Exponential-type solutions to a generalized Drinfel'd-Sokolov equation

    Authors

    E. Sweet;R. A. Van Gorder

    Comments

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    Abbreviated Journal Title

    Phys. Scr.

    Keywords

    HOMOTOPY ANALYSIS METHOD; NONLINEAR DIFFERENTIAL-EQUATIONS; TANH METHOD; W-ALGEBRAS; PERIODIC-SOLUTIONS; WILSON EQUATION; WAVE-EQUATIONS; REDUCTION; OPERATORS; KDV; Physics, Multidisciplinary

    Abstract

    Exact exponential-type solutions to the generalized Drinfel'd-Sokolov (GDS) equations u(t) + alpha(1)uu(x) + beta(1)u(xxx) + gamma (v(delta))(x) = 0 and v(t) + alpha(2)uv(x) + beta(2)v(xxx) = 0 are obtained for the case in which alpha(2) = 0, for various values of the other model parameters. A modification of the homotopy analysis method is then applied to obtain analytical solutions for nonzero values of the parameter alpha(2), in effect extending the exact solutions. In our modification of the standard method, we employ a nonlinear auxiliary operator. In contrast to most standard perturbation methods, in which a nonlinear problem is reduced to 'infinitely many' linear problems, here we reduce a hard nonlinear problem to 'infinitely many' easier nonlinear problems. Indeed, we also provide a solution using a linear auxiliary operator and show that the convergence of obtained solutions is improved (in the sense that fewer terms are required for the approximate solutions to obtain a desired accuracy) when using the auxiliary nonlinear operator, in some cases. An error analysis of the obtained approximate analytical solutions is provided.

    Journal Title

    Physica Scripta

    Volume

    82

    Issue/Number

    3

    Publication Date

    1-1-2010

    Document Type

    Article

    Language

    English

    First Page

    11

    WOS Identifier

    WOS:000281537200007

    ISSN

    0031-8949

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