The fundamental role of solitons in nonlinear dispersive partial differential equations

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This is the final report of a three-year Laboratory Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). Numerical simulations and mathematical analysis have proved crucial to understanding the fundamental role of solitons in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries, nonlinear Schroedinger and the Kadomtsev-Petviashvili equations. These equations have linear dispersion and the solitons have infinite support. Recently, Philip Rosenau and Mac Hyman discovered a new class of solitons with compact support for similar equations with nonlinear dispersion. These compactions display the same modal decompositions ... continued below

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4 p.

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Hyman, J. M.; Camassa, R.; Cooper, F.; Rosenau, P. & Khare, A. November 1, 1998.

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Description

This is the final report of a three-year Laboratory Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). Numerical simulations and mathematical analysis have proved crucial to understanding the fundamental role of solitons in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries, nonlinear Schroedinger and the Kadomtsev-Petviashvili equations. These equations have linear dispersion and the solitons have infinite support. Recently, Philip Rosenau and Mac Hyman discovered a new class of solitons with compact support for similar equations with nonlinear dispersion. These compactions display the same modal decompositions and structural stability observed in earlier integrable partial differential equations. They form from arbitrary initial data, are nonlinearly self stabilizing and maintain their coherence after multiple collisions, even though the equations are not integrable. In related joint research Roberto Camassa and Darryl Holm, made the remarkable discovery that a similar nonlinear dispersive equation can be described by the evolution of solitons with a peaked solution. The equations are Hamiltonian and a subclass is biHamiltonian and, hence, possess an infinite number of conservation laws. This research is the opening for a far reaching and new understanding of the central role of solitons in nonlinear dispersion.

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4 p.

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OSTI as DE99000832

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  • Other Information: PBD: [1998]

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  • Other: DE99000832
  • Report No.: LA-UR--98-1461
  • Grant Number: W-7405-ENG-36
  • DOI: 10.2172/674866 | External Link
  • Office of Scientific & Technical Information Report Number: 674866
  • Archival Resource Key: ark:/67531/metadc712359

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  • November 1, 1998

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  • Sept. 12, 2015, 6:31 a.m.

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  • May 5, 2016, 7:30 p.m.

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Hyman, J. M.; Camassa, R.; Cooper, F.; Rosenau, P. & Khare, A. The fundamental role of solitons in nonlinear dispersive partial differential equations, report, November 1, 1998; New Mexico. (digital.library.unt.edu/ark:/67531/metadc712359/: accessed December 14, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.