Two integrable third-order and fifth-order KdV equations with time-dependent coefficients: Multiple real and multiple complex soliton solutions

Abstract

Purpose

The purpose of this paper is concerned with developing two integrable Korteweg de-Vries (KdV) equations of third- and fifth-orders; each possesses time-dependent coefficients. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for these two equations.

Design/methodology/approach

The integrability of each of the developed models has been confirmed by using the Painlev´e analysis. The author uses the complex forms of the simplified Hirota’s method to obtain two fundamentally different sets of solutions, multiple real and multiple complex soliton solutions for each model.

Findings

The time-dependent KdV equations feature interesting results in propagation of waves and fluid flow.

Research limitations/implications

The paper presents a new efficient algorithm for constructing time-dependent integrable equations.

Practical implications

The author develops two time-dependent integrable KdV equations of third- and fifth-order. These models represent more specific data than the constant equations. The author showed that integrable equation gives real and complex soliton solutions.

Social implications

The work presents useful findings in the propagation of waves.

Originality/value

The paper presents a new efficient algorithm for constructing time-dependent integrable equations.

Citation

Wazwaz, A.-M. (2019), "Two integrable third-order and fifth-order KdV equations with time-dependent coefficients: Multiple real and multiple complex soliton solutions", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 6, pp. 2093-2102. https://doi.org/10.1108/HFF-01-2019-0041