A model system for strong interaction between internal solitary waves
The model for the strong interaction of long internal gravity waves propagating on neighboring pycnoclines in a stratified fluid is described by the system of the equations [\textit{J. A. Gear} and \textit{R. Grimshow}, Stud. Appl. Math. 70, 235-258 (1984; Zbl 0548.76020)], \[ u_ t+uu_ x+u_{xxx}+a_ 3v_{xxx}+a_ 1vv_ x+a_ 2(uv)_ x=0,\tag{1} \] \[ b_ 1v_ t+rv_ x+vv_ x+v_{xxx}+b_ 2a_ 3u_{xxx}+b_ 2a_ 2uu_ x+b_ 2a_ 1(uv)_ x=0, \tag{2} \] where \(a_ 1\), \(a_ 2\), \(a_ 3\), \(b_ 1>0\), \(b_ 2>0\), and \(r\) are real constants. The Cauchy problem for the system (1), (2) is shown to be globally well-posed in suitably strong function spaces. The abstract evolution equations together with somewhat delicate estimates are obtained using techniques from harmonic analysis. The system (1), (2) is shown to be susceptible to the blow-up phenomenon.
scientific article; zbMATH DE number 54714
scientific article