Nonisospectral equations from the Cauchy matrix approach

The Cauchy matrix approach is developed to construct explicit solutions for some nonisospectral equations, including the nonisospectral Korteweg–de Vries (KdV) equation, the nonisospectral modified KdV equation, and the nonisospectral sine-Gordon equation. By means of a Sylvester equation, a set of scalar master functions {S^(i,j)} is defined. We show how nonisospectral dispersion relations are introduced such that the evolutions of {S^(i,j)} can be derived. Some identities of {S^(i,j)} are employed in verifying solutions. Some explicit one-soliton and two-soliton solutions are illustrated together with analysis of their dynamics.