## 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)

### Keywords:

traveling wave solution; Riccati equation
Full Text:

### References:

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