Exact Periodic Wave Solutions to Some Nonlinear Evolution Equations
Exact Periodic Wave Solutions to Some Nonlinear Evolution Equations ~ Investigating the exact solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in uid dynamics, plasma and elastic media are often modeled by the bell-shaped sech solutions and kink-shaped tanh solutions. The exact solution, if available, of those nonlinear equations facilitates the verification of numerical solvers and aids in the stability analysis of solutions. Therefore, in recent years, there has been a growing interest in developing and applying a large variety of analytical methods. Among these methods we can cite.
In fact the Jacobian elliptic functions  degenerate into hyperbolic functions when the modulus ap- proaches 1, has attracted a lot of interest in the investigation of exact solutions. The three basic Jacobian elliptic functions sn(ξ,m), cn(ξ,m) and dn(ξ,m), where m is the modulus of the elliptic function, satisfy the well known type of trigonometric relations such as sn2(ξ) + cn2(ξ) = 1, dn2(ξ) + m2sn2(ξ) = 1, (sn(ξ)) = cn(ξ)dn(ξ), (cn(ξ)) = −sn(ξ)dn(ξ), (dn(ξ)) = −m2sn(ξ)cn(ξ). When m → 0, the Ja- cobi elliptic functions degenerate to the triangular functions, i. e. , sn(ξ) → sin(ξ), cn(ξ) → cos(ξ), dn(ξ) → 1 and when m → 1, the Jacobian elliptic functions degenerate to the hyperbolic functions i. e. , sn(ξ) → tanh(ξ), cn(ξ) → sech(ξ), dn(ξ) → sech(ξ).
A mapping method and its extensions have been successfully applied to derive a variety of Jacobian elliptic function solutions for nonlinear equations [25 − 27]. For a given nonlinear evolution equation, say, in three independent variables, N(u, ut,ux,uy, …)=0, (1) we seek its travelling wave solution of the form u(x, y, t) = u(ξ),ξ = kx + ly − wt, (2) where k, l and w are constants to be determined later. Substituting Eq.(2) into Eq.(1) yields an ordinary differential equation of u(ξ).
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