Trilinear equations, Bell polynomials, and resonant solutions

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.

Key words： Trilinear differential equation; Bell polynomial; superposition principle

References

1 Bell E T. Exponential polynomials. Ann Math, 1934, 35: 258-277

2 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

3 Broer L J F. Approximate equations for long wave equations. Appl Sci Res, 1975, 31(5): 377-395

4 Craik A D D. Prehistory of Faà di Bruno’s formula. Amer Math Monthly, 2005, 112: 217-234

5 Delzell C N. A continuous, constructive solution to Hilbert’s 17th problem. Invent Math, 1984, 76(3): 365-384

6 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

7 Grammaticos B, Ramani A, Hietarinta J. Multilinear operators: the natural extension of Hirota’s bilinear formalism. Phys Lett A, 1994, 190(1): 65-70

8 Hietarinta J. Hirota’s bilinear method and soliton solutions. Phys AUC, 2005, 15(part 1): 31-37

9 Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett, 1971, 27: 1192-1194

10 Hirota R. A new form of Bäcklund transformations and its relation to the inverse scattering problem. Progr Theoret Phys, 1974, 52: 1498-1512

11 Hirota R. Soliton solutions to the BKP equations—I. The Pfaffian technique. J Phys Soc Jpn, 1989, 58: 2285-2296

12 Hirota R. The Direct Method in Soliton Theory. Cambridge: Cambridge University Press, 2004

13 Hu X B, Wang H Y. Construction of dKP and BKP equations with self-consistent sources. Inverse Problems, 2006, 22: 1903-1920

14 Kaup D J. A higher-order water-wave equation and the method for solving it. Progr Theoret Phys, 1975, 54(2): 396-408

15 Lambert F, Springael J. Construction of Bäcklund transformations with binary Bell polynomials. J Phys Soc Jpn, 1997, 66: 2211-2213

16 Lambert F, Springael J, Willox R. On a direct bilinearization method: Kaup’s higher- order water wave equation as a modified nonlocal Boussinesq equation. J Phys A: Math Gen, 1994, 27: 5325-5334

17 Ma W X. Complexiton solutions to the Korteweg-de Vries equation. Phys Lett A, 2002, 301: 35-44

18 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

19 Ma W X. Generalized bilinear differential equations. Stud Nonlinear Sci, 2011, 2: 140-144

20 Ma W X. Bilinear equations, Bell polynomials and the linear superposition principle. J Phys: Conf Ser, 2013, 411-012021

21 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

22 Ma W X, Fan E G. Linear superposition principle applying to Hirota bilinear equations. Comput Math Appl, 2011, 61: 950-959

23 Ma W X, Li C X, He J S. A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal, 2009, 70: 4245-4258

24 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

25 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

26 Matsukidaira J, Satsuma J. Integrable four-dimensional nonlinear lattice expressed by trilinear form. J Phys Soc Jpn, 1990, 59(10): 3413-3416

27 Matsukidaira J, Satsuma J. Exactly solvable four-dimensional discrete equation expressed by trilinear form. Phys Lett A, 1991, 154: 366-372

28 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

29 Matsukidaira J, Satsuma J, Strampp W. Soliton equations expressed by trilinear forms and their solutions. Phys Lett A, 1990, 147: 467-471

30 Terng C-L, Uhlenbeck K. Geometry of solitons. Notices Amer Math Soc, 2000, 47(1): 17-25

31 Terng C-L, Uhlenbeck K. The n × n KdV hierarchy. J Fixed Point Theory Appl, 2011, 10(1): 37-61