The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations
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Publication:2570785
DOI10.1016/j.amc.2004.08.005zbMath1082.65585OpenAlexW2125518390MaRDI QIDQ2570785
Publication date: 28 October 2005
Published in: Applied Mathematics and Computation (Search for Journal in Brave)
Full work available at URL: https://doi.org/10.1016/j.amc.2004.08.005
Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) KdV equations (Korteweg-de Vries equations) (35Q53) Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs (65M70)
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Uses Software
Cites Work
- An automated \(\tanh\)-function method for finding solitary wave solutions to nonlinear evolution equations
- Compactons and solitary patterns structures for variants of the KdV and the KP equations
- Compactons dispersive structures for variants of the \(K(n,n)\) and the KP equations
- The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations
- A computational approach to soliton solutions of the Kadomtsev-Petviashvili equation
- General compactons solutions for the focusing branch of the nonlinear dispersive \(K(n,n)\) equations in higher-dimensional spaces
- General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive \(K(n,n)\) equations in higher dimensional spaces
- Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method
- The tanh method for traveling wave solutions of nonlinear equations
- On the numerical solution of the sine-Gordon equation. I: Integrable discretizations and homoclinic manifolds
- Exact solutions for a compound KdV-Burgers equation
- Discrete singular convolution for the sine-Gordon equation
- Solitary wave solutions of nonlinear wave equations
- A model unified field equation
- Link between solitary waves and projective Riccati equations
- The tanh method: II. Perturbation technique for conservative systems
- The tanh method: I. Exact solutions of nonlinear evolution and wave equations
- Exact solutions for two nonlinear equations. I
- A study of nonlinear dispersive equations with solitary-wave solutions having compact support
- Exact special solutions with solitary patterns for the nonlinear dispersive \(K(m,n)\) equations
- New solitary-wave special solutions with compact support for the nonlinear dispersive \(K(m,n)\) equations
- Nonlinear preconditioning for diffuse interfaces
