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.     

Long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions

The long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions at infinity is investigated by the nonlinear steepest descent method of Deift and Zhou. Three asymptotic sectors in space–time plane are found: the plane wave sector I, plane wave sector II and an intermediate sector with a modulated one-phase elliptic wave. The asymptotic solutions of the three sectors are proposed by successively deforming the corresponding Riemann–Hilbert problems to solvable model problems. Moreover, a time-dependent g-function mechanism is introduced to remove the exponential growths of the jump matrices in the modulated one-phase elliptic wave sector. Finally, the modulational instability is studied to reveal the criterion for the existence of modulated elliptic waves in the central region.     

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.     

Inverse Scattering Transform for the Nonlocal Reverse Space–Time Nonlinear Schrödinger Equation

Nonlocal reverse space–time equations of the nonlinear Schrödinger (NLS) type were recently introduced. They were shown to be integrable infinite-dimensional dynamical systems, and the inverse scattering transform (IST) for rapidly decaying initial conditions was constructed. Here, we present the IST for the reverse space–time NLS equation with nonzero boundary conditions (NZBCs) at infinity. The NZBC problem is more complicated because the branching structure of the associated linear eigenfunctions is complicated. We analyze two cases, which correspond to two different values of the phase at infinity. We discuss special soliton solutions and find explicit one-soliton and two-soliton solutions. We also consider spatially dependent boundary conditions.     

Bound-state soliton and rogue wave solutions for the sixth-order nonlinear Schrödinger equation via inverse scattering transform method

In this work, inverse scattering transform for the sixth-order nonlinear Schrödinger equation with both zero and nonzero boundary conditions at infinity is given, respectively. For the case of zero boundary conditions, in terms of the Laurent's series and generalization of the residue theorem, the bound-state soliton is derived. For nonzero boundary conditions, using the robust inverse scattering transform, we present a matrix Riemann–Hilbert problem of the sixth-order nonlinear Schrödinger equation. Then, based on the obtained Riemann–Hilbert problem, the rogue wave solutions are derived through a modified Darboux transformation. Besides, according to some appropriate parameters choices, several graphical analysis are provided to discuss the dynamical behaviors of the rogue wave solutions and analyze how the higher-order terms affect the rogue wave.     

The Derivative Nonlinear Schrödinger Equation with Zero/Nonzero Boundary Conditions: Inverse Scattering Transforms and N-Double-Pole Solutions

In this paper, we report a rigorous theory of the inverse scattering transforms (ISTs) for the derivative nonlinear Schrödinger (DNLS) equation with both zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs) at infinity and double zeros of analytical scattering coefficients. The scattering theories for both ZBCs and NZBCs are addressed. The direct scattering problem establishes the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix, and properties of discrete spectra. The inverse scattering problems are formulated and solved with the aid of the matrix Riemann–Hilbert problems, and the reconstruction formulae, trace formulae and theta conditions are also posed. In particular, the IST with NZBCs at infinity is proposed by a suitable uniformization variable, which allows the scattering problem to be solved on a standard complex plane instead of a two-sheeted Riemann surface. The reflectionless potentials with double poles for the ZBCs and NZBCs are both carried out explicitly by means of determinants. Some representative semi-rational bright–bright soliton, dark–bright soliton, and breather–breather solutions are examined in detail. These results and idea can also be extended to other types of DNLS equations such as the Chen–Lee–Liu-type DNLS equation, Gerdjikov–Ivanov-type DNLS equation, and Kundu-type DNLS equation and will be useful to further explore and apply the related nonlinear wave phenomena.

     

Semiclassical Limit of Focusing NLS for a Family of Square Barrier Initial Data

The small dispersion limit of the focusing nonlinear Schrödinger equation (NLS) exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales. The non‐self‐adjoint scattering problem and ill‐posed limiting Whitham equations associated to focusing NLS make rigorous asymptotic results difficult. Previous studies have focused on special classes of analytic initial data for which the limiting elliptic Whitham equations are wellposed. In this paper we consider another exactly solvable family of initial data,the family of square barriers,ψ 0(x) = qχ[?L,L] for real amplitudes q. Using Riemann‐Hilbert techniques, we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing NLS globally in space and up to an O(1) maximal time. In particular, we show that the discontinuities in our initial data regularize by the immediate generation of genus‐one oscillations emitted into the support of the initial data. To the best of our knowledge, this is the first case in which the genus structure of the semiclassical asymptotics for focusing NLS have been calculated for nonanalytic initial data. © 2013 Wiley Periodicals, Inc.     

Riemann–Hilbert problem for the modified Landau–Lifshitz equation with nonzero boundary conditions

Theoretical and Mathematical Physics - We study a matrix Riemann–Hilbert (RH) problem for the modified Landau–Lifshitz (mLL) equation with nonzero boundary conditions at infinity. In...     

Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems

The effective and efficient numerical solution of Riemann‐Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann‐Hilbert problems, the resulting numerical methods have been shown, in practice, to retain accuracy as values of certain parameters become arbitrarily large. Remarkably, this numerical approach does not require knowledge of local parametrices; rather, the deformed contour is scaled near stationary points at a specific rate. The primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved. To do so, we first construct a general theoretical framework for the numerical solution of Riemann‐Hilbert problems. Second, we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes. In particular, we prove sufficient conditions for numerical methods to retain accuracy. Finally, we compute solutions to the homogeneous Painlevé II equation and the modified Korteweg–de Vries equation to explicitly demonstrate the practical validity of the theory. © 2014 Wiley Periodicals, Inc.     

Riemann–Hilbert problem for the Kundu-type nonlinear Schrödinger equation with $$N$$ distinct arbitrary-order poles

Theoretical and Mathematical Physics - We use the Riemann–Hilbert (RH) method to study the Kundu-type nonlinear Schrödinger (Kundu–NLS) equation with a zero boundary condition in...     

Initial Boundary Value Problems for Scalar and Vector Burgers Equations

In this article we study Burgers equation and vector Burgers equation with initial and boundary conditions. First we consider the Burgers equation in the quarter plane x >0, t >0 with Riemann type of initial and boundary conditions and use the Hopf–Cole transformation to linearize the problems and explicitly solve them. We study two limits, the small viscosity limit and the large time behavior of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gradient of a scalar function. We investigate the asymptotic behavior of this solution as time tends to infinity and generalize a result of Hopf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an initial value problem extending the previous work of Sachdev et al. (1994). We also study the limit as viscosity parameter goes to 0.Finally, we get an explicit solution for a boundary value problem in a cylinder.     

A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary-order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher-order “rogue wave” solutions in an inverse-scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc.     

The Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation: Long‐time dynamics of nonzero boundary conditions

We consider the Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation on the line. The initial value q(x,0) is given and satisfies the symmetric, nonzero boundary conditions at infinity, that is, q(x,0)→q_± as x→±∞, and |q_±|=q₀>0. The goal of this paper is to study the asymptotic behavior of the solution of this initial value problem as t→∞. The main tool is the asymptotic analysis of an associated matrix Riemann‐Hilbert problem by using the steepest descent method and the so‐called g‐function mechanism. We show that the solution q(x,t) of this initial value problem has a different asymptotic behavior in different regions of the xt‐plane. In the regions and , the solution takes the form of a plane wave. In the region , the solution takes the form of a modulated elliptic wave.     

N solutions for a derivative nonlinear Schrödinger‐type equation via Riemann‐Hilbert approach

The inverse scattering transform for the derivative nonlinear Schrödinger‐type equation is studied via the Riemann‐Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann‐Hilbert problem is established for the derivative nonlinear Schrödinger‐type equation. In the inverse scattering process, N‐soliton solutions of the derivative nonlinear Schrödinger‐type equation are obtained by solving Riemann‐Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.     

Long-time behavior for a nonlinear fourth-order parabolic equation

We study the asymptotic behavior of solutions of the initial- boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.

     

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit

In the small‐dispersion limit, solutions to the Korteweg—de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg—de Vries solution near the leading edge of the oscillatory zone up to second‐order corrections. This expansion involves the Hastings‐McLeod solution of the Painlevé II equation. We prove our results using the Riemann‐Hilbert approach. © 2009 Wiley Periodicals, Inc.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

Reverse Space‐Time Nonlocal Sine‐Gordon/Sinh‐Gordon Equations with Nonzero Boundary Conditions

Nonlocal reverse space‐time Sine/Sinh‐Gordon type equations were recently introduced. They arise from a remarkably simple nonlocal reduction of the well‐known AKNS scattering problem, hence, they constitute an integrable evolution equations. Furthermore, the inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, the IST for these novel nonlocal equations corresponding to nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complex due to the intricate branching structure of the associated linear eigenfunctions. Two cases are analyzed, which correspond to two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1‐soliton and 2‐soliton solutions are found. Both spatially independent and spatially dependent boundary conditions are considered.     

Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

The semiclassical (zero‐dispersion) limit of solutions $q=q(x,t,\epsilon)$ to the one‐dimensional focusing nonlinear Schrödinger equation (NLS) is studied in a scaling neighborhood D of a point of gradient catastrophe ($x_0,t_0$) . We consider a class of solutions, specified in the text, that decay as $|x| \rightarrow \infty$ . The neighborhood D contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast‐amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions q near the point of gradient catastrophe: (i) each spike has height $3|q{_0}(x_0,t_0)|$ and uniform shape of the rational breather solution to the NLS, scaled to the size ${\cal O}(\epsilon)$ ; (ii) the location of the spikes is determined by the poles of the tritronquée solution of the Painlevé I (P1) equation through an explicit map between D and a region of the Painlevé independent variable; (iii) if $(x,t)\in D$ but lies away from the spikes, the asymptotics of the NLS solution $q(x,t, \epsilon)$ is given by the plane wave approximation $q_0(x,t, \epsilon)$ , with the correction term being expressed in terms of the tritronquée solution of P1. The relation with the conjecture of Dubrovin, Grava, and Klein about the behavior of solutions to the focusing NLS near a point of gradient catastrophe is discussed. We conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. Our technique is based on the nonlinear steepest‐descent method for matrix Riemann‐Hilbert problems and discrete Schlesinger isomonodromic transformations. © 2013 Wiley Periodicals, Inc.