Solitonic attractors in the coupled nonlinear Schrödinger equations with weak dissipations

We use the Lagrangian perturbation method to investigate the properties of soliton solutions in the coupled nonlinear Schrödinger equations subject to weak dissipation.Our study reveals that the two-component soliton solutions act as fixed-point attractors,where the numerical evolution of the system always converges to a soliton solution,regardless of the initial conditions.Interestingly,the fixed-point attractor appears as a soliton solution with a constant sum of the two-component intensities and a fixed soliton velocity,but each component soliton does not exhibit the attractor feature if the dissipation terms are identical.This suggests that one soliton attractor in the coupled systems can correspond to a group of soliton solutions,which is different from scalar cases.Our findings could inspire further discussions on dissipative-soliton dynamics in coupled systems.

GLOBAL EXISTENCE FOR THE NONHOMOGENEOUS QUASILINEAR WAVE EQUATION WITH A LOCALIZED WEAKLY NONLINEAR DISSIPATION IN EXTERIOR DOMAINS

It is proved that the global existence for the nonhomogeneous quasilinear wave equation with a localized weakly nonlinear dissipation in exterior domains.

Weak Dissipative Structure for Compressible Navier-Stokes Equations

This paper concerns the Cauchy problem for compressible Navier-Stokes equations.The weak dissipative structure is explored and a new proof for the classical solutions are shown to exist globally in time if the initial data is sufficiently small.

THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton （gμHS2） system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative （gμHS2） system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative （gμHS2） system, we show that the semi-group of the solution operator {S（t）}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S（t）}t≥0 possesses a global attractor in the Sobolev space H2（S） × H2（S）.

A NOTE ON NONAUTONOMOUS KLEIN-GORDON-SCHRDINGER EQUATIONS WITH HOMOGENEOUS DIRICHLET BOUNDARY CONDITION

This note discusses the long time behavior of solutions for nonautonomous weakly dissipative Klein-Gordon-Schrodinger equations with homogeneous Dirichlet boundary condition.The authors prove the existence of compact kernel sections for the associated process by using a suitable decomposition of the equations.

A DISSIPATION-PRESERVING INTEGRATOR FOR DAMPED OSCILLATORY HAMILTONIAN SYSTEMS

In this paper,based on discrete gradient,a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established.The solution of this system is a damped nonlinear oscillator.Basically,lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach.The new integrator gives a discrete analogue of the dissipation property of the original system.Meanwhile,since the integrator is based on the variation-of-constants formula for oscillatory systems,it preserves the oscillatory structure of the system.Some properties of the new integrator are derived.The convergence is analyzed for the implicit iterations based on the discrete gradient integrator,and it turns out that the convergence of the implicit iterations based on the new integrator is independent of k Mk,where M governs the main oscillation of the system and usually k Mk≫1.This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system.Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature。

Global Weak Solutions for the Weakly Dissipative Camassa-Holm Equation

In this paper we prove the existence and uniqueness of global weak solutions to the weakly dissipative Camassa-Holm equation.

EXISTENCE OF WEAK SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACES

In this paper, by using weak dissipative type conditions and weak noncompact conditions, we give the existence of weak solutions of differential equations in weak complete Banach spaces.