The authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations. The physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system. The emphasis is, however, on the mathematical aspects, i.e. the formal procedure necessary to find single solitary wave solutions. By means of examples it is shown how various cases of pulse-type and kink-type solutions are to be obtained by this method. An exhaustive list of equations so treated is presented in tabular form, together with the particular intermediate relations necessary for deriving their solutions. The extension of the technique to construct N-soliton solutions and indicate connections with other existing methods is outlined.

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Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method

Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method

The propagation of longitudinal deformation waves in an elastic rod is modelled by the nonlinear partial differential equation 






with p = 3 or 5. This equation is first derived under a range of possible constraints. We then show that this equation and even certain generalizations do not pass the Painleve test, and hence are probably not completely integrable. Finally, we study the head-on collision of two equal solitary waves numerically and also asymptotically for small and large amplitude.

Solitary‐Wave Interactions in Elastic Rods

Galerkin method for the numerical solution of the RLW equation using quintic B-splines

Approximation of the RLW equation by the least square cubic B-spline finite element method

The combined approach of linearisation and finite difference method is used to solve the improved Boussinesq equation. A three-level iterative scheme having second order accuracy and constant coefficients matrix is devised and used in discussing the dynamics of waves having various initial wave packets. The results are in good agreement with the available results.

Padam C. Jain

Papers

Numerical solutions of the improved Boussinesq equation

Numerical solutions of the regularized long-wave equation