Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.     

The Davey-Stewartson Equation on the Half-Plane

The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrödinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI.     

Breathers and solitons for the coupled nonlinear Schr?dinger system in three-spine α-helical protein

We mainly investigate the variable-coefficient 3-coupled nonlinear Schr?dinger(NLS) system, which describes soliton dynamics in the three-spine α-helical protein with inhomogeneous effect. The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly. The Hirota method is used to solve the constant coefficient NLS equation, and then we get the one-and two-breather solutions of the variable-coefficient NLS equation. The results show that, in the background of plane waves and periodic waves, the breather can be transformed into some forms of combined soliton solutions. The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail. Our results are helpful to study the soliton dynamics inα-helical protein.     

The Three-Component Defocusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions

We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrödinger (NLS) equation with initial conditions approaching constant values with the same amplitude as \({x\to\pm\infty}\). The theory combines and extends to a problem with non-zero boundary conditions three fundamental ideas: (i) the tensor approach used by Beals, Deift and Tomei for the n-th order scattering problem, (ii) the triangular decompositions of the scattering matrix used by Novikov, Manakov, Pitaevski and Zakharov for the N-wave interaction equations, and (iii) a generalization of the cross product via the Hodge star duality, which, to the best of our knowledge, is used in the context of the IST for the first time in this work. The combination of the first two ideas allows us to rigorously obtain a fundamental set of analytic eigenfunctions. The third idea allows us to establish the symmetries of the eigenfunctions and scattering data. The results are used to characterize the discrete spectrum and to obtain exact soliton solutions, which describe generalizations of the so-called dark-bright solitons of the two-component NLS equation.     

Symbolic-computation construction of transformations for a more generalized nonlinear Schrödinger equation with applications in inhomogeneous plasmas,optical fibers,viscous fluids and Bose-Einstein condensates

Currently, the variable-coefficient nonlinear Schr?dinger (NLS)-typed models have attracted considerable attention in such fields as plasma physics, nonlinear optics, arterial mechanics and Bose-Einstein condensates. Motivated by the recent work of Tian et al. [Eur. Phys. J. B 47, 329 (2005)], this paper is devoted to finding all the cases for a more generalized NLS equation with time- and space-dependent coefficients to be mapped onto the standard one. With the computerized symbolic computation, three transformations and relevant constraint conditions on the coefficient functions are obtained, which turn out to be more general than those previously published in the literature. Via these transformations, the Lax pairs are also derived under the corresponding conditions. For physical applications, our transformations provide the feasibility for more currently-important inhomogeneous NLS models to be transformed into the homogeneous one. Applications of those transformations to several example models are illustrated and some soliton-like solutions are also graphically discussed.     

Solitary waves in the resonant nonlinear Schrödinger equation: Stability and dynamical properties

The stability and dynamical properties of the so-called resonant nonlinear Schrödinger (RNLS) equation, are considered. The RNLS is a variant of the nonlinear Schrödinger (NLS) equation with the addition of a perturbation used to describe wave propagation in cold collisionless plasmas. We first examine the modulational stability of plane waves in the RNLS model, identifying the modifications of the associated conditions from the NLS case. We then move to the study of solitary waves with vanishing and nonzero boundary conditions. Interestingly the RNLS, much like the usual NLS, exhibits both dark and bright soliton solutions depending on the relative signs of dispersion and nonlinearity. The corresponding existence, stability and dynamics of these solutions are studied systematically in this work.     

On Solutions of the Integrable Boundary Value Problem for KdV Equation on the Semi-Axis

This paper is concerned with the Korteweg–de Vries (KdV) equation on the semi-axis. The boundary value problem with inhomogeneous integrable boundary conditions is studied. We establish some characteristic properties of solutions of the problem. Also we construct a wide class of solutions of the problem using the inverse spectral method.     

Soliton and Rogue Wave Solution of the New Nonautonomous Nonlinear Schrödinger Equation

In this paper, a new type (or the second type) of transformation which is used to map the variable coefficient nonlinear Schrödinger (VCNLS) equation to the usual nonlinear Schrödinger (NLS) equation is given. As a special case, a new kind of nonautonomous NLS equation with a t-dependent potential is
introduced. Further, by using the new transformation and making full use of the known soliton and rogue wave solutions of the usual NLS equation, the corresponding kinds of solutions of a special model of the new nonautonomous NLS equation are discussed respectively. Additionally, through using the new transformation, a new expression, i.e., the non-rational formula, of the rogue wave of a special VCNLS equation is given analytically.
The main differences between the two types of transformation mentioned above are listed by three items.     

Darboux Transformations,Higher-Order Rational Solitons and Rogue Wave Solutions for a(2+1)-Dimensional Nonlinear Schrdinger Equation

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.     

Transformation from the nonautonomous to standard NLS equations

In this paper we show a systematical method to obtain exact solutions of the nonautonomous nonlinear Schrödinger (NLS) equation. An integrable condition is first obtained by the Painlevé analysis, which is shown to be consistent with that obtained by the Lax pair method. Under this condition, we present a general transformation, which can directly convert all allowed exact solutions of the standard NLS equation into the corresponding exact solutions of the nonautonomous NLS equation. The method is quite powerful since the standard NLS equation has been well studied in the past decades and its exact solutions are vast in the literature. The result provides an effective way to control the soliton dynamics. Finally, the fundamental bright and dark solitons are taken as examples to demonstrate its explicit applications.     

