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

Purpose The purpose of this paper is concerned with developing new integrable Vakhnenko-Parkes equations with time-dependent coecients. 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 coecient. Research limitations/implications - The paper presents an efficient algorithm for handling timedependent 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 ?nding integrable equations with time-dependent coecients. Originality/value The paper gives new integrable Vakhnenko-Parkes equations, which give a variety of multiple real and complex soliton solutions.