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The Cauchy problem and stability of solitary-wave solutions for RLW–KP-type equations. (English) Zbl 1023.35085
Summary: The Kadomtsev Petviashvilli (KP) equation, \[ (u_t+u_x+ uu_x+u_{xxx})_x +\varepsilon u_{yy}=0, \tag{1} \] arises in various contexts where nonlinear dispersive waves propagate principally along the \(x\)-axis, but with weak dispersive effects being felt in the direction parallel to the \(y\)-axis perpendicular to the main direction of propagation. We propose and analyze here a class of evolution equations of the form \[ (u_t+u_x+ u^pu_x+Lu_t)_x+ \varepsilon u_{yy}=0, \tag{2} \] which provides an alternative to (1) in the same way the regularized long-wave equation is related to the classical Korteweg-de Vries (KdV) equation. The operator \(L\) is a pseudo-differential operator in the \(x\)-variable, \(p\geq 1\) is an integer and \(\varepsilon=\pm 1\).
After discussing the underlying motivation for the class (2), a local well-posedness theory for the initial-value problem is developed. With assumptions on \(L\) and \(p\) that include conditions pertaining to models of interesting physical phenomenon, the solutions defined locally in time \(t\) are shown to be smoothly extendable to the entire time-axis. In the particularly interesting case where \(L=-\partial^2_x\) and \(\varepsilon=-1\), (1) possesses travelling-wave solutions \(u(x,y,t)= \varphi_c(x-ct,y)\) provided \(c<1\) and \(0<p<4\). It is shown here that these solitary waves are stable for \(0<p< {4\over 3}\) and \(c>1\) and for \({4\over 3}<p< 4\) if \(c>(4p)/(4+p)\).
The paper concludes with commentary on extensions of the present theory to more than two space dimensions.

35Q53 KdV equations (Korteweg-de Vries equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
76B25 Solitary waves for incompressible inviscid fluids
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