dorsal/arxiv
View SchemaShadowing matching errors for wave-front-like solutions
| Authors | Xiao-Biao Lin |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9604001 |
| URL | https://arxiv.org/abs/patt-sol/9604001 |
Abstract
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 $\epsilon u_t=-(-\epsilon^2 D_{xx})^m u+ \dots.$
{
"annotation_id": "964a58ab-a311-404a-82c9-82cfff2ef38c",
"date_created": "2026-03-02T18:00:28.609000Z",
"date_modified": "2026-03-02T18:00:28.609000Z",
"file_hash": "d9dc02e48cc34ea1d7fe25db3c032ba7b3518f2943e9b7fb6ce9809000d9f23c",
"private": false,
"record": {
"abstract": "Consider a singularly perturbed system $$\\epsilon u_t=\\epsilon^2 u_{xx} +\nf(u,x,\\epsilon),\\quad u\\in {\\Bbb R}^n,x\\in{\\Bbb R},t\\geq 0. $$ Assume that the\nsystem has a sequence of regular and internal layers occurring alternatively\nalong the $x$-direction. These ``multiple wave\u0027\u0027 solutions can formally be\nconstructed by matched asymptotic expansions. To obtain a genuine solution, we\nderive a {\\em Spatial Shadowing Lemma} which assures the existence of an exact\nsolution that is near the formal asymptotic series provided (1) the residual\nerrors are small in all the layers, and (2) the matching errors are small along\nthe lateral boundaries of the adjacent layers. The method should work on some\nother systems like $\\epsilon u_t=-(-\\epsilon^2 D_{xx})^m u+ \\dots.$",
"arxiv_id": "patt-sol/9604001",
"authors": [
"Xiao-Biao Lin"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"title": "Shadowing matching errors for wave-front-like solutions",
"url": "https://arxiv.org/abs/patt-sol/9604001"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "faf04318-77d2-4270-96bf-834e1ab46e49",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}