Hints on the Hirota Bilinear Method

P.P. Goldstein The Andrzej Soltan Institute for Nuclear Studies HoŻa 69, 00-681 Warsaw, Poland

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Received: 11 10 2007;

We discuss four stages of the Hirota bilinear method, for construction of soliton solutions to partial differential equations: the proper substitution to express the equation in the bilinear variables (1), reduction of the excess degrees of freedom (2), the perturbation scheme (3), and solution of the system of equations at the successive orders of magnitude (4). For the first stage we suggest an extension of the well-known singularity analysis. In the second stage we point out the need for caution to avoid overdetermined systems. In the third one we suggest a path to proper assumptions on the orders of magnitude of the unknown functions. Finally, we summarize the question of the choice of appropriate solutions. For the expansions at the stages (1) and (3) we suggest a "renormalization", i.e. completion of the lower order terms with higher order ones to achieve the desired form of the coefficients.

DOI: 10.12693/APhysPolA.112.1171 PACS numbers: 02.30.Ik, 42.81.Dp