Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations

We consider the nonlinear elliptic system

The Linear Stability of Shock Waves for the Nonlinear Schrödinger–Inviscid Burgers System

We investigate the coupling between the nonlinear Schrödinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schrödinger–Burgers system, when the reference solution is an entropy-satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.     

On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging

We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree–Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray free electron laser. We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schrödinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential.     

Riemann–Hilbert method and multi-soliton solutions of the Kundu-nonlinear Schrödinger equation

In this work, we study the Kundu-nonlinear Schrödinger (Kundu-NLS) equation (so-called the extended NLS equation), which can describe the propagation of the waves in dispersive media. A Lax spectral problem is used to construct the Riemann–Hilbert problem, via a matrix transformation. Based on the inverse scattering transformation, the general solutions of the Kundu-NLS equation are calculated. In the reflection-less case, the special matrix Riemann–Hilbert problem is carefully proposed to derive the multi-soliton solutions. Finally, some novel dynamics behaviors of the nonlinear system are theoretically and graphically discussed.

     

Asymptotics of the Solutions of the Random Schrödinger Equation

We consider solutions of the Schrödinger equation with a weak time-dependent random potential. It is shown that when the two-point correlation function of the potential is rapidly decaying, then the Fourier transform \({\hat\zeta_\epsilon(t,\xi)}\) of the appropriately scaled solution converges point-wise in ξ to a stochastic complex Gaussian limit. On the other hand, when the two-point correlation function decays slowly, we show that the limit of \({\hat\zeta_\epsilon(t,\xi)}\) has the form \({\hat\zeta_0(\xi){\rm exp}(iB_\kappa(t,\xi))}\) where B _κ (t, ξ) is a fractional Brownian motion.     

On the Darboux transformation of a generalized inhomogeneous higher-order nonlinear Schrödinger equation

Recently, a paper about the Nth-order rogue waves for an inhomogeneous higher-order nonlinear Schrödinger equation using the generalized Darboux transformation is published. Song et al. (Nonlinear Dyn 82(1):489–500. doi: 10.1007/s11071-015-2170-6, 2015). However, the inhomogeneous equation which admits a nonisospectral linear eigenvalue problem is mistaken for having a constant spectral parameter by the authors. This basic error causes the results to be wrong, especially regarding the Darboux transformation (DT) in Sect. 2 when the inhomogeneous terms are dependent of spatial variable x. In fact, the DT for inhomogeneous equation has an essential difference from the isospectral case, and their results are correct only in the absence of inhomogeneity which was already discussed in detail before. Consequently, we firstly modify the DT based on corresponding nonisospectral linear eigenvalue problem. Then, the nonautonomous solitons are obtained from zero seed solutions. Properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. Finally, the failure of finding breather and rogue wave solutions from this modified DT is also discussed.     

KdV Limit of the Euler–Poisson System

Consider the scaling ${\varepsilon^{1/2}(x-Vt) \to x, \varepsilon^{3/2}t \to t}$ in the Euler–Poisson system for ion-acoustic waves (1). We establish that as ${\varepsilon \to 0}$ , the solutions to such Euler–Poisson systems converge globally in time to the solutions of the Korteweg–de Vries equation.     

Long Time Behavior of Weak Solutions to Navier–Stokes–Poisson System

Ducomet et?al. (Discrete Contin Dyn Syst 11(1): 113?C130, 2004) showed the existence of global weak solutions to the Navier?CStokes?CPoisson system. We study the global behavior of such a solution. This is done by (1) proving uniqueness of a solution to the stationary system; (2) by showing convergence of a weak solution to the stationary solution. In (1) we consider only the case with repulsion. We prove our result in the case of a bounded domain with smooth boundary in ${\mathbb{R}^3}$ and also in the case of the whole space ${\mathbb{R}^3}$ .     

Classical Limit for a System of Non-Linear Random Schrödinger Equations

This work is concerned with the semi-classical analysis of mixed state solutions to a Schrödinger–Position equation perturbed by a random potential with weak amplitude and fast oscillations in time and space. We show that the Wigner transform of the density matrix converges weakly and in probability to solutions of a Vlasov–Poisson–Boltzmann equation with a linear collision kernel.Aconsequence of this result is that a smooth non-linearity such as the Poisson potential (repulsive or attractive) does not change the statistical stability property of the Wigner transform observed in linear problems.We obtain, in addition, that the local density and current are self-averaging, which is of importance for some imaging problems in random media. The proof brings together the martingale method for stochastic equations with compactness techniques for non-linear PDEs in a semi-classical regime. It relies partly on the derivation of an energy estimate that is straightforward in a deterministic setting but requires the use of a martingale formulation and well-chosen perturbed test functions in the random context.     

