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On the initial value problem for the Davey-Stewartson systems. (English) Zbl 0727.35111

This paper is concerned with the nonlinear Davey-Stewartson equations

$i{u}_{t}+\delta {u}_{xx}+{u}_{yy}={\kappa |u|}^{2}u+bu{\phi }_{x};\phantom{\rule{1.em}{0ex}}{\phi }_{xx}+m{\phi }_{yy}={\left(|u|}^{2}{\right)}_{x}$

which are of different type according to the signs of $\delta$ and m respectively. There are several special cases of this system which can be treated by an inverse scattering method, but this paper deals with the general case. In a first chapter there is given a brief motivation of these equations as a model for the propagation of water waves in an impermeable and horizontal bed of infinite extent. Moreover some relevant conservation laws - like conservation of mass, momentum and energy - are derived and also the interesting relation

${I}^{\text{'}\text{'}}\left(t\right)=8E\left(u\left(t\right)\right),\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}I\left(t\right)={\int }_{{ℝ}^{2}}\left(\delta {x}^{2}+{y}^{2}\right){|u|}^{2}dxdy,$

which has already been noticed by Ablowitz and Segur. Here E(u(t)) (the energy) is a specific constant of motion. This relation is used for the proof of a blow up result for solutions.

In the second chapter the Cauchy problem in the case $m>0$ is studied. Here is an existence and uniqueness result in a finite time interval which may be infinite if the Cauchy data are sufficiently small. Furthermore regularity and continuous dependence of solutions from data are studied. In the third chapter nonexistence results are emphasized as for example in the case $\delta >0$ for $\kappa \ge max\left(-b,0\right)$ the solutions exist global in time and for $\kappa there is a blow up in finite time. Chapter four is devoted to prove existence in the case $\delta =1$ and $m<0$. The proofs are sometimes short and only indicated. The results of the paper had been announced by the authors [C. R. Acad. Sci., Paris, Sér. I 308, No.4, 115-120 (1989; Zbl 0656.76012)].

Reviewer: G.Jank (Aachen)

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35Q55 NLS-like (nonlinear Schrödinger) equations 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35B40 Asymptotic behavior of solutions of PDE