Inverse Scattering Transform for the Defocusing Manakov System with Nonzero Boundary Conditions

Under investigation in this work is a newly proposed nonlocal long-wave–short-wave resonance interaction (LSRI) equation with the self-induced parity-time (PT) symmetric potential. This equation offers PT symmetry analogues of the classical integrable LSRI equation and may be important for the occurence of such equations in nonlinear optics as the nonlocal NLS equation. General soliton solutions to the nonlocal LSRI equation with nonzero boundary condition are derived by using the Hirota’s bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method. These solutions are expressed in terms of Gramian determinants and include dark–dark solitons, dark–antidark solitons and antidark–antidark solitons. Three typical cases of the two solitons, namely two dark–dark solitons, two dark–antidark solitons and two antidark–antidark solitons, are demonstrated.

We extend the Riemann–Hilbert (RH) method to study the Fokas–Lenells (FL) equation with nonzero boundary conditions at infinity and successfully find its multiple soliton solutions with one high-order pole and N high-order poles. The mathematical structures of the FL equation are constructed, including global conservation laws and local conservation laws. Then, the conditions (analytic, symmetric, and asymptotic properties) needed to construct the RH problem are obtained by analyzing the spectral problem. The reflection coefficient r(z) with two cases appearing in the RH problem is considered, including one high-order pole and N high-order poles. In order to overcome the difficulty of establishing the residue expressions corresponding to high-order poles, we introduce the generalized residue formula. Finally, the expression of exact soliton solutions with reflectionless potential is further derived by a closed algebraic system.

The discrete modified Korteweg–de Vries equation under nonzero boundary conditions

The nonlinear Schrödinger equation is a classical nonlinear evolution equation with wide applications. This paper explores the asymptotic behavior of solutions to the nonlinear Schrödinger equation with non-zero boundary conditions in the presence of a pair of second-order discrete spectra. We analyze the Riemann–Hilbert problem in the inverse scattering transform by the Deift–Zhou nonlinear steepest descent method. Then we propose a proper deformation to deal with the growing time term and give the conditions for the series in the process of deformation by the Laurent expansion. Finally, we provide the characterization of the interactions between the solitary waves corresponding to second-order discrete spectra and the coherent oscillations produced by the perturbation. Numerical verifications are also performed.

In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems \(-\Delta _p \,u=f_\lambda (x,u) \) in \(\Omega , u = \varphi \ \text{ on } ~\partial \Omega \), where \(\lambda > 0\) is a parameter. The family we consider includes in particular the Pohozaev type equation \(-\Delta _p \,u = \lambda u^{p^{*}-1}\). The main new feature is the consideration of the p-Laplacian \(-\Delta _p\) together with a nonzero boundary condition \(\varphi \). In order to deal with these nonhomogeneous problems, it has been important to extend to this new context several basic results such as the Brezis-Nirenberg theorem on local minimization in \(W^{1,p}\) and \(C^1\), a \(C^{1,\alpha }\) estimate for a family of equations with critical growth, and a variational approach to the method of upper–lower solutions. These extensions have an independent interest for applications in other situations.

On the lifespan of nonzero background solutions to a class of focusing nonlinear Schrödinger equations

In this paper, the diagnosis of soft faults in lossy electric transmission lines is studied through the inverse scattering approach, extending a recently reported study on lossless transmission lines. The considered soft faults are modeled as continuous spatial variations of distributed characteristic parameters of transmission lines. The diagnosis of such faults from reflection and transmission coefficients measured at the ends of a line can be formulated as an inverse problem. The relationship between this inverse problem and the inverse scattering theory has been studied by Jaulent in 1982 through transformations from the telegrapher's equations of transmission lines to Zakharov-Shabat equations. The present paper clarifies and completes the computation of the theoretic scattering data required by the inverse scattering transform from the practically measured engineering scattering data. The inverse scattering method is then applied to numerically simulated lossy transmission lines to confirm the feasibility of the studied approach to soft fault diagnosis.

