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Register a new userStability of the solitary wave solutions to a coupled BBM system
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In this work, we present a stability criteria for the solitary wave solutions to a BBM system that contains coupled nonlinear terms. Using the idea by Bona, Chen and Karakashian and exploiting the accurate point spectrum information of the associated Schrodinger operator, we improve the stability results previously got by Pereira.
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In this work, we consider the electromechanical density pulse as a coupled solitary waves represented by a longitudinal compression wave and an out-of-plane transversal wave (i.e., perpendicular to the membrane surface). We analyzed using, the variational approach, the characteristics of the coupled solitary waves in the presence of damping within the framework of coupled nonlinear Burger-Korteweg-de Vries-Benjamin-Bona-Mahony (BKdV-BBM) equation. It is shown that, the inertia parameter increases the stability of coupled solitary waves while the damping parameter decreases it. Moreover, the presence of damping term induces a discontinuity of stable regions in the inertia-speed parameter space, appearing in he form of an island of points. Bell shape and solitary-shock like wave profiles were obtained by varying the propagation speed and their linear stability spectrum computed. It is shown that bell shape solitary wave exhibit bound state eigenvalue spectrum, therefore stable. On the other hand, the solitary-shock like wave profiles exhibit unbound state eigenvalue spectrum and are therefore generally unstable.
The well-posedness for the supersonic solutions of the Euler-Poisson system for hydrodynamical model in semiconductor devices and plasmas is studied in this paper. We first reformulate the Euler-Poisson system in the supersonic region into a second order hyperbolic-elliptic coupled system together with several transport equations. One of the key ingredients of the analysis is to obtain the well-posedness of the boundary value problem for the associated linearized hyperbolic-elliptic coupled system, which is achieved via a delicate choice of multiplier to gain energy estimate. The nonlinear structural stability of supersonic solution in the general situation is established by combining the iteration method with the estimate for hyperbolic-elliptic system and the transport equations together.
We study the long-time behavior an extended Navier-Stokes system in $R^2$ where the incompressibility constraint is relaxed. This is one of several reduced models of Grubb and Solonnikov 89 and was revisited recently (Liu, Liu, Pego 07) in bounded domains in order to explain the fast convergence of certain numerical schemes (Johnston, Liu 04). Our first result shows that if the initial divergence of the fluid velocity is mean zero, then the Oseen vortex is globally asymptotically stable. This is the same as the Gallay Wayne 05 result for the standard Navier-Stokes equations. When the initial divergence is not mean zero, we show that the analogue of the Oseen vortex exists and is stable under small perturbations. For completeness, we also prove global well-posedness of the system we study.
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrodinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it a $textit{fundamental solitary wave}$. By using a result of one of the authors and his collaborator, the bifurcations of the fundamental solitary wave are detected. We utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral or orbital stability of the bifurcated solitary waves as well as as that of the fundamental one under some nondegenerate conditions which are easy to verify, compared with those of the previous results. We apply our theory to CNLS with a cubic nonlinearity and give numerical evidences for the theoretical results.
We consider a system of two coupled non-linear Klein-Gordon equations. We show the existence of standing waves solutions and the existence of a Lyapunov function for the ground state.
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