dorsal/arxiv
View SchemaMultidimensional Pattern Formation Has an Infinite Number of Constants of Motion
| Authors | Mark B. Mineev--Weinstein |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9302001 |
| URL | https://arxiv.org/abs/patt-sol/9302001 |
| DOI | 10.1103/PhysRevE.47.R2241 |
Abstract
Extending our previous work on 2D growth for the Laplace equation we study here {\it multidimensional} growth for {\it arbitrary elliptic} equations, describing inhomogeneous and anisotropic pattern formations processes. We find that these nonlinear processes are governed by an infinite number of conservation laws. Moreover, in many cases {\it all dynamics of the interface can be reduced to the linear time--dependence of only one ``moment" $M_0$} which corresponds to the changing volume while {\it all higher moments, $M_l$, are constant in time. These moments have a purely geometrical nature}, and thus carry information about the moving shape. These conserved quantities (eqs.~(7) and (8) of this article) are interpreted as coefficients of the multipole expansion of the Newtonian potential created by the mass uniformly occupying the domain enclosing the moving interface. Thus the question of how to recover the moving shape using these conserved quantities is reduced to the classical inverse potential problem of reconstructing the shape of a body from its exterior gravitational potential. Our results also suggest the possibility of controlling a moving interface by appropriate varying the location and strength of sources and sinks.
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"abstract": "Extending our previous work on 2D growth for the Laplace equation we study\nhere {\\it multidimensional} growth for {\\it arbitrary elliptic} equations,\ndescribing inhomogeneous and anisotropic pattern formations processes. We find\nthat these nonlinear processes are governed by an infinite number of\nconservation laws. Moreover, in many cases {\\it all dynamics of the interface\ncan be reduced to the linear time--dependence of only one ``moment\" $M_0$}\nwhich corresponds to the changing volume while {\\it all higher moments, $M_l$,\nare constant in time. These moments have a purely geometrical nature}, and thus\ncarry information about the moving shape. These conserved quantities (eqs.~(7)\nand (8) of this article) are interpreted as coefficients of the multipole\nexpansion of the Newtonian potential created by the mass uniformly occupying\nthe domain enclosing the moving interface. Thus the question of how to recover\nthe moving shape using these conserved quantities is reduced to the classical\ninverse potential problem of reconstructing the shape of a body from its\nexterior gravitational potential. Our results also suggest the possibility of\ncontrolling a moving interface by appropriate varying the location and strength\nof sources and sinks.",
"arxiv_id": "patt-sol/9302001",
"authors": [
"Mark B. Mineev--Weinstein"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1103/PhysRevE.47.R2241",
"title": "Multidimensional Pattern Formation Has an Infinite Number of Constants of Motion",
"url": "https://arxiv.org/abs/patt-sol/9302001"
},
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