The integrable Vakhnenko–Parkes (VP) and the modified Vakhnenko–Parkes (MVP) equations: Multiple real and complex soliton solutions

Abstract

In this work, we show that the integrable Vakhnenko–Parkes (VP) equation passes the Painlevé test and admits multiple real and multiple complex soliton solutions. We also present, for the first time, the modified Vakhnenko-Parkes (MVP) equation, show its complete integrability, and formally derive its multiple real and multiple complex soliton solutions. To achieve the goal set for this work, we introduce two complex forms of the simplified Hirota’s method, the first works effectively for the VP equation, and the other form is nicely applicable for the MVP equation. We believe that establishing the complex forms will shed light on complex solitons of other integrable equations.