The Cauchy matrix approach for generating localized waves arising from non-isospectral integrable systems

Abstract This paper presents a unified and systematic framework for constructing explicit solutions to a class of non-isospectral integrable systems using the Cauchy matrix approach. Starting from a Sylvester equation that links the spectral data with the nonlinear evolution equations, we derive a set of dispersion relations and introduce a master function that covers the dynamics of the systems. Using this approach, we first derive the non-isospectral KdV (nKdV) equation and express its solution in a compact form related to the determinant of the dressed Cauchy matrix. The framework is then extended to the unreduced non-isospectral modified KdV (nmKdV) and the non-isospectral Hirota(nHE) systems and their corresponding reductions, by formulating the problem in terms of block matrices and suitable dispersion relations. Imposing specific constraints on the vectors r , s and the matrix K , the N-soliton solutions are obtained for both cases through two distinct formulations: one with a diagonal matrix yielding classical solitons, and one using a lower triangular Toeplitz matrix (LTTM), which also leads to multiple-pole soliton solutions. Further analysis of the solution dynamics are provided to reveal the influence of time-dependent media properties on amplitude, trajectory, and soliton interactions.