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Linear superposition principle applying to Hirota bilinear equations. (English) Zbl 1217.35164
Summary: A linear superposition principle of exponential traveling waves is analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of N-soliton solutions formed by linear combinations of exponential traveling waves. Applications are made for the 3+1 dimensional KP, Jimbo-Miwa and BKP equations, thereby presenting their particular N-wave solutions. An opposite question is also raised and discussed about generating Hirota bilinear equations possessing the indicated N-wave solutions, and a few illustrative examples are presented, together with an algorithm using weights.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35C08Soliton solutions of PDE
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
References:
[1]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 · doi:10.1016/j.physleta.2003.10.030
[2]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 · doi:10.1016/j.physleta.2005.11.047
[3]Hirota, R.: The direct method in soliton theory, (2004)
[4]Fordy, A. P.: Soliton theory: A survey of results. Nonlinear science: theory and applications, (1990)
[5]Hietarinta, J.: Hirota’s bilinear method and soliton solutions, Phys. AUC 15, No. part 1, 31-37 (2005)
[6]Ma, W. X.; Srampp, W.: Bilinear forms and Bäcklund transformations of the perturbation systems, Phys. lett. A 341, 441-449 (2005) · Zbl 1171.37332 · doi:10.1016/j.physleta.2005.05.013
[7]Ma, W. X.: Integrability, Encyclopedia of nonlinear science, 450-453 (2005)
[8]Ablowitz, M. J.; Segur, H.: On the evolution of packets of water waves, J. fuid mech. 92, 691-715 (1979) · Zbl 0413.76009 · doi:10.1017/S0022112079000835
[9]Infeld, E.; Rowlands, G.: Three-dimensional stability of Korteweg–de Vries waves and solitons II, Acta phys. Polon. A 56, 329-332 (1979)
[10]Jimbo, M.; Miwa, T.: Solitons and infinite dimensional Lie algebras, Publ. res. Inst. math. Sci. 19, 943-1001 (1983) · Zbl 0557.35091 · doi:10.2977/prims/1195182017
[11]Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T.: Transformation groups for soliton equations VI–KP hierarchies of orthogonal and symplectic type, J. phys. Soc. Japan 50, 3813-3818 (1981)
[12]Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T.: Transformation groups for soliton equations IV–a new hierarchy of soliton equations of KP-type, Physica D 4, 343-365 (1981/1982) · Zbl 0571.35100 · doi:10.1016/0167-2789(82)90041-0
[13]Ma, W. X.; Zhou, R. G.; Gao, L.: Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2+1) dimensions, Modern phys. Lett. A 21, 1677-1688 (2009) · Zbl 1168.35426 · doi:10.1142/S0217732309030096
[14]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 · doi:10.1088/1751-8113/42/9/095206
[15]Fan, E. G.: Supersymmetric KdV–Sawada–Kotera–Ramani equation and its quasi-periodic wave solutions, Phys. lett. A 374, 744-749 (2010)
[16]Hirota, R.; Ohta, Y.; Satsuma, J.: Wronskian structures of solutions for soliton equations, Progr. theoret. Phys. suppl. 94, 59-72 (1988)
[17]Ma, W. X.; You, Y.: Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions, Trans. amer. Math. soc. 357, 1753-1778 (2005) · Zbl 1062.37077 · doi:10.1090/S0002-9947-04-03726-2
[18]Hu, X. B.; Zhao, J. X.: Commutativity of pfaffianization and Bäcklund transformations: the KP equation, Inverse problems 21, 1461-1472 (2005) · Zbl 1086.35091 · doi:10.1088/0266-5611/21/4/016
[19]Konopelchenko, B.; Strampp, W.: The AKNS hierarchy as symmetry constraint of the KP hierarchy, Inverse problems 7, L17-L24 (1991) · Zbl 0728.35111 · doi:10.1088/0266-5611/7/2/002
[20]Lin, R. L.; Zeng, Y. B.; Ma, W. X.: Solving the KdV hierarchy with self-consistent sources by inverse scattering method, Physica A 291, 287-298 (2001) · Zbl 0972.35128 · doi:10.1016/S0378-4371(00)00519-7
[21]Liu, X. J.; Zeng, Y. B.; Lin, R. L.: A new extended KP hierarchy, Phys. lett. A 372, 3819-3823 (2008) · Zbl 1220.37055 · doi:10.1016/j.physleta.2008.02.070
[22]Ma, W. X.: An extended harry dym hierarchy, J. phys. A: math. Theor. 43, 165202 (2010)
[23]Ma, W. X.: Commutativity of the extended KP flows, Commun. nonlinear sci. Numer. simul. 16, 722-730 (2011)