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

Xie, Fuding; Wang, Jingquan

Chaos Solitons Fractals 27, No. 4, 1067-1071 (2006).

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.

Reviewer: Klaus R. Schneider (Berlin)

MSC:

34K99

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

Keywords:

traveling wave solution; Riccati equation

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