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Conservation Laws, Hamiltonian Structure, Modulational Instability Properties and Solitary Wave Solutions for a Higher‐Order Model Describing Nonlinear Internal Waves
Author(s) -
Swaters G. E.,
Dosser H. V.,
Sutherland B. R.
Publication year - 2012
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2011.00533.x
Subject(s) - modulational instability , conservation law , hamiltonian (control theory) , nonlinear schrödinger equation , nonlinear system , physics , classical mechanics , conserved quantity , hamiltonian system , conservation of mass , mathematical analysis , mathematics , mechanics , quantum mechanics , mathematical optimization
Recent theoretical advances in connecting the wave‐induced mean flow with the conserved pseudomomentum per unit mass has permitted the first rational derivation of a model that describes the weakly nonlinear propagation of internal gravity plane waves in a continuously stratified fluid. Depending on the particular parameter regime examined the new model corresponds to an extended bright or dark derivative nonlinear Schrödinger equation or an extended complex‐valued modified Korteweg‐de Vries or Sasa–Satsuma equation. Mass, momentum, and energy conservation laws are derived. A noncanonical infinite‐dimensional Hamiltonian formulation of the model is introduced. The modulational stability characteristics associated with the Stokes wave solution of the model are described. The bright and dark solitary wave solutions of the model are obtained.