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.
AKNS Formalism and Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients
