Abstract
Theexp(-Φ(ξ))-expansion method is improved by presenting a new auxiliary ordinary differential equation forΦ(ξ). By using this method, new exact traveling wave solutions of two important nonlinear evolution equations, i.e., the ill-posed Boussinesq equation and the unstable nonlinear Schrödinger equation, are constructed. The obtained solutions contain Jacobi elliptic function solutions which can be degenerated to the hyperbolic function solutions and the trigonometric function solutions. The present method is very concise and effective and can be applied to other types of nonlinear evolution equations.
Highlights
Nonlinear evolution equations have widely applied in various areas of science and engineering, such as nonlinear optics, fluid dynamics, biophysics, and plasma physics
The traveling wave solutions of the nonlinear evolution equations solved by the exp(−Φ(ξ))-expansion method have the form n u (ξ) = ∑li exp (−Φ (ξ))i, (1)
Many traveling wave solutions have been derived by different methods, such as the semi-inverse variational principle [40], the sinecosine method [25], the new Jacobi elliptic function rational expansion method and exponential rational function method [26], the (G/G)-expansion method [41], the tan(Φ(ξ)/2)expansion method [42], and the extended simple equation method [43]
Summary
Nonlinear evolution equations have widely applied in various areas of science and engineering, such as nonlinear optics, fluid dynamics, biophysics, and plasma physics. The traveling wave solutions of the nonlinear evolution equations solved by the exp(−Φ(ξ))-expansion method have the form n u (ξ) = ∑li exp (−Φ (ξ))i , (1) We improve the exp(−Φ(ξ))-expansion method by presenting a new auxiliary equation: (Φ (ξ))2 (3)
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