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Analytical solutions to a generalized Drinfel’d-Sokolov equation related to DSSH and KdV6. (English) Zbl 1195.35271
Summary: Analytical solutions to the generalized Drinfel’d-Sokolov (GDS) equations
\[ u_t+\alpha_1uu_x+ \beta_1u_{xxx}+ \gamma(v^\delta)_x=0 \quad\text{and}\quad v_t+ \alpha_2 uv_x+\beta_2v_{xxx}=0 \]
are obtained for various values of the model parameters. In particular, we provide perturbation solutions to illustrate the strong influence of the parameters \(\beta_1\) and \(\beta_2\) on the behavior of the solutions. We then consider a Miura-type transform which reduces the gDS equations into a sixth-order nonlinear differential equation under the assumption that \(\delta=1\). Under such a transform the GDS reduces to the sixth-order Drinfel’d-Sokolov-Satsuma-Hirota (DSSH) equation (also known as KdV6) in the very special case \(\alpha_1=-\alpha_2\). The method of homotopy analysis is applied in order to obtain analytical solutions to the resulting equation for arbitrary \(\alpha_1\) and \(\alpha_2\). An error analysis of the obtained approximate analytical solutions is provided.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B20 Perturbations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35A35 Theoretical approximation in context of PDEs
35C05 Solutions to PDEs in closed form
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