Solutions for the propagation of light in nonlinear optical media with spatially inhomogeneous nonlinearities

In this paper, we present solutions for the nonlinear Schrödinger (NLS) equation with spatially inhomogeneous nonlinearities describing propagation of light in nonlinear media, under two sets of transverse modulation forms of inhomogeneous nonlinearity. The bright soliton solution and Gaussian solution have been obtained for one set of inhomogeneous nonlinearity modulation. For the other, bright soliton solution, black soliton solution and the train solution have been presented. Stability of the solutions has been determined by exact soliton solutions under certain conditions.     

Painlevé Integrability of Nonlinear Schrödinger Equations with bothSpace- and Time-Dependent Coefficients

We investigate the Painlevé integrability of nonautonomous nonlinearSchrödinger (NLS) equations with both space- and time-dependent dispersion, nonlinearity, and external potentials. The Painlevé analysis is carried out without using the Kruskal's simplification, which results in more generalized form of inhomogeneous equations. The obtained equations are shown to be reducible to the standard NLS equation by using a point transformation. We also construct the corresponding Lax pair and carry out its Kundu-type reduction to the standard Lax pair. Special cases of equations from choosing limited form of coefficients coincide with the equations from the previous Painlevé analyses and/or become unknown new equations.     

Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data: Announcement of Results

We consider the one-dimensional focusing nonlinear Schrödinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a Bäcklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann?CHilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, the earlier work on the problem by Holmer and Zworski.     

Computation of discrete transparent boundary conditions for the 2D Helmholtz equation

We present a family of non-local transparent boundary conditions for the 2D Helmholtz equation. The whole domain, on which the Helmholtz equation is defined, is decomposed into an interior and an exterior domain. The corresponding interior Helmholtz problem is formulated as a variational problem in a standard manner, representing a boundary value problem, whereas the exterior problem is posed as an initial value problem in the radial variable. This problem is then solved approximately by means of the Laplace transformation. The derived boundary conditions are asymptotically correct, model inhomogeneous exterior domains and are simple to implement.     

The Riemann-Hilbert Formalism for Certain Linear and Nonlinear Integrable PDEs

Abstract

We show that by deforming the Riemann-Hilbert (RH) formalism associated with certain linear PDEs and using the so-called dressing method, it is possible to derive in an algorithmic way nonlinear integrable versions of these equations.

In the usual Dressing Method, one first postulates a matrix RH problem and then constructs dressing operators. Here we present an algorithmic construction of matrix Riemann-Hilbert (RH) problems appropriate for the dressing method as opposed to postulating them ad hoc. Furthermore, we introduce two mechanisms for the construction of the relevant dressing operators: The first uses operators with the same dispersive part, but with different decay at infinity, while the second uses pairs of operators corresponding to different Lax pairs of the same linear equation. As an application of our approach, we derive the NLS, derivative NLS, KdV, modified KdV and sine-Gordon equations.     

Scattering of Gaussian beam by arbitrarily shaped inhomogeneous particles

A hybrid finite element-boundary integral method is applied to characterize the scattering of an arbitrarily incident-focused Gaussian beam by arbitrarily shaped inhomogeneous particles. Specifically, the Davis–Barton fifth-order approximation in combination with rotation Euler angles is used to represent the arbitrarily incident Gaussian beams. The finite element method is employed to formulate the fields in the interior region of the inhomogeneous particle, while the boundary integral equation is applied to represent the fields in the exterior region. The interior and exterior fields are coupled by means of the field continuity conditions. To reduce the computational burden, the frontal method and the multilevel fast multipole algorithm are adopted to solve the resultant matrix equation. Numerical results for differential scattering cross sections of several selected inhomogeneous particles are presented and can be served as further study on this subject.     

Soliton solutions for a generalized fifth-order KdV equation with t-dependent coefficients

We consider a generalized fifth-order KdV equation with time-dependent coefficients exhibiting higher-degree nonlinear terms. This nonlinear evolution equation describes the interaction between a water wave and a floating ice cover and gravity-capillary waves. By means of the subsidiary ordinary differential equation method, some new exact soliton solutions are derived. Among these solutions, we can find the well known bright and dark solitons with sech and tanh function shapes, and other soliton-like solutions. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous KdV system supporting high-order dispersive and nonlinear effects.     

Dark Solitons for the Defocusing Cubic Nonlinear Schrödinger Equation with the Spatially Periodic Potential and Nonlinearity

We study the existence of dark solitons of the defocusing cubic nonlinear Schrödinger (NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincaré map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.     

Vector solitons of a one-dimensional spatially inhomogeneous coupled nonlinear Schrödinger equation with a double well potential

Vector soliton solutions of a coupled nonlinear Schr?dinger equation with spatially inhomogeneous nonlinearities and a double well potential are studied. A type of non-auto-B?cklund transformations is established to cast the investigated system to a couple of constant coefficient NLS equations under general conditions associating the inhomogeneous nonlinearities with the external potential. It is seen that the judicious choice of the inhomogeneous nonlinear interactions and the external potential is critical in the transformation work. In detail, three types of vector solitons are explicitly presented, and their structures and stability properties are also discussed.     

Exact multi-soliton solutions in nonlinear optical systems

In this paper, we present the (1 + 1)-dimensional inhomogeneous nonlinear Schrödinger (NLS) equation, which describes propagation of optical waves in nonlinear optical systems exhibiting spatial inhomogeneity, inhomogeneous nonlinearity and gain or loss at the same time. Exact multi-soliton solutions are presented by the simple Darboux transformation based on the Lax Pair, and the exact one- and two-soliton solutions in explicit forms are also generated. As two examples, we consider two nonlinear optical systems. In the systems, based on the exact solutions, a series of interesting properties of optical waves are displayed.