Two new integrable modified KdV equations, of third-and fifth-order, with variable coefficients: multiple real and multiple complex soliton solutions

In this work, we develop two new integrable modified KdV equations, of third and fifth orders, with time-dependent coefficients, which can be used to describe many nonlinear phenomena in fluid dynamics and solitary waves theory. We systematically investigate the complete integrability of each equation by exhibiting the Painleve test. With the aid of new complex forms of the simplified Hirota's method, we show that each equation admits multiple real and multiple complex soliton solutions. The influence of the new terms with variable coefficients on solitonic structures and interaction properties are investigated. The two developed equations are likely to be of applicative relevance, because it may be considered an application of a large class of nonlinear equations with variable coefficients.