Motivated by the fact that the generalized Benney-Luke equation is a formally valid approximation for describing two-way water wave propagation, the authors study the linear instability of its solitary wave solutions.
The method employed starts with the use of Fourier transforms which provide a description of solutions in terms of a Fourier multiplier. It is noted that, in small amplitude, the one-dimensional Benney-Like equation is asymptotically related to the generalized KdV equation. An operator generalization of Rouché’s theorem due to Gohberg and Sigal is used to establish the existence of an unstable eigenvalue. This method hence avoids the use of Evans functions.
The last part of the paper validates numerically the above results for the generalized Benney-Luke equation through a finite difference numerical scheme which combines an explicit predictor and an implicit corrector step to compute solutions.