We consider a Hilbert boundary value problem with an unknown parametric function on arbitrary infinite straight line passing through the origin. We propose to transform the Hilbert boundary value problem to Riemann boundary value problem, and address it by defining symmetric extension for holomorphic functions about an arbitrary straight line passing through the origin. Finally, we develop the general solution and the solvable conditions for the Hilbert boundary value problem.

The inverse scattering problem (ISP) on the whole line for a Dirac system is considered. The reflection coefficient (FC) is represented as a rational function with an arbitrary number of poles. The method of solving for the Gel'fand-Levitan-Marchenko (GLM) equation generated by a rational reflection coefficient (RFC) is extended to n poles, when a spectral gap is present. The explicit solution in the case of three poles is presented. Graphs of the potential as a function of distance are displayed for cases having up to four poles.

This article addresses the mathematical study of a mean-field model of superconducting vortices in a II-type superconductor, previously introduced in [S. J. Chapman, SIAM J. Appl. Math., 55 (1995), pp. 1259–1279]. We investigate a hyperbolic-elliptic type system of PDEs in a given domain. Motivated by physical experiments, we consider nonzero and nonconstant boundary conditions, which describe a flux of superconducting vortices through the domain. We prove the existence of the regular solutions of a parabolic-elliptic approximated system and establish a uniform $L_ \infty $-bound for the vorticity and the convergence to the initial system. Finally, we analyze the regularity of weak solutions.

: We are continuing our research in the development of the Inverse Scattering Transform (IST). IST is a method which allow's one to solve nonlinear wave equations by solving certain related direct and inverse scattering problems. Research is really two pronged. It is necessary for us to understand and effectively solve both classical and new direct and inverse scattering problems. We use these results to find solutions to nonlinear wave equations much like one uses Fourier analysis for linear problems. Moreover the nonlinear wave equations arise naturally in physical problems. We are particularly interested in fluid dynamical applications. Although original discoveries employed IST in one spatial dimension, we have developed effective procedures to carry forth the method for multidimensional problems. In one spatial dimension solving the inverse scattering problem requires one to solve a vector Riemann-Hilbert boundary value problem. In multidimensions we have shown that the DBAR method is essential to solve the increase problem. In a special case it reduces to a Riemann-Hilbert problem (sometimes a nonlocal Riemann-Hilbert problem). The method applies to the multidimensional Schrodinger scattering problem (i.e. the Helmboltz equation) higher order scalar differential operators, multidimensional first order systems and even discrete equations i.e. difference equations. The multidimensional DBAR method is an extremely powerful method to analyze and solve inverse scattering problems. The DBAR method to solve a variety of novel and important inverse scattering problems in multidimensions.

The inverse scattering problem for Sturm–Liouville operators on the line with a matrix transfer condition at the origin is considered. We show that the transfer matrix can be reconstructed from the eigenvalues and reflection coefficient. In addition, for potentials with compact essential support, we show that the potential can be uniquely reconstructed.

Efficient and accurate mathematical codes for the prediction of underwater sound propagation are a critical component of SONAR system development and operation. The goal of the research presented herein is to develop, implement, and verify an efficient and rigorous coupled-mode solution for acoustic wave propagation in shallow water range-dependent environments. A theoretical framework involving a range-expanded inner product for capturing the coupling between modes as they propagate through a horizontally-variable medium is presented. This framework includes a novel discretization of the range-dependent acoustic medium. A difference equations approach, which implements the inner product to account for non-adiabatic energy transfer between modes, is used to recursively compute reflection and transmission coefficients throughout the discretized environment. Increased efficiency is gained in the method due to the ability to compute coupling via closed-form algebraic expressions and in the application of asymptotic analysis to simplify the transmission and reflection coefficients. Results will be shown for a sample of waveguides with horizontally-variable bottom depth, along with comparable results from R. Evans’ COUPLE model and an FEM solution produced by ARL-UT. [This work was funded by the Naval Undersea Warfare Center, Newport Rhode Island.]

This paper studies the N-soliton solutions of the generalized nonlinear Schrödinger equation. Firstly, the spectral analysis of a Lax pair is carried out. Secondly, the Riemann–Hilbert (RH) problem is established. Finally, the RH problem without reflection is considered and the expression of general N-soliton solutions of the equation is obtained.