Soliton solutions and long-time asymptotics for variable-coefficient Mel’nikov system

Abstract We consider a (2+1)-dimensional variable-coefficient version of the Mel’nikov system and analyze soliton dynamics under time-dependent parameter modulation. Variable coefficients are of interest in physical settings involving nonuniform media or managed dispersion/nonlinearity, where the evolution of coherent structures can differ markedly from that in constant-parameter models. Within Hirota’s bilinear formalism we derive the bilinear equations and obtain the associated coefficient requirement that ensures the bilinearization (including γ 1 ( t ) = γ 2 ( t )). This enables us to construct explicit one and two-soliton solutions in closed form by means of perturbation expansions. Using these expressions, we illustrate typical spatiotemporal behaviors for representative coefficient choices and show how the coefficients tune soliton trajectories and amplitude evolution. Moreover, we derive long-time asymptotic expressions of the two-soliton solution as t → ± ∞ , which clearly separate the incoming and outgoing states and describe the interaction outcome analytically. Our results provide a concise framework for exploring managed soliton interactions in variable environments.