Stochastic Nonlinear Schrodinger Equation Research Articles

Strichartz and Local Smoothing Estimates for Stochastic Dispersive Equations with Linear Multiplicative Noise

We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schrodinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both the conservative and nonconservative case. In particular, we obtain the P-integrability of constants in these estimates, where P is the underlying probability measure. Several applications are given to nonlinear problems, including local well-posedness of stochastic nonlinear Schrodinger equations with variable coefficients and lower order perturbations, integrability of global solutions to stochastic nonlinear Schrodinger equations with constant coefficients. As another consequence, we prove as well the large deviation principle for the small noise asymptotics.

Формирование неклассических многофотонных состояний света со сжатыми квантовыми флуктуациями в волокнах из модифицированного висмутом теллуритного стекла

An optical fiber with a high Kerr nonlinearity coefficient is proposed and produced from a bismuth-modified tellurite glass for creation of nonclassical multiphoton states of light. Specifically, we propose to use those fibers to squeeze the quantum fluctuations of one of the quadratures of light in the 20 W signal significantly below -10 dB compared to the standard quantum noise limit, which is important for various practical applications. Using numerical modeling based on stochastic nonlinear Schrodinger equation, we demonstrate noise squeezing stronger than -16 dB for lengths of tellurite fiber 6-14 m, while squeezing of -14 dB is expected in silica fibers for lengths of 120-300 m. Analytical formulas were used to analyse the physical factors which limit the squeezing achievable.

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

We study optimal control problems for stochastic nonlinear Schrodinger equations in both the mass subcritical and critical case. For general initial data of the minimal $$L^2$$ regularity, we prove the existence and first order Lagrange condition of an open loop control. In particular, these results apply to the stochastic nonlinear Schrodinger equations with the critical quintic and cubic nonlinearities in dimensions one and two, respectively. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type $$U^2$$ and $$V^2$$ , adapted to evolution operators related to linear Schrodinger equations with lower order perturbations. These estimates yield a new temporal regularity of (backward) stochastic solutions, which is crucial for the tightness of approximating controls induced by Ekeland’s variational principle.

Study on the behavior of weakly nonlinear water waves in the presence of random wind forcing

Specific solutions of the nonlinear Schrodinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schrodinger equation is obtained for the case of wind-forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schrodinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather-type solutions such as the Peregrine breather occur even in strong gusty wind conditions.

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

We construct a martingale solution of the stochastic nonlinear Schrodinger equation (NLS) with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in $$H^1$$ on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood–Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of cadlag functions and Jakubowski’s generalization of the Skorohod-Theorem to nonmetric spaces.

Blow-up for the stochastic nonlinear Schrödinger equations with quadratic potential and additive noise

We study the dynamics of a stochastic nonlinear Schrodinger equation with both a quadratic potential and an additive noise. We show that in both cases of repulsive potential and attractive one, any initial data with finite variance gives birth to a solution that blows up in arbitrarily small time. This is in contrast to the deterministic case when the potential is repulsive, where strong potentials could prevent the solutions from blowing up. Our result hence indicates that the additive noise rather than the potential dominates the dynamical behaviors of the solutions to the stochastic nonlinear Schrodinger equations.

Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach

We prove well-posedness results for stochastic nonlinear Schrodinger equations with linear multiplicative Wiener noise, including the nonconservative case. Our approach is different from the standard literature on stochastic nonlinear Schrodinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrodinger equation with lower-order terms and treat the resulting equation by a fixed point argument based on generalizations of Strichartz estimates proved by Marzuola et al. (J Funct Anal 255(6):1479–1553, 2008). This approach makes it possible to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrodinger equation. In contrast to the latter, we obtain well-posedness in the full range \([1, 1 + 4/d)\) of admissible exponents in the nonlinear part (where \(d\) is the dimension of the underlying Euclidean space), i.e., in exactly the same range as in the deterministic case.

