Bilinearization and Exact Soliton Dynamics of an Integrable Variable-Coefficient Korteweg–de Vries Model for Shallow-Water Waves and Its Boundary with Variable-Depth Shoaling

The variable-coefficient Korteweg–de Vries equation ut+f(t)uux+g(t)uxxx+h(t)ux+σ(t)u=0 is bilinearized by the Hirota method after gauge and Galilean reductions, and exact N-soliton solutions are shown to exist precisely under the integrability condition σ+ddtln(g/f)=0, equivalent to reducibility to the constant-coefficient equation. Closed-form laws are obtained for the soliton amplitude A=3k12g/f, width, velocity, and trajectory; the two-soliton collision is proved strictly elastic, with a coefficient-independent phase shift and no fusion or fission; the conservation laws are recast as exactly modulated invariants; and a constructive coefficient-programming design realizes prescribed-amplitude and shape-preserving solitons. The physically derived variable-depth equation satisfies the condition only at constant depth, so that a generic sloping bottom admits only the adiabatic single-soliton regime (A∝D−1), compared structurally with the classical shoaling laws. Every solution is verified by direct symbolic substitution.