Inverse Scattering Transform for the Focusing Ablowitz‐Ladik System with Nonzero Boundary Conditions

The purpose of this paper is to construct the inverse scattering transform for the focusing Ablowitz‐Ladik equation with nonzero boundary conditions at infinity. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann‐Hilbert problem on a doubly connected curve in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz‐Ladik potential.     

Inverse Scattering Transform and Solitons for Square Matrix Nonlinear Schrödinger Equations

The inverse scattering transform (IST) is developed for a class of matrix nonlinear Schrödinger‐type systems whose reductions include two equations that model certain hyperfine spin spinor Bose–Einstein condensates, and two novel equations that were recently shown to be integrable, and that have applications in nonlinear optics and four‐component fermionic condensates. In addition, the general behavior of the soliton solutions for all four reductions is analyzed in detail, and some novel solutions are presented.     

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 the Nonlinear Schrödinger Equation on the Half Line with Homogeneous Robin Boundary Conditions

Boundary value problems for the nonlinear Schrödinger equations on the half line with homogeneous Robin boundary conditions are revisited using Bäcklund transformations. In particular: relations are obtained among the norming constants associated with symmetric eigenvalues; a linearizing transformation is derived for the Bäcklund transformation; the reflection‐induced soliton position shift is evaluated and the solution behavior is discussed. The results are illustrated by discussing several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self‐symmetric eigenvalues.     

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.     

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.     

On the Inverse Scattering of the Time-Dependent Schrödinger Equation and the Associated Kadomtsev-Petviashvili (I) Equation

The Kadomtsev-Petviashvili equation, a two-spatial-dimensional analogue of the Korteweg-deVries equation, arises in physical situations in two different forms depending on a certain sign appearing in the evolution equation. Here we investigate one of the two cases. The initial-value problem, associated with initial data decaying sufficiently rapidly at infinity, is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation of a nonlocal Riemann-Hilbert problem in terms of scattering data expressible in closed form in terms of given initial data. The lump solutions, algebraically decaying solitons, are given a definite spectral characterization. Pure lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t. Many of the above results are also relevant to the problem of inverse scattering for the so-called time-dependent Schrödinger equation.     

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.     

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.     

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.     

Scattering for Schrödinger Operators with Magnetic Fields

We study the existence and completeness of the wave operators W_ω(A(b),-Δ) for general Schrodinger operators of the form is a magnetic potential.     

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.     

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.     

A Generalization of the Direct and Inverse Problem for the Radial Schrödinger Equation

We consider the operators H₀= ?d²/dr² and H₁ = ?d²/dr² + V(r) (0< r< ∞) acting on a Hilbert space of complex functions f(r) such that the subspaces in which the operators are defined consist of twice differentiable functions which satisfy the boundary condition (d/dr)f(0) = αf(0). H₁ and H₀ are Hermitian in this subspace. Assuming V(r)→0 as r→∞ sufficiently rapidly, the scattering operator formalism is set up for the direct scattering problem. Next we consider the inverse problem of determining V(r) from H₀ and the spectral measure function for the spectrum of H₁ through the use of an appropriate Gelfand-Levitan equation. It is shown that generally the value of α associated with H₁ differs from that for H₀, i.e., H₁ and H₀ generally operate in different subspaces. Thus scattering cannot be defined. However, by changing the spectral measure function, one obtains a new Gelfand-Levitan equation such that H₁ is the same as before [i.e., α and V(r) are the same] from the operator H₀, which uses the same value of α as H₁. Thus H₁ and the new H₀ operate in the same subspace of Hilbert space, and scattering can be defined. The process of obtaining the new H₀ after finding H₁ from the old H₀ is somewhat analogous to renormalization in field theory, where a new H₀ is picked to have properties compatible with H₁. A necessary and sufficient condition on the spectral data is given which makes the domains of H₀ and H₁ coincide and thus makes “renormalization” unnecessary. The direct problem is a generalization of the usual l=0 radial Schrödinger equation. The inverse problem is a generalization of the corresponding inverse problem. It is also a generalization of the case α=0 for H₀ considered by Gelfand and Levitan in their early work on the inverse spectral problem. An incompletely understood connection of the inverse problem for the radial equation to solutions of the Korteweg-deVries equation in the half space is discussed. The existence of such a connection is one of the motivations for studying the generalized radial Schrodinger equation.     

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

For the following singularly perturbed problem we construct a solution which concentrates at several given isolated positive local minimum components of V as . Here, the nonlinearity f is of critical growth. Moreover, the monotonicity of and the so‐called Ambrosetti–Rabinowitz condition are not required.     

The Inverse Scattering Transform for the Benjamin–Ono Equation

We extend the inverse scattering transform (IST) for the Benjamin–Ono (BO) equation, given by A. S. Fokas and M. J. Ablowitz ( Stud. Appl. Math. 68:1, 1983), in two important ways. First, we restrict the IST to purely real potentials, in which case the scattering data and the inverse scattering equations simplify. Second, we extend the analysis of the asymptotics of the Jost functions and the scattering data to include the nongeneric classes of potentials, which include, but may not be limited to, all N -soliton solutions. In the process, we also study the adjoint equation of the eigenvalue problem for the BO equation, from which, for real potentials, we find a very simple relation between the two reflection coefficients (the functions β(λ) and f (λ)) introduced by Fokas and Ablowitz. Furthermore, we show that the reflection coefficient also defines a phase shift, which can be interpreted as the phase shift between the left Jost function and the right Jost function. This phase shift leads to an analogy of Levinson's theorem, as well as a condition on the number of possible bound states that can be contained in the initial data. For both generic and nongeneric potentials, we detail the asymptotics of the Jost functions and the scattering data. In particular, we are able to give improved asymptotics for nongeneric potentials in the limit of a vanishing spectral parameter. We also study the structure of the scattering data and the Jost functions for pure soliton solutions, which are examples of nongeneric potentials. We obtain remarkably simple solutions for these Jost functions, and they demonstrate the different asymptotics that nongeneric potentials possess. Last, we show how to obtain the infinity of conserved quantities from one of the Jost functions of the BO equation and how to obtain these conserved quantities in terms of the various moments of the scattering data.     

Scattering theory below energy for the cubic fourth‐order Schrödinger equation

We investigate the global existence and scattering for the cubic fourth‐order Schrödinger equation in a low regularity space with . We provide an alternative approach to obtain a new interaction Morawetz estimate which extends the range of the dimension of the interaction Morawetz estimate in Pausader 29 . We utilize the interaction Morawetz estimates and the I‐method to prove the global well‐posed and scattering result.     

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.     

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.