Two integrable third-order and fifth-order KdV equations with time-dependent coefficients

Abstract

PurposeThe 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/approachThe 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.FindingsThe time-dependent KdV equations feature interesting results in propagation of waves and fluid flow.Research limitations/implicationsThe paper presents a new efficient algorithm for constructing time-dependent integrable equations.Practical implicationsThe 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 implicationsThe work presents useful findings in the propagation of waves.Originality/valueThe paper presents a new efficient algorithm for constructing time-dependent integrable equations.