Weak Second Order Multirevolution Composition Methods for Highly Oscillatory Stochastic Differential Equations with Additive or Multiplicative Noise

We introduce a class of numerical methods for highly oscillatory systems of stochastic differential equations with general noncommutative noise. We prove global weak error bounds of order two uniformly with respect to the stiffness of the oscillations, which permits to use large time steps. The approach is based on the micro-macro framework of multi-revolution composition methods recently introduced for deterministic problems and inherits its geometric features, in particular to design integrators preserving exactly quadratic first integral. Numerical experiments, including the stochastic nonlinear Schrodinger equation with space-time multiplicative noise, illustrate the performance and versatility of the approach.

Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation

In this article we analyze the error of a semidiscrete scheme for the stochastic nonlinear Schrodinger equation with power nonlinearity. We consider supercritical or subcritical nonlinearity and the equation can be either focusing or defocusing. Allowing sufficient spatial regularity we prove that the numerical scheme has strong order $\frac 12$ in general and order 1 if the noise is additive. Furthermore, we also prove that the weak order is always 1.

Stochastic Nonlinear Schrödinger Equations Driven by a Fractional Noise. Well-Posedness, Large Deviations and Support

We consider stochastic nonlinear Schrodinger equations driven by an additive noise. The noise is fractional in time with Hurst parameter $H \in (0,1)$ and colored in space with a nuclear space correlation operator. We study local well-posedness. Under adequate assumptions on the initial data, the space correlations of the noise and for some saturated nonlinearities, we prove sample path large deviations and support results in a space of Holder continuous in time until blow-up paths. We consider Kerr nonlinearities when $H > 1/2$.

Ergodicity for a weakly damped stochastic non-linear Schrödinger equation

We study a damped stochastic non-linear Schrodinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markov transition semi-group toward a unique invariant probability measure. This kind of method was originally developed to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schrodinger equation in the one-dimensional cubic case. We prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power.

A semi-discrete scheme for the stochastic nonlinear Schr�dinger equation

We study the convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrodinger equation. The nonlinear term is a power law and the noise is multiplicative with a Stratonovich product. Our scheme is implicit in the deterministic part of the equation as is usual for conservative equations. We also use an implicit discretization of the noise which is better suited to Stratonovich products. We consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate. However, in the semi discrete case, no Ito formula is available and we have to use a discrete form of this tool. Also, in the course of the proof we need to introduce a cut-off of the diffusion coefficient, which allows to treat the nonlinearity. Then, we prove convergence by a compactness argument. Due to the presence of noise and to the implicit discretization of the noise, this is rather complicated and technical. We finally obtain convergence of the discrete solutions in various topologies.

Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber

Broadband supercontinuum spectra are generated in a microstructured fiber using femtosecond laser pulses. Noise properties of these spectra are studied through experiments and numerical simulations based on a generalized stochastic nonlinear Schrodinger equation. In particular, the relative intensity noise as a function of wavelength across the supercontinuum is measured over a wide range of input pulse parameters, and experimental results and simulations are shown to be in good quantitative agreement. For certain input pulse parameters, amplitude fluctuations as large as 50% are observed. The simulations clarify that the intensity noise on the supercontinuum arises from the amplification of two noise inputs during propagation – quantum-limited shot noise on the input pulse, and spontaneous Raman scattering in the fiber. The amplification factor is a sensitive function of the input pulse parameters. Short input pulses are critical for the generation of very broad supercontinua with low noise.

Dispersion-managed solitons in a periodically and randomly inhomogeneous birefringent optical fiber

The propagation of dispersion-managed vector solitons in optical fibers with periodic and random birefringence is studied. With the help of a variational approach, the equations that describe the evolution of pulse parameters are derived. Numerical modeling is performed for variational equations and for fully coupled periodic and stochastic nonlinear Schrodinger equations. It is shown that variational equations can be effectively used to describe the averaged dynamics of dispersion-managed vector solitons with stochastic perturbations. It is shown, analytically and numerically, that dispersion-managed (DM) solitons have the same resistance to random birefringence as do ordinary solitons. The dependence of the mean decay length of a DM vector soliton on the strength of random birefringence and on the energy of the initial pulse is found.

