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.


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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[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
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