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Exponential-type solutions to a generalized Drinfel’d-Sokolov equation. (English) Zbl 1206.37043
The paper considers the generalized Drinfel’d-Sokolov equation
\[ u_{tt}+\alpha_1uu_x+\beta u_{xxx} + \gamma(v^{\delta})_x=0, \quad v_t + \alpha_2uv_x + \beta_2v_{xxx}=0. \]
First exact exponential solutions are given for the following combination of parameters: \(\alpha_2=0\), \(\gamma=-2\), \(\delta=2\). In order to obtain exact exponential solutions under general parameter regimes, it is used the homotopy method. The homotopy analysis is firstly “tried” on a problem with linear auxiliary operator, then on a problem with nonlinear auxiliary operator. In this way the results obtained for the starting combination of parameters are extended to other combinations. Numerical results are displayed and convergence discussed. An impressive list of references ends this paper.

MSC:
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
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