×

A new method for solving nonlinear differential-difference equations. (English) Zbl 1094.34058

Consider nonlinear differential-difference equations of the form \[ \begin{split} P(u_{n+ p_1}(t), u_{n+ p_2}(t),\dots, u_{n+ p_s}(t), u_{n+p_1}'(t),\dots, u_{n+ p_s}'(t),\dots,\\ u^{(r)}_{n+ p_1}(t),\dots, u^{(r)}_{n+ p_s}(t))= 0,\end{split}\tag{\(*\)} \] where \(P\) is a polynom in its arguments, and \(u_n(t)= u(n,t)\).
The authors look for conditions such that \((*)\) has a travelling wave solution of the type \[ u_n(t)= \sum^n_{j=1} a_j\phi^j(\xi_n)+ \sum^m_{k=1} b_k\phi^{-k}(\xi_n), \] where \(\xi_n= dn+ ct+\xi_0\), and \(\phi\) satisfies the Riccati equation \[ {d\phi(\xi_n)\over d\xi_n}= 1+ \mu\phi^2(\xi_n),\quad \mu=\pm 1, \] and where the solution can be exactly determined.
The authors describe an algorithm how to solve this problem and illustrate it by means of an example.

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Toda, M.; Wadati, M.; Toda, M., Theory of nonlinear lattices, J phys soc jpn, 39, 1204, (1988), Springer-Verlag Berlin Heidelberg
[2] Wadati, M.; Toda, M.; Wadati, M.; Sanuki, H.; Konno, K.; Wadati, M.; Wadati, M.; Watanabe, M., J phys soc jpn, Prog theor phys, J phys soc jpn, Prog theor phys, 57, 808, (1977)
[3] Levi, D.; Yamilov, R., J math phys, 38, 6648, (1997)
[4] Hereman W, Sanders JA, Sayers J, Wang JP. In: Levi D, Winternitz P, editors, Symbolic computation of conserved densities, gerneralized symmetries, and recursion operators for nonlinear differential-difference equations, CRM Proc and Lect Notes, vol. 39, Amer Math Soc, Providence, RI, 2004, in press. · Zbl 1080.65119
[5] Hu, X.B.; Ma, W.X., Phys lett A, 293, 161, (2002)
[6] Ablowtiz, M.J.; Ladik, J.F., J math phys, 16, 598, (1975)
[7] Baldwin, D.; Göktas, Ü.; Hereman, W., Comput phys commun, 162, 203, (2004)
[8] Fan, E.G., Phys lett A, 277, 212, (2000)
[9] Wadati, M., Prog theor phys suppl, 59, 36-63, (1976)
[10] Ablowtiz, M.J.; Ladik, J.F., Stud appl math, 57, 1, (1977)
[11] Adler, V.E.; Svinolupov, S.I.; Yamilov, R.I., Phys lett A, 254, 24, (1999)
[12] Wu, W.T., Polynomial equation-solving and its application, Algorithms and computation, (1994), Springer Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.