dorsal/arxiv
View SchemaShock waves in the dissipative Toda lattice
| Authors | J. Hietarinta, T. Kuusela, B. Malomed |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9502001 |
| URL | https://arxiv.org/abs/solv-int/9502001 |
| DOI | 10.1088/0305-4470/28/11/007 |
Abstract
We consider the propagation of a shock wave (SW) in the damped Toda lattice. The SW is a moving boundary between two semi-infinite lattice domains with different densities. A steadily moving SW may exist if the damping in the lattice is represented by an ``inner'' friction, which is a discrete analog of the second viscosity in hydrodynamics. The problem can be considered analytically in the continuum approximation, and the analysis produces an explicit relation between the SW's velocity and the densities of the two phases. Numerical simulations of the lattice equations of motion demonstrate that a stable SW establishes if the initial velocity is directed towards the less dense phase; in the opposite case, the wave gradually spreads out. The numerically found equilibrium velocity of the SW turns out to be in a very good agreement with the analytical formula even in a strongly discrete case. If the initial velocity is essentially different from the one determined by the densities (but has the correct sign), the velocity does not significantly alter, but instead the SW adjusts itself to the given velocity by sending another SW in the opposite direction.
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"abstract": "We consider the propagation of a shock wave (SW) in the damped Toda lattice.\nThe SW is a moving boundary between two semi-infinite lattice domains with\ndifferent densities. A steadily moving SW may exist if the damping in the\nlattice is represented by an ``inner\u0027\u0027 friction, which is a discrete analog of\nthe second viscosity in hydrodynamics. The problem can be considered\nanalytically in the continuum approximation, and the analysis produces an\nexplicit relation between the SW\u0027s velocity and the densities of the two\nphases. Numerical simulations of the lattice equations of motion demonstrate\nthat a stable SW establishes if the initial velocity is directed towards the\nless dense phase; in the opposite case, the wave gradually spreads out. The\nnumerically found equilibrium velocity of the SW turns out to be in a very good\nagreement with the analytical formula even in a strongly discrete case. If the\ninitial velocity is essentially different from the one determined by the\ndensities (but has the correct sign), the velocity does not significantly\nalter, but instead the SW adjusts itself to the given velocity by sending\nanother SW in the opposite direction.",
"arxiv_id": "solv-int/9502001",
"authors": [
"J. Hietarinta",
"T. Kuusela",
"B. Malomed"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1088/0305-4470/28/11/007",
"title": "Shock waves in the dissipative Toda lattice",
"url": "https://arxiv.org/abs/solv-int/9502001"
},
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