On Strongly Interacting Internal Solitary Waves

Abstract

The Cauchy problem and global well-posedness for a mathematical model of the strong interaction of two-dimensional, long, internal gravity waves propogating on neighboring pycnoclines in a stratified fluid have been studied by Bona, Ponce, Saut, Tom, and others. We show that global well-posedness occurs even when the initial data is rough.

Coupled system of Korteweg–de Vries equations type in domains with moving boundaries

“…They proved that the system is globally well-posed in H s (R) × H s (R) for s 1, provided that |a 3 | √ b 2 < 1. This result was improved by Ash et al [1]. They proved that it is globally well-posed in L 2 (R) × L 2 (R), if p =1, provided that |a 3 | √ b 2 < 1.…”

“…Therefore, choosing appropriate ε i and taking into account (4.32) we obtain 1] |z ε |c(ε 5 ) Integrating (4.33) in t ∈ (0, T ), performing straightforward calculations and using the Gronwall inequality we have such that c does not depend on ε > 0.…”

Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity

“…Later on, this result was improved by J.M. Ash et al [2], showing that the system (1.1)-(1.3) is globally well-posed in L 2 (R) × L 2 (R) with √ b 2 a 3 = 1. In 2004, F. Linares and M. Panthee [18] In 2001, O. Vera [19], following the idea of W. Craig et al [7] shown that C ∞ solutions (u(·, t), v(·, t)) to (1.1)-(1.3) are obtained for t > 0 if the initial data (u 0 (x), v 0 (x)) belong to a suitable Sobolev space satisfying reasonable conditions as |x| → ∞.…”

Stability of Solitary-Wave Solutions of Systems of Dispersive Equations

“…A particular system of the type displayed above, but with BBM-type dispersion, was studied by Hakkaev [23]. Recently, theory for the pure initial-value problem posed on the entire real line R and for the periodic initial-value problem for such systems has been developed (see [2,3,11,15,29], and with BBM-type dispersion, [22]). One of the hallmarks of equations featuring both nonlinear and dispersive effects, as these systems do, is the existence of solitary-wave solutions.…”