New integrable Vakhnenko–Parkes (VP) equations with time-dependent coefficients: Multiple real and multiple complex soliton solution

Abstract

Purpose

The purpose of this paper is concerned with developing new integrable Vakhnenko–Parkes equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for the time-dependent equations.

Design/methodology/approach

The developed time-dependent models have been handled by using the Hirota’s direct method. The author also uses Hirota’s complex criteria for deriving multiple complex soliton solutions.

Findings

The developed integrable models exhibit complete integrability for any analytic time-dependent coefficient.

Research limitations/implications

The paper presents an efficient algorithm for handling time-dependent integrable equations with time-dependent coefficients.

Practical implications

The author develops two Vakhnenko–Parkes equations with time-dependent coefficients. These models represent more specific data than the related equations with constant coefficients. The author showed that integrable equations with time-dependent coefficients give real and complex soliton solutions.

Social implications

The work presents useful techniques for finding integrable equations with time-dependent coefficients.

Originality/value

The paper gives new integrable Vakhnenko–Parkes equations, which give a variety of multiple real and complex soliton solutions.

Citation

Wazwaz, A.-M. (2019), "New integrable Vakhnenko–Parkes (VP) equations with time-dependent coefficients: Multiple real and multiple complex soliton solution", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 12, pp. 4598-4606. https://doi.org/10.1108/HFF-04-2019-0358