On the characterization of breather and rogue wave solutions and modulation instability of a coupled generalized nonlinear Schrödinger equations

We construct Darboux transformation of a coupled generalized nonlinear Schrödinger (CGNLS) equations and obtain exact analytic expressions of breather and rogue wave solutions. We also formulate the conditions for isolating these solutions. We show that the rogue wave solution can be found only when the four wave mixing parameter becomes real. We also investigate the modulation instability of the steady state solution of CGNLS system and demonstrate that it can occur only when the four wave mixing parameter becomes real. Our results give an evidence for the connection between the occurrence of rogue wave solution and the modulation instability.     

Global Attractor of a Dissipative Fractional Klein Gordon Schrödinger System

Journal of Dynamics and Differential Equations - In this paper we study the local and global well posedness of a fractional dissipative Klein–Gordon–Schrödinger type system in...     

On the Dynamics of Solitons in the Nonlinear Schr?dinger Equation

We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12W_e^￠(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi},     

Analysis and comparative study of non-holonomic and quasi-integrable deformations of the nonlinear Schrödinger equation

Nonlinear Dynamics - The non-holonomic deformation of the nonlinear Schrödinger equation, uniquely obtained from both the Lax pair and Kupershmidt’s bi-Hamiltonian (Kupershmidt in Phys...     

Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach

Nonlinear Dynamics - We systematically develop a Riemann–Hilbert approach for the quartic nonlinear Schrödinger equation on the line with both zero boundary condition and nonzero...     

On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with a Quadratic Trap

We consider the dynamics of the three-dimensional N-body Schrödinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N ^3β-1 V(N ^β x). We justify the mean-field approximation and offer a rigorous derivation of the three-dimensional cubic nonlinear Schrödinger equation (NLS) with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon (Commun Math Phys 279:169–185, 2008) for ${\beta \in (0, 2/7]}$ by adapting and simplifying an argument in Chen and Pavlovi? (Annales Henri Poincaré, 2013) which solves the problem for ${\beta \in (0, 1/4)}$ in the absence of a trap.     

Breathers,solitons and rogue waves of the quintic nonlinear Schrödinger equation on various backgrounds

Nonlinear Dynamics - We investigate the generation of breathers, solitons, and rogue waves of the quintic nonlinear Schrödinger equation (QNLSE) on uniform and elliptical backgrounds. The...     

Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

Nonlinear Dynamics - All possible nonlocal versions of the derivative nonlinear Schrödinger equations are derived by the nonlocal reduction from the Chen–Lee–Liu equation, the...     

Global Behavior of Spherically Symmetric Navier–Stokes–Poisson System with Degenerate Viscosity Coefficients

In this paper, we study a free boundary problem for compressible spherically symmetric Navier–Stokes–Poisson equations with degenerate viscosity coefficients and without a solid core. Under certain assumptions that are imposed on the initial data, we obtain the global existence and uniqueness of the weak solution and give some uniform bounds (with respect to time) of the solution. Moreover, we obtain some stabilization rate estimates of the solution. The results show that such a system is stable under small perturbations, and could be applied to the astrophysics. This work is supported by NSFC 10571158, Zhejiang Provincial NSF of China (Y605076) and China Postdoctoral Science Foundation 20060400335.     

Uniqueness of Positive Ground State Solutions of the Logarithmic Schrödinger Equation

We prove the uniqueness of positive ground state solutions of the problem \({ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln(|u|) = 0}\), \({u(r) > 0~\forall r \ge 0}\), and \({(u(r),u'(r)) \to (0, 0)}\) as \({r \to \infty}\). This equation is derived from the logarithmic Schrödinger equation \({{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left(|u|^{2}\right)}\), and also from the classical equation \({{\frac {\partial u}{\partial t}} = {\Delta} u +u \left(|u|^{p-1}\right) -u}\). For each \({n \ge 1}\), a positive ground state solution is \({ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}\). We combine \({u_{0}(r)}\) with energy estimates and associated Ricatti equation estimates to prove that, for each \({n \in \left[1, 9 \right]}\), \({u_{0}(r)}\) is the only positive ground state. We also investigate the stability of \({u_{0}(r)}\). Several open problems are stated.