Soliton collision, modulation instability and long-time asymptotic for non-autonomous solutions in the variable-coefficient coupled Schrödinger-KdV equations

This paper extends the coupled Schrödinger-KdV equations by incorporating variable time-dependent coefficients. The Hirota bilinear method is employed to systematically derive the bilinear form of the variable-coefficient coupled Schrödinger-KdV equations. Explicit expressions for non-autonomous one- and two-soliton solutions are obtained with time-dependent variations in amplitude, width, and velocity. The principal advantage of the variable-coefficient model lies in its ability to achieve precise theoretical control over soliton generation, transmission, shaping, and interactions through appropriate design of the time-evolution of coefficient functions such as a(t) and b(t). Then, the asymptotic behaviors of the two-soliton solutions are analyzed in two distinct cases. Moreover, modulational instability is also investigated, demonstrating how small perturbations can grow under variable-coefficient conditions, which is crucial for evaluating soliton stability. This controllability indicates potential applications in optical communication systems.