Inverse Scattering Transform for the Multi‐Component Nonlinear Schrödinger Equation with Nonzero Boundary Conditions

The Inverse Scattering Transform (IST) for the defocusing vector nonlinear Schrödinger equations (NLS), with an arbitrary number of components and nonvanishing boundary conditions at space infinities, is formulated by adapting and generalizing the approach used by Beals, Deift, and Tomei in the development of the IST for the N ‐wave interaction equations. Specifically, a complete set of sectionally meromorphic eigenfunctions is obtained from a family of analytic forms that are constructed for this purpose. As in the scalar and two‐component defocusing NLS, the direct and inverse problems are formulated on a two‐sheeted, genus‐zero Riemann surface, which is then transformed into the complex plane by means of an appropriate uniformization variable. The inverse problem is formulated as a matrix Riemann‐Hilbert problem with prescribed poles, jumps, and symmetry conditions. In contrast to traditional formulations of the IST, the analytic forms and eigenfunctions are first defined for complex values of the scattering parameter, and extended to the continuous spectrum a posteriori.     

Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line

Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.     

The Self-Focusing Singularity in the Nonlinear Schrödinger Equation

Under certain circumstances, solutions of the cylindrically symmetric nonlinear Schrödinger equation collapse to a singularity in a finite time. An asymptotic series for the solution near the singularity is derived here. At leading order, the central amplitude of the spike grows like[(log Δt)/Δt]^1/2, where Δt is the time remaining to the appearance of the singularity.     

Asymptotic Behavior of the Nonlinear Schrödinger Equation with Harmonic Trapping

We consider the cubic nonlinear Schrödinger equation with harmonic trapping on ?^D (1 ≤ D ≤ 5). In the case when all directions but one are trapped (aka “cigar‐shaped trap”), we prove modified scattering and construct modified wave operators for small initial and final data, respectively. The asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the NLS equation with full trapping on ?D?1. In the physical dimension D = 3, this system turns out to be exactly the (CR) equation derived by Faou, Germain, and the first author as the large box limit of the resonant NLS equation in the homogeneous (zero potential) setting. The special dynamics of the latter equation, combined with the above modified scattering results, allow us to justify and extend some physical approximations in the theory of Bose‐Einstein condensates in cigar‐shaped traps.© 2016 Wiley Periodicals, Inc.     

Effective Dynamics of the Nonlinear Schrödinger Equation on Large Domains

We consider the nonlinear Schrödinger (NLS) equation posed on the box [0,L]^d with periodic boundary conditions. The aim is to describe the long‐time dynamics by deriving effective equations for it when L is large and the characteristic size ɛ of the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in the theory of statistical physics of dispersive waves, which goes by the name of “wave turbulence.” Our main result is deriving a new equation, the continuous resonant (CR) equation, which describes the effective dynamics for large L and small ɛ over very large timescales. Such timescales are well beyond the (a) nonlinear timescale of the equation, and (b) the euclidean timescale at which the effective dynamics are given by (NLS) on ℝ^d. The proof relies heavily on tools from analytic number theory, such as a relatively modern version of the Hardy‐Littlewood circle method, which are modified and extended to be applicable in a PDE setting.© 2018 Wiley Periodicals, Inc.     

Asymptotic Solution of the Weakly Nonlinear Schrödinger Equation with Variable Coefficients

A perturbation method based on Fourier analysis and multiple scales is introduced for solving weakly nonlinear, dispersive wave propagation problems with Fourier-transformable initial conditions. Asymptotic solutions are derived for the weakly nonlinear cubic Schrödinger equation with variable coefficients, and verified by comparison with numerical solutions. In the special case of constant coefficients, the asymptotic solution agrees to leading order with previously derived results in the literature; in general, this is not true to higher orders. Therefore previous asymptotic results for the strongly nonlinear Schrödinger equation can be valid only for restricted initial conditions.     

Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

Initial‐boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so‐called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well‐posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so‐called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrödinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet‐to‐Neumann map supplied by the defocusing nonlinear Schrödinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial‐boundary value problem on the half‐line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter‐plane space‐time domain.     

Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

We characterize the long‐time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity by using a variant of the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest‐descent method of Deift and Zhou for oscillatory Riemann‐Hilbert problems. First, we formulate the IST over a single sheet of the complex plane without introducing the uniformization variable that was used by Biondini and Kova?i? in 2014. The solution of the focusing NLS equation with nonzero boundary conditions is thereby associated with a matrix Riemann‐Hilbert problem whose jumps grow exponentially with time for certain portions of the continuous spectrum. This growth is the signature of the well‐known modulational instability within the context of the IST. We then eliminate this growth by performing suitable deformations of the Riemann‐Hilbert problem in the complex spectral plane. The results demonstrate that the solution of the focusing NLS equation with nonzero boundary conditions remains bounded at all times. Moreover, we show that, asymptotically in time, the xt ‐plane decomposes into two types of regions: a left far‐field region and a right far‐field region, where the solution equals the condition at infinity to leading order up to a phase shift, and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. Finally, we show how, in the latter region, the modulus of the leading‐order solution, initially obtained as a ratio of Jacobi theta functions, can be reduced to the well‐known elliptic solutions of the focusing NLS equation. These results provide the first characterization of the long‐time behavior of generic perturbations of a constant background in a modulationally unstable medium. © 2017 Wiley Periodicals, Inc.     

On Scattering for the Quintic Defocusing Nonlinear Schrödinger Equation on R × T2

We consider the problem of large‐data scattering for the quintic nonlinear Schrödinger equation on R × T ². This equation is critical both at the level of energy and mass. Most notably, we exhibit a new type of profile (a “large‐scale profile”) that controls the asymptotic behavior of the solutions. © 2014 Wiley Periodicals, Inc.     

On the Global Behavior of Solutions of a Coupled System of Nonlinear Schrödinger Equation

We mainly study a system of two coupled nonlinear Schrödinger equations where one equation includes gain and the other one includes losses. This model constitutes a generalization of the model of pulse propagation in birefringent optical fibers. We aim in this study at partially answering a question of some authors in [1]: “Is the H¹‐norm of the solution globally bounded in the Manakov case, when ?” We found that in the Manakov case, and when , the solution stays in , and also that the H¹‐norm of the solution cannot blow up in finite time. In the Manakov case, an estimate of the total energy is provided, which is different from that has been given in [1]. These results are corroborated by numerical results that have been obtained with a finite element solver well adapted for that purpose.     

Weakly Nonlocal Solitary Waves in a Singularly Perturbed Nonlinear Schrödinger Equation

We consider the nonlinear Schrödinger equation perturbed by the addition of a third-derivative term whose coefficient constitutes a small parameter. It is known from the work of Wai et al. [1] that this singular perturbation causes the solitary wave solution of the nonlinear Schrödinger equation to become nonlocal by the radiation of small-amplitude oscillatory waves. The calculation of the amplitude of these oscillatory waves requires the techniques of exponential asymptotics. This problem is re-examined here and the amplitude of the oscillatory waves calculated using the method of Borel summation. The results of Wai et al. [1] are modified and extended.     

On the Integrability of the Poisson Driven Stochastic Nonlinear Schrödinger Equations

We consider the Cauchy problem for the dissipative nonlinear Schrödinger equations driven by a Poisson noise, namely (1) where γ_n > 0 and 0 < t₁ < ? < t_n < ? are certain sequences of random numbers and is the deterministic loss coefficient. This perturbation incorporates the possibility of sudden changes in the field that occur randomly. If Γ= 0, we prove that the resulting equation can be piece‐wise related to the unperturbed NLS equation and show how to solve the initial value problem. We also determine a complete set of conserved quantities. When Γ≠ 0 the equation is nonintegrable. Nevertheless, we determine the random evolution of physically relevant quantities like the field’s Energy E(t) ≡∫dx|u|²(t, x) and momentum. By considering a joint z‐Laplace transform we obtain the mean Energy decay. A naturally related quantity is the “half‐life”, or the time before the Energy degrades below a given value E₁. We show that the mean of this random quantity satisfies an integral equation and solve it by Laplace transformation. In particular cases we also determine the complete probability distribution of Energy and half life.     

Probabilistic Methods for Discrete Nonlinear Schrödinger Equations

We show that the thermodynamics of the focusing cubic discrete nonlinear Schrödinger equation are exactly solvable in dimension 3 and higher. A number of explicit formulas are derived. © 2012 Wiley Periodicals, Inc.     

On Schrödinger operators with multipolar inverse-square potentials

Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established.