Solitons for the modified (2 + 1)-dimensional Konopelchenko\u2013Dubrovsky equations

Xiumei Lyu,Wei Gu

doi:10.1186/s13662-019-2370-3

Abstract

In the paper, we consider the modified (2 + 1)-dimensional Konopelchenko–Dubrovsky equations which possess high order nonlinear terms. Under the aid of Maple, we derive the exact traveling wave solutions of the mKDs by the auxiliary equation approach. Under some special conditions, Jacobi elliptic function solutions, degenerated triangular function solutions, and solitons for the mKD equations are constructed.

Highlights

IntroductionKonopelchenko and Dubrovsky [1] ever presented a (2 + 1)-dimensional Konopelchenko and Dubrovsky (KD) model as follows:
Konopelchenko and Dubrovsky [1] ever presented a (2 + 1)-dimensional Konopelchenko and Dubrovsky (KD) model as follows: ⎧ ⎨ut uxxx 6b0uux + a20u2ux 3vy 3a0uxv = ⎩uy = vx, (1)
Based on the similarity transformations approach together with Lie group theory, the KD system has been reshaped into another system of ordinary differential equations, new closed form solutions of Eq (1) have been obtained [21]

Summary

Introduction

Konopelchenko and Dubrovsky [1] ever presented a (2 + 1)-dimensional Konopelchenko and Dubrovsky (KD) model as follows:. Based on the similarity transformations approach together with Lie group theory, the KD system has been reshaped into another system of ordinary differential equations, new closed form solutions of Eq (1) have been obtained [21]. The tanh-sech method, coshsinh method, and the exponential method were employed to discuss Eq (1) by Wazwaz [22], and some exact solutions of distinct physical structures, solitons, kinks, and periodic wave solutions were derived Motivated by these works [1, 2, 19,20,21,22], the presented paper will study the following modified Konopelchenko and Dubrovsky (mKD) equations:. The paper will make use of the auxiliary differential equation approach to solve Eq (2) analytically and obtains abundant new exact solutions of Eq (2) in general and special cases.

WW W

When the modulus r

Conclusion