On the convergence of solutions of some evolution differential equations

Two conjectures concerning the asymptotic behaviour of solutions of two types of evolution differential equations are presented.     

Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application

We establish the uniqueness of ground states of some coupled nonlinear Schrödinger systems in the whole space. We firstly use Schwartz symmetrization to obtain the existence of ground states for a more general case. To prove the uniqueness of ground states, we use the radial symmetry of the ground states to transform the systems into an ordinary differential system, and then we use the integral forms of the system. More interestingly, as an application of our uniqueness results, we derive a sharp vector-valued Gagliardo-Nirenberg inequality.     

Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors

In this paper we study the stability of transonic strong shock solutions of the steady-state one-dimensional unipolar hydrodynamic model for semiconductors. The approach is based on the construction of a pseudo-local symmetrizer and on the paradifferential calculus with parameters, which combines the work of Bony-Meyer and the introduction of a large parameter.     

On the Boussinesq/Full dispersion systems and Boussinesq/Boussinesq systems for internal waves

In this paper, we consider some asymptotic models for internal waves in the small amplitude/small amplitude regime, which were derived recently by Bona, Lannes and Saut. We first prove that the Boussinesq/Full dispersion systems and the Boussinesq/Boussinesq systems can be derived from the Full dispersion/Full dispersion systems. Then using a contraction-mapping argument and the energy method, we will prove that the derived systems that are linearly well-posed are in fact locally nonlinearly well-posed in suitable Sobolev classes. In particular, we improve and extend some known results on the well-posedness of Boussinesq systems for surface waves.     

Asymptotic regularity for some dissipative equations

This paper is devoted to proving some asymptotic regularity, for both reaction-diffusion equation with a polynomially growing nonlinearity of arbitrary order and strongly damped wave equation with critical nonlinearity, which excel the sharp regularity allowed by the corresponding stationary equations (equilibrium points). Based on this regularity, the existence of the finite-dimensional global and exponential attractors can be obtained easily.     

Remarks on the derivation of the hydrostatic Euler equations

The motion of an inviscid incompressible fluid between two horizontal plates is studied in the limit when the plates are infinitesimally close. The convergence of the solutions of the Euler equations to those of their formal ‘hydrostatic’ limit can be established in the case when the initial velocity field satisfies a local Rayleigh conditions. This result, originally obtained by Grenier through weighted energy estimates based on Arnold's stability analysis of the Euler equations, is proven here by a more straightforward method even closer to Arnold's method.     

Bounded traveling waves of the Burgers-Huxley equation

In order to investigate bounded traveling waves of the Burgers-Huxley equation, bifurcations of codimension 1 and 2 are discussed for its traveling wave system. By reduction to center manifolds and normal forms we give conditions for the appearance of homoclinic solutions, heteroclinic solutions and periodic solutions, which correspondingly give conditions of existence for solitary waves, kink waves and periodic waves, three basic types of bounded traveling waves. Furthermore, their evolutions are discussed to investigate the existence of other types of bounded traveling waves, such as the oscillatory traveling waves corresponding to connections between an equilibrium and a periodic orbit and the oscillatory kink waves corresponding to connections of saddle-focus.     

Finitary isomorphism of some renewal processes to Bernoulli schemes

Using the marker and filler methods of Keane and Smorodinsky, we prove that entropy is a complete finitary isomorphism invariant for r-processes. It is conjectured that entropy is a complete finitary isomorphism invariant for finitary factors of Bernoulli schemes. We present a weaker version of this conjecture with hope that its proof is more attainable with present methods. In doing so, we define a one-way finitary isomorphism and prove one-way finitary results for random walks. We will also extend the marker and filler methods of Keane and Smorodinsky to a class of countable state processes.     

Wave breaking for the periodic weakly dissipative Dullin-Gottwald-Holm equation

In this paper, we consider the periodic weakly dissipative Dullin-Gottwald-Holm equation. The present work is mainly concerned with blow-up phenomena for the Cauchy problem for this new kind of equation. We apply the optimal constant to give sufficient conditions via an appropriate integral form of the initial data, which guarantee the finite-time singularity formation for the corresponding solution.     

Note on the flux problem for stationary incompressible Navier-Stokes equations in domains with a multiply connected boundary

In this paper we present a new proof for the existence of Navier-Stokes flows with nonhomogeneous boundary conditions in domains with multiply connected boundaries. For sufficiently small fluxes, we show the existence of at least one stationary solution with an improved smallness assumption.     

Positive clustered layered solutions for the Gierer-Meinhardt system

We consider the stationary Gierer-Meinhardt system in a ball of RN:

     

Strong instability of solitary waves for nonlinear Klein–Gordon equations and generalized Boussinesq equations

We study here instability problems of standing waves for the nonlinear Klein–Gordon equations and solitary waves for the generalized Boussinesq equations. It is shown that those special wave solutions may be strongly unstable by blowup in finite time, depending on the range of the wave's frequency or the wave's speed of propagation and on the nonlinearity.     

Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces

In this paper we prove the local and global well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces under certain conditions of s, q, and σ.     

Parametric exponential energy decay for dissipative electron-ion plasma waves

We consider the following evolution system of Klein-Gordon-Schrödinger type

Stability of post-fertilization traveling waves

This paper studies the stability of a family of traveling wave solutions to the system proposed by Lane et al. [D.C. Lane, J.D. Murray, V.S. Manoranjan, Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs, IMA J. Math. Appl. Med. Biol. 4 (4) (1987) 309-331], to model a pair of mechanochemical phenomena known as post-fertilization waves on eggs. The waves consist of an elastic deformation pulse on the egg's surface, and a free calcium concentration front. The family is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration. This work establishes the spectral, linear and nonlinear orbital stability of these post-fertilization waves for small values of the coupling parameter. The usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation. Nonetheless, exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established. We show that small perturbations of the waves yield solutions to the nonlinear equations decaying exponentially to a phase-modulated traveling wave.     

Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations

Zero curvature formulations, pseudo-potentials, modified versions, “Miura transformations”, conservation laws, and nonlocal symmetries of the Korteweg–de Vries, Camassa–Holm and Hunter–Saxton equations are investigated from a unified point of view: these three equations belong to a two-parameter family of equations describing pseudo-spherical surfaces, and therefore their basic integrability properties can be studied by geometrical means.      

Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e2iωtφ,0,0)$(e^{2i\omega t}\phi ,0,0)$, (0,e2iωtφ,0)$(0,e^{2i\omega t}\phi ,0)$, (0,0,e2iωtφ)$(0,0,e^{2i\omega t}\phi )$, where φ is a ground state of the scalar nonlinear Schrödinger equation.