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Zbl 1094.34058
Xie, Fuding; Wang, Jingquan
A new method for solving nonlinear differential-difference equations. (English)
[J] Chaos Solitons Fractals 27, No. 4, 1067-1071 (2006). ISSN 0960-0779

Consider nonlinear differential-difference equations of the form $$\multline P(u_{n+ p_1}(t), u_{n+ p_2}(t),\dots, u_{n+ p_s}(t), u_{n+p_1}'(t),\dots, u_{n+ p_s}'(t),\dots,\\ u^{(r)}_{n+ p_1}(t),\dots, u^{(r)}_{n+ p_s}(t))= 0,\endmultline\tag{$*$}$$ where $P$ is a polynom in its arguments, and $u_n(t)= u(n,t)$.\par 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.\par The authors describe an algorithm how to solve this problem and illustrate it by means of an example.
[Klaus R. Schneider (Berlin)]

MSC 2000:: *34K99 Functional-differential equations

Keywords: traveling wave solution; Riccati equation

Cited in: Zbl 1096.34552

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