Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
1996-04-11
Nonlinear Sciences
Pattern Formation and Solitons
52 pages in a dvi file
Scientific paper
Consider a singularly perturbed system $$\epsilon u_t=\epsilon^2 u_{xx} + f(u,x,\epsilon),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0. $$ Assume that the system has a sequence of regular and internal layers occurring alternatively along the $x$-direction. These ``multiple wave'' solutions can formally be constructed by matched asymptotic expansions. To obtain a genuine solution, we derive a {\em Spatial Shadowing Lemma} which assures the existence of an exact solution that is near the formal asymptotic series provided (1) the residual errors are small in all the layers, and (2) the matching errors are small along the lateral boundaries of the adjacent layers. The method should work on some other systems like $\epsilon u_t=-(-\epsilon^2 D_{xx})^m u+ \dots.$
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