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Hirota bilinear equations with linear subspaces of solutions. (English) Zbl 1245.35109
Summary: We explore when Hirota bilinear equations possess linear subspaces of solutions. First, we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to Hirota bilinear equations. Second, we show that multivariate polynomials whose zeros form a vector space can generate the desired Hirota bilinear equations with given linear subspaces of solutions, and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero. Third, applying an algorithm using weights, we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired Hirota bilinear equations.

35Q53 KdV equations (Korteweg-de Vries equations)
11D09 Quadratic and bilinear Diophantine equations
35C07 Traveling wave solutions
Full Text: DOI
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