is of interest, in this paper. This KP equation is studied in various areas of Applied Mathematics and Physics, in the context of dusty plasmas [8]. In Applied Mathematics, it appears in the study of wave propagation in two-dimensions. If a plane wave propagates in the x-direction, a question arises, when these plane waves cross at infinity, about the nonlinear interaction of these crossing waves. In that case, the y-dependences will not be trivial. When the wave propagates predominantly in the x-direction, with the appropriate balance of nonlinearity and dispersion, also in the x-direction, the relevant dependence on the y-variable is also included. This contribution appearing at the same order as nonlinearity and dispersion, leads to the formulation of the KP equation [9], [10].

Section snippets

Mathematical analysis

The dimensionless form of the KP equation, with power law nonlinearity, which is studied in this paper, is given by [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] ${(q_{t} + {aq}^{n} q_{x} + q_{xxx})}_{x} + {bq}_{yy} = 0 .$ Here in (1) a and b are real valued constants. The first term of (1) represents the evolution term, the coefficient of a is the nonlinear term with the power law dictated by the exponent n and the third term represents the dispersion in the x-direction. Finally, the coefficient of b is the y-dependence term and

Integrals of motion

The KP equation, with power law nonlinearity, permits at least two conserved quantities [1], [9], [10]. They are the mass (M) and linear momentum (P) that are respectively given by $M = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} q (x, y, t) dx dy = \frac{A}{B_{1} B_{2}} \frac{Γ (\frac{1}{n}) Γ (\frac{1}{2})}{Γ (\frac{1}{n} + \frac{1}{2})},$ $P = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} q^{2} (x, y, t) dx dy = \frac{A^{2}}{B_{1} B_{2}} \frac{Γ (\frac{2}{n}) Γ (\frac{1}{2})}{Γ (\frac{2}{n} + \frac{1}{2})},$ where in (14), (15), $Γ (x)$ represents Euler’s gamma function. In order to compute these conserved quantities, the 1-soliton solution, given by (13), is used.

Conclusions

This paper obtains the 1-soliton solution of the KP equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration of this equation. As noticed that by using this technique, it is not possible to obtain soliton radiation. In future, the fifth order KP equation [2] will be studied and also the perturbations terms of this equation will be considered and will be studied using the soliton perturbation theory. In addition to the deterministic perturbation terms,

Acknowledgements

The research work of the first author was fully supported by NSF-CREST Grant No: HRD-0630388 and Army Research Office (ARO) along with the Air Force Office of Scientific Research (AFOSR) under the award number: W54428-RT-ISP and these supports are genuinely and sincerely appreciated.

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