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

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. [ABSTRACT FROM AUTHOR]