A Stochastic Nonlinear Schrödinger Equation¶with Multiplicative Noise

We study a conservative stochastic nonlinear Schrodinger equation with a multiplicative noise. We show the global existence and uniqueness of square integrable solutions for subcritical nonlinearities, the critical exponent being the same, in dimension 1 or 2, as the critical exponent of the deterministic equation.

Two approaches to coupling classical and quantum variables

We address the issue of coupling variables whichare essentially classical to variables that are quantum.Two approaches are discussed. In the first, continuousquantum measurement theory is used to construct a phenomenological description of theinteraction of a quasiclassical variable X with aquantum variable x, where the quasiclassical nature ofX is assumed to have come about as a result ofdecoherence. The state of the quantum subsystem evolvesaccording to the stochastic nonlinear Schrodingerequation of a continuously measured system, and theclassical system couples to a stochastic c-number\(\overline x \left( t \right)\) representing the imprecisely measured value of x. The theorygives intuitively sensible results even when the quantumsystem starts out in a superposition of well-separatedlocalized states. The second approach involves a derivation of an effective theory from theunderlying quantum theory of the combinedquasiclassical-quantum system, and uses the decoherenthistories approach to quantum theory.

Coupling Classical and Quantum Variables using Continuous Quantum Measurement Theory

We propose a system of equations to describe the interaction of a quasiclassical variable $X$ with a set of quantum variables $x$ that goes beyond the usual mean field approximation. The idea is to regard the quantum system as continuously and imprecisely measured by the classical system. The effective equations of motion for the classical system therefore consist of treating the quantum variable $x$ as a stochastic c-number $\x (t) $ the probability distibution for which is given by the theory of continuous quantum measurements. The resulting theory is similar to the usual mean field equations (in which $x$ is replaced by its quantum expectation value) but with two differences: a noise term, and more importantly, the state of the quantum subsystem evolves according to the stochastic non-linear Schrodinger equation of a continuously measured system. In the case in which the quantum system starts out in a superposition of well-separated localized states, the classical system goes into a statistical mixture of trajectories, one trajectory for each individual localized state.

Quantum state diffusion, density matrix diagonalization, and decoherent histories: A model.

We analyze the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival. The QSD picture exploits mathematical connection between the usual Markovian master equation for the evolution of the density operator and a class of stochastic nonlinear Schr\"odinger equations (Ito equations) for a pure state \ensuremath{\Vert}\ensuremath{\psi}〉, and appears to supply a good description of individual systems and processes. We find approximate stationary solutions to the Ito equation (exact, for the case of quadratic potentials). The solutions are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show, for quadratic potentials, that every initial state approaches these stationary solutions in the long time limit. We recover the density operator corresponding to these solutions, and thus show, for this particular model, that the QSD picture effectively supplies a prescription for approximately diagonalizing the density operator in a basis of phase space localized states. We show that the rate of localization is related to the decoherence time, and also to the time scale on which thermal and quantum fluctuations become comparable. We use these results to exemplify the general connection between the QSD picture and the decoherent histories approach to quantum mechanics, discussed previously by Di\'osi, Gisin, Halliwell, and Percival. \textcopyright{} 1995 The American Physical Society.

Inhomogenious Model for DNA Dynamics

The Peyrard-Bishop model for DNA dynamics is generalized so as to include the differences among bases. The model reduces to a nonlinear Schrodinger type equation, which we name a stochastic nonlinear Schrodinger equation. Propagations of solitons are analysed under an assumption that masses and spring-constants obey Gaussian white noises.

Propagation of Solitons in Random Lattices

Wave propagations in nonlinear lattices with random mass distribution are investigated. The following two cases which have important applications are studied; 1) slowly varying waves in a lattice with a cubic nonlinearity, and 2) carrier wave modulations in a lattice with quadratic and cubic nonlinearities. The former and the latter cases reduces to stochastic modified Korteweg-de Vries equation and stochastic nonlinear Schrodinger equation, respectively. Propagations of solitons are analysed for both cases under an assumption that mass distribution is Gaussian and white.