On the stability of internal waves

  title={On the stability of internal waves},
  author={Henrik Kalisch and Nguyet Thanh Nguyen},
  journal={Journal of Physics A: Mathematical and Theoretical},
The extended KdV equation ut + uux + αu2ux + uxxx = 0 is widely used as a model describing internal waves in ideal fluids. The equation admits a family of negative and positive solitary waves Φc. These solitary waves exhibit the typical broadening effect seen in internal waves. It is shown here that all solitary-wave solutions of the extended KdV equation are orbitally stable. The proof of stability is based on the general theory of Grillakis et al (1987 J. Funct. Anal. 74 160) for equations of… 

Existence of Solitary Internal Waves in a Two-Layer Fluid of Infinite Height

This paper concerns the existence of internal solitary waves moving with a constant speed at the interface of a two-layer fluid with infinite height. The fluids are immiscible, inviscid, and

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Studied here is the generalized Benjamin-Ono--Zakharov-Kuznetsov equation $u_t+u^pu_x+\alpha\mathscr{H}u_{xx}+\varepsilon u_{xyy}=0, \quad (x,y)\in\rr^2\!,\;\;t\in \rr^+\!$ in two space dimensions.

Remarks on solitary waves of the generalized two dimensional Benjamin-Ono equation

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The study of hyperbolic waves involves various notions which help characterise how these structures evolve. One important facet is the notion of genuine nonlinearity, namely the ability for shocks

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ABSTRACT Mohapatra, S.C.; Fonseca, R.B., and Guedes Soares, C., 2018. Comparison of analytical and numerical simulations of long nonlinear internal solitary waves in shallow water. This study

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New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave

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The stability of solitary waves

  • T. Benjamin
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1972
The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary

The stability of internal solitary waves

A theory is developed relating to the stability of solitary-wave solutions of the so-called Benjamin-Ono equation. This equation was derived by Benjamin (5) as a model for the propagation of internal

Well-posedness of the initial value problem for the Korteweg-de Vries equation

(1.1) &ItU + axu + U1xU = O, x, t E R { u(x, 0) = uo(x). The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been

Model equations for long waves in nonlinear dispersive systems

  • T. BenjaminJ. BonaJ. Mahony
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1972
Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind.

Weakly-Nonlinear, Long Internal Gravity Waves in Stratified Fluids of Finite Depth

This paper presents an analytical investigation of the propagation of a weakly-nonlinear, long internal gravity wave in a stratified medium of finite total depth. The governing equation is derived

Evolution equations for strongly nonlinear internal waves

This paper is concerned with shallow-water equations for strongly nonlinear internal waves in a two-layer fluid, and comparison of their solitary solutions with the results of fully nonlinear

Stability and instability of solitary waves of Korteweg-de Vries type

Considered herein are the stability and instability properties of solitary-wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of

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A new type of solitary wave motion in incompressible fluids of non-uniform density has been investigated experimentally and theoretically. If a fluid is stratified in such a manner that there are two

Internal waves of permanent form in fluids of great depth

  • T. Benjamin
  • Environmental Science
    Journal of Fluid Mechanics
  • 1967
This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these

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Abstract Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology