The nonlinear Schroedinger equation: Solitons dynamics

In this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter h→0.We prove that under suitable assumptions, the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenterqh(t) satisfies the equation

     

Standing waves for the NLS on the double-bridge graph and a rational–irrational dichotomy

We study standing waves of NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, $L_1$ and $L_2$. We study the solutions with possibly nontrivial components on the half-lines and a cnoidal component on the circle. The problem is equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter ω. After classifying the solutions with rational $L_1/L_2$, we turn to $L_1/L_2$ irrational showing that there exist standing waves only in correspondence to a countable set of negative frequencies ${\omega }_n$. Moreover we show that the frequency sequence admits cluster points and any negative real number can be a limit point of frequencies choosing a suitable irrational geometry $L_1/L_2$. These results depend on basic properties of diophantine approximation of real numbers.     

Quasi-periodic solutions in a nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equation

iut−uxx+mu+⁴|u|u=0     

A KAM algorithm for the resonant non-linear Schrödinger equation

We prove, through a KAM algorithm, the existence of large families of stable and unstable quasi-periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a non-degenerate integrable normal form proved in  and  and a generalization of the quasi-Töplitz functions defined in [31].     

On the existence of ground states for nonlinear Schrödinger-Poisson equation

This paper is concerned with the existence of ground states for the Schrödinger-Poisson equation , where V(u) is a Hartree type nonlinearity, stemming from the coupling with the Poisson equation, which includes the so-called doping profile or impurities. By means of variational methods in the energy space we show that ground states exist and belong to the Schwartz space of rapidly decreasing functions whenever total charge not exceed some critical value, it is also shown that for values of the total charge greater than this critical value, energy is not bounded from below. In addition, we show that this critical value is the total charge given by the impurities.     

Traveling wave solutions of the Camassa-Holm equation

All weak traveling wave solutions of the Camassa-Holm equation are classified. We show that, in addition to smooth solutions, there are a multitude of traveling waves with singularities: peakons, cuspons, stumpons, and composite waves.     

Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation

The purpose of the paper is to study properties of solutions of the Cauchy problem for the equation under the assumption . General selfsimilar solutions are constructed. Moreover, for initial data with some decay at infinity, we determine the leading term of the asymptotics of solutions in which is described by either solutions of the linear heat equation or by particular selfsimilar solutions of the original equation.     

On fluid limit for the semiconductors Boltzmann equation

This paper is devoted to the derivation of (non-linear) drift-diffusion equations from the semiconductor Boltzmann equation. Collisions are taken into account through the non-linear Pauli operator, but we do not assume relation on the cross section such as the so-called detailed balance principle. In turn, equilibrium states are implicitly defined. This article follows and completes the contribution of Mellet (Monatsh. Math. 134 (4) (2002) 305-329) where the electric field is given and does not depend on time. Here, we treat the self-consistent problem, the electric potential satisfying the Poisson equation. By means of a Hilbert expansion, we shall formally derive the asymptotic model in the general case. We shall then rigorously prove the convergence in the one-dimensional case by using a modified Hilbert expansion.     

Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind

The nonlinear Schrödinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The ‘derivative’ NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen-Lee-Liu equation, are studied. More precisely, the complex envelope is separated into the absolute value and the phase. The absolute value is solved in terms of a polynomial in elliptic functions while the phase is expressed in terms of elliptic integrals of the third kind. The exact periodicity condition will imply that only a countable set of elliptic function moduli is allowed. This feature contrasts sharply with other periodic solutions of envelope equations, where a continuous range of elliptic function moduli is permitted.     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

Asymptotic behavior of the solution to the non-isothermal phase field equation

We consider the Penrose–Fife phase field model [Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62] with homogeneous Neumann boundary condition to the nonlinear heat flux q=∇(1/θ)$\mathbf{q}=\nabla (1/\theta )$, i.e., q=0$\mathbf{q}=0$ on the boundary, where θ>0$\theta >0$ is the temperature. There is a unique H¹$H^1$ solution globally in time with the non-empty, connected, compact ω$\omega$-limit set composed of stationary solutions, and the linearized stable stationary solution is dynamically stable.     

Global nonexistence for a semilinear Petrovsky equation

In this paper we consider a semilinear Petrovsky equation with damping and source terms. It is proved that the solution blows up in finite time if the positive initial energy satisfies a suitable condition. Moreover for the linear damping case, we show that the solution blows up in finite time even for vanishing initial energy. This is an important breakthrough, since it is only well known that the solution blows up in finite time if the initial energy is negative from all the previous literature.     

Asymptotic equipartition and long time behavior of solutions of a thin-film equation

We investigate the large-time behavior of classical solutions to the thin-film type equation ut=−x(uuxxx). It was shown in previous work of Carrillo and Toscani that for non-negative initial data u₀ that belongs to H¹(R) and also has a finite mass and second moment, the strong solutions relax in the L¹(R) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H¹(R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H¹(R) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates—due to the asymptotic equipartition—and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Time-discretization and global solution for a doubly nonlinear Volterra equation

This note addresses the analysis of an abstract doubly nonlinear Volterra equation with a nonsmooth kernel and possibly unbounded and degenerate operators. By exploiting a suitable implicit time-discretization technique, we obtain the existence of a global strong solution. As a by-product, the discrete scheme is proved to be conditionally stable and convergent.     

A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation

In this paper we consider the Gross-Pitaevskii equation iu _t = Δu + u(1 − |u|²), where u is a complex-valued function defined on , N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x ₁ − ct, x ₂, …, x _N ), where is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence result for non-constant travelling waves of fixed speed having small energy.     

Variational iteration method for solving cubic nonlinear Schrödinger equation

The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.     

Global well-posedness for the generalized 2D Ginzburg-Landau equation

The local well-posedness for the generalized two-dimensional (2D) Ginzburg-Landau equation is obtained for initial data in Hs(R²)(s>1/2). The global result is also obtained in Hs(R²)(s>1/2) under some conditions. The results on local and global well-posedness are sharp except the endpoint s=1/2. We mainly use the Tao's [k;Z]-multiplier method to obtain the trilinear and multilinear estimates.