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

Consider nonlinear differential-difference equations of the form

P(u n+p 1 (t),u n+p 2 (t),,u n+p s (t),u n+p 1 ' (t),,u n+p s ' (t),,u n+p 1 (r) (t),,u n+p s (r) (t))=0,(*)

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)= j=1 n a j ϕ j (ξ n )+ k=1 m b k ϕ -k (ξ n ),

where ξ n =dn+ct+ξ 0 , and ϕ satisfies the Riccati equation

dϕ(ξ n ) dξ n =1+μϕ 2 (ξ n ),μ=±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.

34K99Functional-differential equations