The modified KdV equation with variable coefficients: Exact uni/bi-variable travelling wave-like solutions

In this paper, the modified Korteweg–de Vries (mKdV) equation with variable coefficients (vc-mKdV equation) is investigated via two kinds of approaches and symbolic computation. On the one hand, we firstly reduce the vc-mKdV equation to a second-order nonlinear nonhomogeneous ODE using travelling wave-like similarity transformation. And then we obtain its many types of exact fractional solutions with one travelling wave-like variable by applying some fractional transformations to the obtained nonlinear ODE. On the other hand, we reduce the vc-mKdV equation to two nonlinear PDEs with variable coefficients using the anti-tangent and anti-hypertangent function transformations, respectively. And then we given its many types of exact solutions with two different travelling wave-like variables by studying the obtained nonlinear PDE with variable coefficients.