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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.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
11D09 Quadratic and bilinear Diophantine equations
35C07 Traveling wave solutions
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[1] Hirota, R., The direct method in soliton theory, (2004), Cambridge University Press
[2] Ma, W.X.; Huang, T.W.; Zhang, Y., A multiple exp-function method for nonlinear differential equations and its application, Phys. scr., 82, 065003, (2010) · Zbl 1219.35209
[3] Ma, W.X.; Fan, E.G., Linear superposition principle applying to Hirota bilinear equations, Comput. math. appl., 61, 950-959, (2011) · Zbl 1217.35164
[4] Hirota, R., A new form of Bäcklund transformations and its relation to the inverse scattering problem, Prog. theor. phys., 52, 1498-1512, (1974) · Zbl 1168.37322
[5] Hietarinta, J., Hirota’s bilinear method and soliton solutions, Phys. AUC, 15, part 1, 31-37, (2005)
[6] Ma, W.X.; You, Y., Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans. am. math. soc., 357, 1753-1778, (2005) · Zbl 1062.37077
[7] Ma, W.X.; Li, C.X.; He, J.S., A second Wronskian formulation of the Boussinesq equation, Nonlinear anal., 70, 4245-4258, (2009) · Zbl 1159.37425
[8] Zhang, Y.; Yan, J.J., Soliton resonance of the NI-BKP equation, (), 231-242 · Zbl 1223.37094
[9] Zhang, Y.; Ye, L.Y.; Lv, Y.N.; Zhao, H.Q., Periodic wave solutions of the Boussinesq equation, J. phys. A: math. theor., 40, 5539-5549, (2007) · Zbl 1138.35406
[10] Ma, W.X.; Zhou, R.G.; Gao, L., Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2+1) dimensions, Mod. phys. lett. A, 21, 1677-1688, (2009) · Zbl 1168.35426
[11] Fan, E.G., Quasi-periodic waves and an asymptotic property for the asymmetrical nizhnik – novikov – veselov equation, J. phys. A: math. theor., 42, 095206, (2009) · Zbl 1165.35044
[12] Hirota, R.; Ito, M., Resonance of solitons in one dimension, J. phys. soc. jpn., 52, 744-748, (1983)
[13] Ma, W.X.; Fuchssteiner, B., Explicit and exact solutions to a kolmogorov – petrovskii – piskunov equation, Int. J. non-linear mech., 31, 329-338, (1996) · Zbl 0863.35106
[14] Ma, W.X., Diversity of exact solutions to a restricted boiti – leon – pempinelli dispersive long-wave system, Phys. lett. A, 319, 325-333, (2003) · Zbl 1030.35021
[15] Hu, H.C.; Tong, B.; Lou, S.Y., Nonsingular positon and complexiton solutions for the coupled KdV system, Phys. lett. A, 351, 403-412, (2006) · Zbl 1187.35196
[16] Tang, Y.N.; Xu, W.; Gao, L.; Shen, J.W., An algebraic method with computerized symbolic computation for the one-dimensional generalized BBM equation of any order, Chaos soliton fract., 32, 1846-1852, (2007) · Zbl 1225.35206
[17] Zhang, S.; Xia, T.C., A generalized new auxiliary equation method and its applications to nonlinear partial differential equations, Phys. lett. A, 363, 356-360, (2007) · Zbl 1197.35008
[18] Ma, W.X., Integrability, (), 450-453
[19] Hirota, R.; Ohta, Y.; Satsuma, J., Wronskian structures of solutions for soliton equations: recent developments in soliton theory, Prog. theor. phys. suppl., 94, 59-72, (1988)
[20] Hu, X.B.; Zhao, J.X., Commutativity of pfaffianization and Bäcklund transformations: the KP equation, Inverse probl., 21, 1461-1472, (2005) · Zbl 1086.35091
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