Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic interaction of these pulses.     

Double scattering channels for 1D NLS in the energy space and its generalization to higher dimensions

We consider a class of 1D NLS perturbed with a steplike potential. We prove that the nonlinear solutions satisfy the double scattering channels in the energy space. The proof is based on concentration-compactness/rigidity method. We prove moreover that in dimension higher than one, classical scattering holds if the potential is periodic in all but one dimension and is steplike and repulsive in the remaining one.     

The nonlinear Schrödinger equation with a potential

We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.     

Spots and stripes in nonlinear Schrödinger‐type equations with nearly one‐dimensional potentials

We consider the existence of spots and stripes for a class of nonlinear Schrödinger‐type equations in the presence of nearly one‐dimensional localized potentials. Under suitable assumptions on the potential, we construct various types of waves that are localized in the direction of the potential and have single‐ or multihump, or periodic profile in the perpendicular direction. The analysis relies upon a spatial dynamics formulation of the existence problem, together with a center manifold reduction. This reduction procedure allows these waves to be realized as unipulse or multipulse homoclinic orbits, or periodic orbits in a reduced system of ordinary differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Piecewise homotopy analysis method and convergence analysis for formally well-posed initial value problems

In this paper, we apply the Schwarz waveform relaxation (SWR) method to the one-dimensional Schrödinger equation with a general linear or a nonlinear potential. We propose a new algorithm for the Schrödinger equation with time-independent linear potential, which is robust and scalable up to 500 subdomains. It reduces significantly computation time compared with the classical algorithms. Concerning the case of time-dependent linear potential or the nonlinear potential, we use a preprocessed linear operator for the zero potential case as a preconditioner which leads to a preconditioned algorithm. This ensures high scalability. In addition, some newly constructed absorbing boundary conditions are used as the transmission conditions and compared numerically.     

Multiplicity results for superlinear boundary value problems with impulsive effects

We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non‐zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

Evans Function for Lax Operators with Algebraically Decaying Potentials

We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singularities in the essential spectrum, which may include embedded eigenvalues with algebraically decaying eigenfunctions. We construct a renormalized Evans function and study bifurcations of embedded eigenvalues, when an algebraically decaying potential is perturbed by a generic potential with a faster decay at infinity. We show that the bifurcation problem for embedded eigenvalues can be reduced to cubic or quadratic equations, depending on whether the algebraic potential decays to zero or approaches a nonzero constant. Roots of the bifurcation equations define eigenvalues which correspond to nonlinear waves that are formed from unstable algebraic solitons. Our results provide precise information on the transformation of unstable algebraic solitons in the time-evolution problem associated with the integrable nonlinear equation. Algebraic solitons of the modified KdV equation are shown to transform to either travelling solitons or time-periodic breathers, depending on the sign of the perturbation. Algebraic solitons of the derivative NLS and massive Thirring equations are shown to transform to travelling and rotating solitons for either sign of the perturbation. Finally, algebraic homoclinic orbits of the focusing NLS equation are destroyed by the perturbation and evolve into time-periodic space-decaying solutions.     

Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential.     

Dissipation effects in infinite-dimensional Hamiltonian systems

We show that the potential coupling of classical mechanical systems (an oscillator and a heat bath), one of which (the heat bath) is linear and infinite-dimensional, can provoke energy dissipation in a finitedimensional subsystem (the oscillator). Under natural assumptions, the final dynamics of an oscillator thus reduces to a tendency toward equilibrium. D. V. Treschev previously obtained results concerning the dynamics of an oscillator with one degree of freedom and a quadratic or (under some additional assumptions) polynomial potential. Later, A. V. Dymov considered the case of a linear oscillator with an arbitrary (finite) number of degrees of freedom. We generalize these results to the case of a heat bath (consisting of several components) and a multidimensional oscillator (either linear or nonlinear).     

Coherent States for Systems of L 2-Supercritical Nonlinear Schrödinger Equations

We consider the propagation of wave packets for a nonlinear Schrödinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, at leading order, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.     

Coupled nonlinear waves in two-dimensional lattice

For one-dimensional nonlinear lattices, such as Toda lattice, it has been extensively studied. By considering the nonlinear effects of two-dimensional lattice, we set up the equation of motion for each particles (atoms, molecules or ions). For small amplitude and long wavelength nonlinear waves in this system, both the linear dispersion relation and the coupled Korteweg de Vries (KdV) equation are obtained. The simple soliton solution is obtained. If the nonlinear lattice is symmetric in the x and y directions, It is noted that there are two kinds of solitons. one is that propagates in either x or y directions, (1, 0) or (0, 1), the other is that propagates in the direction of (1, 1). It is in agreements with that of one-dimensional lattice. The different properties are investigated for different nonlinear interacting potentials, such as Toda potential, Morse potential and LJ potential.     

