Exact kink and two-front solutions of an extended Benney-Luke equation via logarithmic transformations

In this paper, we investigated an extended Benney-Luke equation with second-order linear modulation, which generalizes the classical inviscid capillary-gravity shallow water wave model by introducing an additional linear second-order spatial derivative term. We first constructed the Hirota-type logarithmic transformation and derived the reduced fourth-order traveling-wave ODE, then obtained explicit single kink/antikink front solutions under the existence constraint $\frac{\alpha}{\lambda} < 0$. The method of undetermined coefficients was adopted to cross-validate the front solutions and construct four symmetric sign variants of the equal-amplitude two-front solution. Algebraic derivation showed that the interaction coefficient uniformly equals $A = 1$ for all cases, which corresponds to transparent superposition of independent wave fronts rather than nonlinear elastic soliton collision. Representative dimensionless test parameters are adopted for qualitative graphical illustration only: 3D spatiotemporal surfaces, 2D temporal sectional curves, and density contours were plotted for the velocity potential $ u $, while supplementary figures visualized the spatially localized pulse of its spatial derivative $u_{x}$. In this work, we systematically constructed closed-form kink and two-front wavefront solutions for the linearly wave-speed modulated generalized Benney-Luke equation, filling the research gap where the literature reported only simple traveling-wave profiles.