Abstract
A class of trilinear differential operators is introduced through a technique of assigning signs to derivatives and used to create trilinear differential equations. The resulting trilinear differential operators and equations are characterized by the Bell polynomials, and the superposition principle is applied to the construction of resonant solutions of exponential waves. Two illustrative examples are made by an algorithm using weights of dependent variables.
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References
Bell E T. Exponential polynomials. Ann Math, 1934, 35: 258–277
Bogdan M M, Kovalev A S. Exact multisoliton solution of one-dimensional Landau-Lifshitz equations for an anisotropic ferromagnet. JETP Lett, 1980, 31(8): 424–427
Broer L J F. Approximate equations for long wave equations. Appl Sci Res, 1975, 31(5): 377–395
Craik A D D. Prehistory of Faà di Bruno’s formula. Amer Math Monthly, 2005, 112: 217–234
Delzell C N. A continuous, constructive solution to Hilbert’s 17th problem. Invent Math, 1984, 76(3): 365–384
Gilson C, Lambert F, Nimmo J, Willox R. On the combinatorics of the Hirota D-operators. Proc R Soc Lond A, 1996, 452: 223–234
Grammaticos B, Ramani A, Hietarinta J. Multilinear operators: the natural extension of Hirota’s bilinear formalism. Phys Lett A, 1994, 190(1): 65–70
Hietarinta J. Hirota’s bilinear method and soliton solutions. Phys AUC, 2005, 15(part 1): 31–37
Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett, 1971, 27: 1192–1194
Hirota R. A new form of Bäcklund transformations and its relation to the inverse scattering problem. Progr Theoret Phys, 1974, 52: 1498–1512
Hirota R. Soliton solutions to the BKP equations-I. The Pfaffian technique. J Phys Soc Jpn, 1989, 58: 2285–2296
Hirota R. The Direct Method in Soliton Theory. Cambridge: Cambridge University Press, 2004
Hu X B, Wang H Y. Construction of dKP and BKP equations with self-consistent sources. Inverse Problems, 2006, 22: 1903–1920
Kaup D J. A higher-order water-wave equation and the method for solving it. Progr Theoret Phys, 1975, 54(2): 396–408
Lambert F, Springael J. Construction of Bäcklund transformations with binary Bell polynomials. J Phys Soc Jpn, 1997, 66: 2211–2213
Lambert F, Springael J, Willox R. On a direct bilinearization method: Kaup’s higherorder water wave equation as a modified nonlocal Boussinesq equation. J Phys A: Math Gen, 1994, 27: 5325–5334
Ma WX. Complexiton solutions to the Korteweg-de Vries equation. Phys Lett A, 2002, 301: 35–44
Ma W X. Variational identities and Hamiltonian structures. In: Ma W X, Hu X B, Liu Q P, eds. Nonlinear and Modern Mathematical Physics. AIP Conf Proc 1212. Melville: Amer Inst Phys, 2010, 1–27
Ma W X. Generalized bilinear differential equations. Stud Nonlinear Sci, 2011, 2: 140–144
Ma W X. Bilinear equations, Bell polynomials and the linear superposition principle. J Phys: Conf Ser, 2013, 411: 012021
Ma W X, Abdeljabbar A, Asaad M G. Wronskian and Grammian solutions to a (3+1)-dimensional generalized KP equation. Appl Math Comput, 2011, 217: 10016–10023
Ma W X, Fan E G. Linear superposition principle applying to Hirota bilinear equations. Comput Math Appl, 2011, 61: 950–959
Ma W X, Li C X, He J S. A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal, 2009, 70: 4245–4258
Ma W X, You Y. Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans Amer Math Soc, 2005, 357: 1753–1778
Ma W X, Zhang Y, Tang Y N, Tu J Y. Hirota bilinear equations with linear subspaces of solutions. Appl Math Comput, 2012, 218: 7174–7183
Matsukidaira J, Satsuma J. Integrable four-dimensional nonlinear lattice expressed by trilinear form. J Phys Soc Jpn, 1990, 59(10): 3413–3416
Matsukidaira J, Satsuma J. Exactly solvable four-dimensional discrete equation expressed by trilinear form. Phys Lett A, 1991, 154: 366–372
Matsukidaira J, Satsuma J. The trilinear equation as a (2+2)-dimensional extension of the (1+1)-dimensional relativistic Toda lattice. Phys Lett A, 1991, 161: 267–273
Matsukidaira J, Satsuma J, Strampp W. Soliton equations expressed by trilinear forms and their solutions. Phys Lett A, 1990, 147: 467–471
Terng C -L, Uhlenbeck K. Geometry of solitons. Notices Amer Math Soc, 2000, 47(1): 17–25
Terng C -L, Uhlenbeck K. The n×n KdV hierarchy. J Fixed Point Theory Appl, 2011, 110(1): 37–61
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Ma, WX. Trilinear equations, Bell polynomials, and resonant solutions. Front. Math. China 8, 1139–1156 (2013). https://doi.org/10.1007/s11464-013-0319-5
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DOI: https://doi.org/10.1007/s11464-013-0319-5