Symmetries and conservation laws of the Yao\u2013Zeng two-component short-pulse equation

Ben Gao,Yao Zhang

doi:10.1186/s13661-019-1156-6

Abstract

This paper uses the Lie group method to analyze the symmetries of the Yao–Zeng two-component short-pulse equation which describes the propagation of polarized ultrashort light pulses in cubically nonlinear anisotropic optical fibers. Similarity reductions and exact solutions are obtained by constructing an optimal system of one-dimensional subalgebras. Moreover, the explicit solutions are constructed by the power series method and the convergence of power series solutions is proved. In addition, nonlinear self-adjointness and conservation laws of this system are discussed.

Highlights

IntroductionThe short pulse equation has attracted much attention. It was introduced as a model equation to describe the propagation of ultrashort optical pules in silica optical fibers [1] which has the form 1 uxt = u + 6
In the last decades, the short pulse equation has attracted much attention
We show the convergence of the power series solution (23) of Eq (19)

Summary

Introduction

The short pulse equation has attracted much attention. It was introduced as a model equation to describe the propagation of ultrashort optical pules in silica optical fibers [1] which has the form 1 uxt = u + 6. The Lie group method [13,14,15] is considered to be one of the most effective methods to obtain exact solutions for lots of nonlinear partial differential equations(PDEs). For arbitrary chosen constant numbers p0 and q0, the other terms of the sequence {pn}∞ n=0 and {qn}∞ n=0 can be determined by (25) and (26) This implies that for Eq (19), there is a power series solution (23) with the coefficients constructed by (25) and (26). By the implicit function theorem [27], we see that R = R(z) and S = S(z) are analytic in a neighborhood of the point (0, r0, s0) and with the positive radius This implies that the two power series (23) converge in a neighborhood of the point (0, r0, s0). Remark 4.1 The power series solutions can greatly enrich the solutions of Eq (4) and converge quickly, so it is convenient for computations in both theory and applications

Preliminaries Consider the rth-order system of m PDEs

Construction of conservation laws

Conclusions