Exact periodic steady solutions for nonlinear wave equations: A new approach (original) (raw)
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Acta Mechanica et Automatica
This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.
1986
We present a systematic and formal approach toward finding solitary wave solutions of non-linear 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 non-linearities in the system. In the present paper the emphasis is, however, on the mathematical aspects, i.e. the formal procedure necessary to find single solitary wave solutions. By means of examples we show 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. We also outline the extension of our technique to construct N-soliton solutions and indicate connections with other existing methods.