Nonlinear stability of the equilibria in a double-bar rotating system

We study the nonlinear stability of the equilibria corresponding to the motion of a particle orbiting around a two finite orthogonal straight segment. The potential is a logarithmic function and may be considered as an approximation to the one generated by irregular celestial bodies. Using Arnold’s theorem for non-definite quadratic forms we determine the nonlinear stability of the equilibria, for all values of the parameter of the problem. Moreover, the resonant cases are determined and the stability is investigated.     

Nonlinear orientational dynamics of a molecular chain

We investigate the nonlinear rotational dynamics of a molecular chain with quadrupole interaction in both the discrete and the continuous cases. Based on a system of nonlinear differential-difference equations, we obtain approximate equations describing the chain excitations and preserving the initial symmetry. We introduce an effective potential and normal coordinates, using which allows decoupling the system into linear and nonlinear parts. As a result of a strong anisotropy of the potential, narrow “valleys” occur in the angle plane. Motion along a valley corresponds to a softer interaction (nonlinear equations). Linear equations describe motion across a valley (hard interaction). We consider cases where the derived nonlinear equations reduce to the sine-Gordon equation. We find integrals of motion and exact solutions of our approximate equations. We uniformly describe the energy interval encompassing the domains of order, of orientational melting, and of rotational motion of the molecules in the chain.     

Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions

Split-step orthogonal spline collocation (OSC) methods are proposed for one-, two-, and three-dimensional nonlinear Schrödinger (NLS) equations with time-dependent potentials. Firstly, the NLS equation is split into two nonlinear equations, and one or more one-dimensional linear equations. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, we propose three approximations by using quadrature formulae, but the split order is not reduced. Discrete-time OSC schemes are applied for the linear subproblems. In numerical experiments, many tests are carried out to prove the reliability and efficiency of the split-step OSC (SSOSC) methods. Solitons in one, two, and three dimensions are well simulated, and conservative properties and convergence rates are demonstrated. We also apply the ways of solving the nonlinear subproblems to the split-step finite difference (SSFD) methods and the time-splitting spectral (TSSP) methods, and the approximate ways still work well. Finally, we apply the SSOSC methods to solve some problems of Bose-Einstein condensates.     

Non-asymptotic and potential Landesman-Lazer conditions for a nonlinear beam equation

Existence of solutions for a nonlinear fourth order ordinary differential equation arising in beam theory is considered. We obtain solutions by a degree argument under a non-asymptotic condition on the nonlinear terms of the problem. Moreover, assuming a potential Landesman-Lazer condition, we prove the existence of at least one solution by variational methods.     

Pulsive feedback control of a quarter car model forced by a road profile

We examine the strange chaotic attractor and its unstable periodic orbits for a one degree of freedom nonlinear oscillator with a non-symmetric potential that models a quarter car forced by the road profile. We propose an efficient method of chaos control that stabilizes these orbits using a pulsive feedback technique. A discrete set of pulses is able to transfer the system from one periodic state to another.     

A sufficient optimality theoriem for protoconvex programs

We establish two first order sufficient optimality theorems; one for unconstrained nonlinear minimization problem, and the other for constrained nonlinear minimization problems, both with non-differentiable protoconvex or quasiconvex data functions that are not necessarily locally Lipschitz     

Stability of Manifold-Valued Subdivision Schemes and Multiscale Transformations

Linear subdivision schemes can be adapted in various ways so as to operate in nonlinear geometries such as Lie groups or Riemannian manifolds. It is well known that along with a linear subdivision scheme a multiscale transformation is defined. Such transformations can also be defined in a nonlinear setting. We show the stability of such nonlinear multiscale transforms. To do this we introduce a new kind of proximity condition which bounds the difference of the differential of a nonlinear subdivision scheme and a linear one. It turns out that—unlike the generic nonlinear case and modulo some minor technical assumptions—in the manifold-valued setting, convergence implies stability of the nonlinear subdivision scheme and associated nonlinear multiscale transformations.     

Remarks on Nonlinear Schrödinger Equations with Harmonic Potential

Bose-Einstein condensation is usually modeled by nonlinear Schrödinger equations with harmonic potential. We study the Cauchy problem for these equations. We show that the local problem can be treated as in the case with no potential. For the global problem, we establish an evolution law, which is the analogue of the pseudo-conformal conservation law for the nonlinear Schrödinger equation. With this evolution law, we give wave collapse criteria, as well as an upper bound for the blow up time. Taking the physical scales into account, we finally give a lower bound for the breaking time. This study relies on two explicit operators, suited to nonlinear Schrödinger equations with harmonic potential, already known in the linear setting.