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New Two-soliton and Periodic Solutions to KdV Equation

There has been an active and exciting search for explicit solutions of nonlinear evolution equations(NLEES) ever since the discovery of the soliton in 1834.Many method have been proposed over these years for finding these solutions [1-10] including a number of asymptotic methods that have been proposed recently by J.He [11,12] and the double-exp function method [13] for finding the exact multi-wave solutions and double-wave solutions of NLPDEs. In this work, we show that the most elementary ansatz of double Exp-function method can be produced by an extension of two-soliton ansatz in a fractional form. Applying this ansatz to KdV equation, new abundant two-solitary solutions and periodic solution are obtained. We consider KdV equation [1]: