Showing papers on "Inverse scattering transform published in 2020"
TL;DR: In this article, a systematical inverse scattering transform for both focusing and defocusing nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity is presented.
85 citations
16 Oct 2020
TL;DR: In this article, the authors presented nonlocal reverse-spacetime PT-symmetric multicomponent nonlinear Schrödinger (NLS) equations under a specific nonlocal group reduction, and generated their inverse scattering transforms and soliton solutions by the Riemann-Hilbert technique.
Abstract: The paper presents nonlocal reverse-spacetime PT-symmetric multicomponent nonlinear Schrödinger (NLS) equations under a specific nonlocal group reduction, and generates their inverse scattering transforms and soliton solutions by the Riemann-Hilbert technique. The Sokhotski-Plemelj formula is used to determine solutions to a class of associated Riemann-Hilbert problems and transform the systems that generalized Jost solutions need to satisfy. A formulation of solutions is developed for the Riemann-Hilbert problems associated with the reflectionless transforms, and the corresponding soliton solutions are constructed for the presented nonlocal reverse-spacetime PT-symmetric NLS equations.
53 citations
TL;DR: In this article, a nonlinear fourth-order differential equation with arbitrary refractive index for description of the pulse propagation in an optical fiber is considered and a method for finding soliton solutions of nonlinear evolution equations is presented.
Abstract: A nonlinear fourth-order differential equation with arbitrary refractive index for description of the pulse propagation in an optical fiber is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we look for solutions of the equation using the traveling wave reduction. We present a novel method for finding soliton solutions of nonlinear evolution equations. The essence of this method is based on the hypothesis about the possible type of an auxiliary equation with an already known solution. This new auxiliary equation is used as a basic equation to look for soliton solutions of the original equation. We have found three forms of soliton solutions of the equation at some constraints on parameters of the equation.
44 citations
TL;DR: A controlled synthesis of a dense soliton gas in deep-water surface gravity waves using the tools of nonlinear spectral theory [inverse scattering transform (IST)] for the one-dimensional focusing nonlinear Schrödinger equation is reported.
Abstract: Soliton gases represent large random soliton ensembles in physical systems that exhibit integrable dynamics at the leading order Despite significant theoretical developments and observational evidence of ubiquity of soliton gases in fluids and optical media, their controlled experimental realization has been missing We report a controlled synthesis of a dense soliton gas in deep-water surface gravity waves using the tools of nonlinear spectral theory [inverse scattering transform (IST)] for the one-dimensional focusing nonlinear Schrodinger equation The soliton gas is experimentally generated in a one-dimensional water tank where we demonstrate that we can control and measure the density of states, ie, the probability density function parametrizing the soliton gas in the IST spectral phase space Nonlinear spectral analysis of the generated hydrodynamic soliton gas reveals that the density of states slowly changes under the influence of perturbative higher-order effects that break the integrability of the wave dynamics
39 citations
TL;DR: In this paper, a detailed study of the inverse scattering transforms (IST) applied to the PT symmetric, RST and RT integrable discrete NLS equations is carried out for rapidly decaying boundary conditions.
Abstract: A number of integrable nonlocal discrete nonlinear Schrodinger (NLS) type systems have been recently proposed. They arise from integrable symmetry reductions of the well-known Ablowitz–Ladik scattering problem. The equations include: the classical integrable discrete NLS equation, integrable nonlocal: PT symmetric, reverse space time (RST), and the reverse time (RT) discrete NLS equations. Their mathematical structure is particularly rich. The inverse scattering transforms (IST) for the nonlocal discrete PT symmetric NLS corresponding to decaying boundary conditions was outlined earlier. In this paper, a detailed study of the IST applied to the PT symmetric, RST and RT integrable discrete NLS equations is carried out for rapidly decaying boundary conditions. This includes the direct and inverse scattering problem, symmetries of the eigenfunctions and scattering data. The general linearization method is based on a discrete nonlocal Riemann–Hilbert approach. For each discrete nonlocal NLS equation, an explicit one soliton solution is provided. Interestingly, certain one soliton solutions of the discrete PT symmetric NLS equation satisfy nonlocal discrete analogs of discrete elliptic function/Painleve-type equations.
36 citations
TL;DR: In this article, the authors consider the nonlinear fourth-order partial differential equation that can be used for describing solitary waves in nonlinear optics and demonstrate that nonlinear ODE for description of the wave packet envelope possesses the Painleve property and is integrable.
Abstract: We consider the nonlinear fourth-order partial differential equation that can be used for describing solitary waves in nonlinear optics. The Cauchy problem for this equation is not solved by the inverse scattering transform. However we demonstrate that nonlinear ordinary differential equation for description of the wave packet envelope possesses the Painleve property and is integrable. The Lax pair to this nonlinear ordinary differential equation is presented. Using the determinant for the Lax pair matrix, we find the first integrals of a nonlinear ordinary differential equation. The general solution of the fourth-order nonlinear differential equation is given via the ultraelliptic integrals. Special cases of exact solutions for the fourth-order equation are expressed in terms of the Jacobi elliptic sine. Optical solitons of the original partial differential equation are found.
26 citations
TL;DR: In this article, the inverse scattering transform for the Gerdjikov-Ivanov equation with nonzero boundary at infinity was studied and the symmetries of the Jost solution and the spectral matrix were derived.
Abstract: In this article, we focus on the inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z. Based on the Lax pair of the Gerdjikov–Ivanov equation, we derive its Jost solutions with nonzero boundary. Further asymptotic behaviors, analyticity and the symmetries of the Jost solutions and the spectral matrix are in detail derived. The formula of N-soliton solutions is obtained via transforming the problem of nonzero boundary into the corresponding matrix Riemann–Hilbert problem. As examples of N-soliton formula, for
$$N=1$$
and
$$N=2$$
, respectively, different kinds of soliton solutions and breather solutions are explicitly presented according to different distributions of the spectrum. The dynamical features of those solutions are characterized in the particular case with a quartet of discrete eigenvalues. It is shown that distribution of the spectrum and non-vanishing boundary also affect feature of soliton solutions.
25 citations
TL;DR: In this paper, the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation is investigated, which can be reduced to several integrable systems.
Abstract: Under investigation in this paper is the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation, which can be reduced to several integrable systems. For the direct scattering problem, the spectral analysis is performed for the equation, from which a Riemann–Hilbert problem is well constructed. For the inverse scattering problem, the Riemann–Hilbert problem corresponding to the reflection-less case is solved. Furthermore, as applications, three types of multi-soliton solutions are found. Finally, some figures are presented to discuss the soliton behaviors of the vmKdV equation.
23 citations
TL;DR: In this article, an extended nonlinear Schrodinger equation with nonzero boundary conditions, which can model the propagation of waves in dispersive media, is investigated, and the general solutions for the potentials, and explicit expressions for the reflectionless potentials are presented.
23 citations
TL;DR: In this paper, a spectral analysis of the Lax pair was performed and a Riemann-Hilbert problem was constructed according to the spectral analysis, and three types of multisoliton solutions were obtained.
Abstract: We investigate higher-order nonlinear Schrodinger-Maxwell-Bloch equations using the Riemann-Hilbert method. We perform a spectral analysis of the Lax pair and construct a Riemann-Hilbert problem according to the spectral analysis. As a result, we obtain three types of multisoliton solutions. Based on the analytic solution and with a choice of corresponding parameter values, we obtain solutions of the breather type and a bell-shaped solution and find an interesting phenomenon of the collision of two soliton solutions. We hope that these results can be useful in modeling the wave propagation of a nonlinear optical field in an erbium-doped fiber medium.
TL;DR: In this article, the Riemann-Hilbert inverse scattering transform (RHST) was extended to the complex case of the complex short-pulse equation (CSP) on the line with zero boundary conditions at space infinity.
Abstract: In this paper, we develop the inverse scattering transform (IST) for the complex short-pulse equation (CSP) on the line with zero boundary conditions at space infinity. The work extends to the complex case the Riemann–Hilbert approach to the IST for the real short-pulse equation proposed by A. Boutet de Monvel, D. Shepelsky and L. Zielinski in 2017. As a byproduct of the IST, soliton solutions are also obtained. Unlike the real SPE, in the complex case discrete eigenvalues are not necessarily restricted to the imaginary axis, and, as consequence, smooth 1-soliton solutions exist for any choice of discrete eigenvalue $$k_1\in {\mathbb {C}}$$
with $$\mathop {{\mathrm{Im}}}
olimits k_1< |\mathop {{\mathrm{Re}}}
olimits k_1|$$
. The 2-soliton solution is obtained for arbitrary eigenvalues $$k_1,k_2$$
, providing also the breather solution of the real SPE in the special case $$k_2=-k_1^*$$
.
TL;DR: In this article, the Simple Equations Method (SEsM) is used to obtain exact solutions of nonlinear partial differential equations and several well-known methods for obtaining exact solution of such equations are connected to SEsM.
Abstract: The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrodinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a "small" parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.
TL;DR: In this article, the existence of a common framework in many integrable systems where Fredholm determinants appear is revealed, which consists in a quasi-universal hierarchy of equations, partly unifying an integro-differential generalization of the Painleve II hierarchy, the finite-time solutions of the Kardar-Parisi-Zhang equation, multi-critical fermions at finite temperature and a notable solution to the Zakharov-Shabat system associated to the largest real eigenvalue in the real Ginibre ensemble.
Abstract: As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This consists in a quasi-universal hierarchy of equations, partly unifying an integro-differential generalization of the Painleve II hierarchy, the finite-time solutions of the Kardar-Parisi-Zhang equation, multi-critical fermions at finite temperature and a notable solution to the Zakharov-Shabat system associated to the largest real eigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the explicit unique solution to the inverse scattering transform of the Zakharov-Shabat system in terms of a Fredholm determinant.
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TL;DR: In this article, an inverse scattering transform formalism for the good Boussinesq equation on the line is developed. But it is not suitable for the evaluation of long-time asymptotics via Deift-Zhou steepest descent arguments.
Abstract: We develop an inverse scattering transform formalism for the "good" Boussinesq equation on the line. Assuming that the solution exists, we show that it can be expressed in terms of the solution of a $3 \times 3$ matrix Riemann-Hilbert problem. The Riemann-Hilbert problem is formulated in terms of two reflection coefficients whose definitions involve only the initial data, and it has a form which makes it suitable for the evaluation of long-time asymptotics via Deift-Zhou steepest descent arguments.
TL;DR: In this article, generalized Fourier transforms (GFTs) for the hierarchies of multi-component models of Manakov type are revisited, and they adapt GFT so that one could treat the whole hierarchy of nonlinear evolution equations simultaneously.
Abstract: The generalized Fourier transforms (GFTs) for the hierarchies of multi-component models of Manakov type are revisited. Our aim is to adapt GFT so that one could treat the whole hierarchy of nonlinear evolution equations simultaneously. To this end we consider the potential Q of the Lax operator L as local coordinate on some symmetric space depending on an infinite number of variables t, and $$z_k$$
, $$k=1, 2, \ldots $$
. The dependence on $$z_k$$
is determined by the k-th higher flow (conserved density) of the hierarchy. Thus we have an infinite set of commuting operators with common fundamental analytic solution. We analyze the properties of the resolvent and thus determine the spectral properties of L. Then we derive the generalized Fourier transforms that linearize this hierarchy of NLEE and establish their fundamental properties as well as dynamical compatibility of each two pairs of such flows. Using the classical R-matrix approach we derive the Poisson brackets between all conserved quantities first assuming that Q is a quasi-periodic function of t. Next taking the limit when the period tends to $$\infty $$
, we derive the Poisson brackets between the scattering data of L. In addition we analyze a possible relation of our approach to the one based on the $$\tau $$
-function that could lead to multi-dimensional integrable equations.
TL;DR: In this paper, the authors present a method to compute dispersive shock wave solutions of the Korteweg-de Vries equation that emerge from initial data with step-like boundary conditions at infinity.
Abstract: We present a method to compute dispersive shock wave solutions of the Korteweg-de Vries equation that emerge from initial data with step-like boundary conditions at infinity. We derive two different Riemann-Hilbert problems associated with the inverse scattering transform for the classical Schrodinger operator with possibly discontinuous, step-like potentials and develop relevant theory to ensure unique solvability of these problems. We then numerically implement the Deift-Zhou method of nonlinear steepest descent to compute the solution of the Cauchy problem for small times and in two asymptotic regions. Our method applies to continuous and discontinuous data.
TL;DR: In this paper, the authors investigate the generation and propagation of solitary waves in the context of the Hertz chain and Toda lattice, with the aim to highlight the similarities, as well as differences between these systems.
Abstract: We investigate the generation and propagation of solitary waves in the context of the Hertz chain and Toda lattice, with the aim to highlight the similarities, as well as differences between these systems. We begin by discussing the kinetic and potential energy of a solitary wave in these systems and show that under certain circumstances the kinetic and potential energy profiles in these systems (i.e., their spatial distribution) look reasonably close to each other. While this and other features, such as the connection between the amplitude and the total energy of the wave, bear similarities between the two models, there are also notable differences, such as the width of the wave. We then study the dynamical behavior of these systems in response to an initial velocity impulse. For the Toda lattice, we do so by employing the inverse scattering transform, and we obtain analytically the ratio between the energy of the resulting solitary wave and the energy of the impulse, as a function of the impulse velocity; we then compare the dynamics of the Toda system to that of the Hertz system, for which the corresponding quantities are obtained through numerical simulations. In the latter system, we obtain a universality in the fraction of the energy stored in the resulting solitary traveling wave irrespectively of the size of the impulse. This fraction turns out to only depend on the nonlinear exponent. Finally, we investigate the relation between the velocity of the resulting solitary wave and the velocity of the impulse. In particular, we provide an alternative proof for the numerical scaling rule of Hertz-type systems.
TL;DR: In this article, the Cauchy problem for the Kundu-Eckhaus equation is solved by the inverse scattering transform, and the general solution of nonlinear ordinary differential equation exist for more general case.
TL;DR: In this article, the semiclassical limit of the dispersionless limited integrable system of hydrodynamic type, defined as dDS (dispersionless Davey-Stewartson) system, arises from the commutation condition of Lax pair of one-parameter vector fields.
Abstract: In this paper, the semiclassical limit of Davey–Stewartson system is studied. It shows that the dispersionless limited integrable system of hydrodynamic type, which is defined as dDS (dispersionless Davey–Stewartson) system, arises from the commutation condition of Lax pair of one-parameter vector fields. The relevant nonlinear Riemann–Hilbert problem with reality constraint for the dDS-II system is also constructed. This kind of Riemann–Hilbert problem is meaningful for applying the formal inverse scattering transform method, recently developed by Manakov and Santini, to study the dDS-II system.
TL;DR: In this paper, a class of square matrix nonlinear Schrodinger (MNLS) systems whose reductions include two equations that model hyperfine spin $$F = 1$$ in the focusing and defocusing dispersion regimes, and two novel (mixed sign) equations that were recently shown to be integrable.
Abstract: This work deals with a class of square matrix nonlinear Schrodinger (MNLS) systems whose reductions include two equations that model hyperfine spin $$F = 1$$
spinor Bose–Einstein condensates in the focusing and defocusing dispersion regimes, and two novel (mixed sign) equations that were recently shown to be integrable. Our main goal is to discuss the bright soliton solutions and their interactions for the focusing MNLS and for the two mixed sign systems within the framework of the inverse scattering transform. The nature of the solitons and their interactions depend on whether the associated norming constants (polarization matrices) are rank-one matrices (giving rise to ferromagnetic solitons) or full rank (corresponding to polar solitons). By computing the long-time asymptotics of the 2-soliton solutions, we determine how the polarization matrix of each soliton changes because of the interaction. Explicit formulas for the soliton interactions are given for all possible types of interacting solitons, namely ferromagnetic–ferromagnetic, polar–polar, and polar–ferromagnetic soliton interactions, and for all three inequivalent reductions of the MNLS systems that admit regular bright soliton solutions. We also present bound states, representing 2 solitons travelling with the same velocity, for all three systems.
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TL;DR: In this article, the theory of inverse scattering is developed to study the initial-value problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the $2m\times2m$ $(m\geq 1)$ Lax pairs under the nonzero boundary conditions at infinity.
Abstract: The theory of inverse scattering is developed to study the initial-value problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the $2m\times2m$ $(m\geq 1)$ Lax pairs under the nonzero boundary conditions at infinity. In the direct problem, by introducing a suitable uniform transformation we establish the proper complex $z$-plane in order to discuss the Jost eigenfunctions, scattering matrix and their analyticity and symmetry of the equation. Moreover the asymptotic behavior of the Jost functions and scattering matrix needed in the inverse problem are analyzed via Wentzel-Kramers-Brillouin expansion. In the inverse problem, the generalized Riemann-Hilbert problem of the mmKdV equation is first established by using the analyticity of the modified eigenfunctions and scattering coefficients. The reconstruction formula of potential function with reflection-less case is derived by solving this Riemann-Hilbert problem and using the scattering data. In addition the dynamic behavior of the solutions for the focusing mmKdV equation including one- and two- soliton solutions are presented in detail under the the condition that the potential is scalar and the $2\times2$ symmetric matrix. Finally, we provide some detailed proofs and weak version of trace formulas to show that the asymptotic phase of the potential and the scattering data.
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TL;DR: In this paper, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established.
Abstract: In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established. Motivated by the ideas of Ablowitz and Musslimani (2016 Nonlinearity 29 915), we successfully derive the inverse scattering transform (IST) of the nonlocal LPD equation. The direct scattering problem of the equation is first constructed, and some important symmetries of the eigenfunctions and the scattering data are discussed. By using a novel Left-Right Riemann-Hilbert (RH) problem, the inverse scattering problem is analyzed, and the potential function is recovered. By introducing the special conditions of reflectionless case, the time-periodic soliton solutions formula of the equation is derived successfully. Take $J=\overline{J}=1,2,3$ and $4$ for example, we obtain some interesting phenomenon such as breather-type solitons, arc solitons, three soliton and four soliton. Furthermore, the influence of parameter $\delta$ on these solutions is further considered via the graphical analysis. Finally, the eigenvalues and conserved quantities are investigated under a few special initial conditions.
TL;DR: The double-pole solution of the initial value problem for the Harry Dym equation (HDE) was obtained by using the inverse scattering transform (IST) method, i.e., solving the associated Gelfand–Levitan–Marchenko (GLM) equation.
TL;DR: In this article, a high-precision numerical approach for DS II type equa- tions is presented, treating initial data from the Schwartz class of smooth, rapidly decreasing functions.
Abstract: We present an efficient high-precision numerical approach for DaveyStewartson (DS) II type equa- tions, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As wit...
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TL;DR: In this article, the inverse scattering transform is considered for the Gerdjikov-Ivanov equation with zero and non-zero boundary conditions by a matrix Riemann-Hilbert (RH) method.
Abstract: In this article, the inverse scattering transform is considered for the Gerdjikov-Ivanov equation with zero and non-zero boundary conditions by a matrix Riemann-Hilbert (RH) method. The formula of the soliton solutions are established by Laurent expansion to the RH problem. The method we used is different from computing solution with simple poles since the residue conditions here are hard to obtained. The formula of multiple soliton solutions with one high-order pole and $N$ multiple high-order poles are obtained respectively. The dynamical properties and characteristic for the high-order pole solutions are further analyzed.
16 May 2020
TL;DR: In this paper, the inverse scattering transform and multi-solition solutions of the sextic nonlinear Schr\"{o}dinger equation are derived directly, and the scattering data with $t = 0$ are obtained according to analyze the symmetry and other related properties of the Jost functions.
Abstract: In this work, we consider the inverse scattering transform and multi-solition solutions of the sextic nonlinear Schr\“{o}dinger equation. The Jost functions of spectrum problem are derived directly, and the scattering data with $t=0$ are obtained according to analyze the symmetry and other related properties of the Jost functions. Then we take use of translation transformation to get the relation between potential and kernel, and recover potential according to Gel’fand-Levitan-Marchenko (GLM) integral equations. Furthermore, the time evolution of scattering data is considered, on the basic of that, the multi-solition solutions are derived. In addition, some solutions of the equation are analyzed and revealed its dynamic behavior via graphical analysis, which could be enriched the nonlinear phenomena of the sextic nonlinear Schr\”{o}dinger equation.
TL;DR: In this paper, the authors considered the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms without spectrum broadening and proposed a matrix Riemann-Hilbert (RH) problem.
Abstract: We consider the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms as the mixed problem for the Maxwell–Bloch equations without spectrum broadening. An approach to the study of such a problem is proposed. We use the inverse scattering transform method in the form of the matrix Riemann–Hilbert (RH) problem, using simultaneous spectral analysis of both the Lax equations. The proposed matrix RH problem solves the problem of the propagation of a sinusoidal signal in an unperturbed stable medium (attenuator). It is proved that this RH problem provides the causality principle for the region t < x, and for the region of the light cone, 0 < x < t allows us to find the asymptotics of the transmitted signal. First, we study the asymptotics of the RH problem for large times, and then, we obtain asymptotic formulas for the mixed problem solution of the Maxwell–Bloch equations when the attenuator is long enough. Three sectors are obtained in the light cone where the asymptotics have essentially different behaviors.
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TL;DR: In this paper, the authors studied the asymptotic behavior of Riemann-Hilbert problems arising in the AKNS hierarchy of integrable equations, based on the $\dbar$-steepest descent method.
Abstract: We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations. Our analysis is based on the $\dbar$-steepest descent method. We consider RHPs arising from the inverse scattering transform of the AKNS hierarchy with $H^{1,1}(\R)$ initial data. The analysis will be divided into three regions: fast decay region, oscillating region and self-similarity region (the Painleve region). The resulting formulas can be directly applied to study the long-time asymptotic of the solutions of integrable equations such as NLS, mKdV and their higher-order generalizations.
TL;DR: The collision of two plane gravitational waves in Einstein's theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain this paper.
Abstract: The collision of two plane gravitational waves in Einstein’s theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann–Hilbert problem. The formulation of the Riemann–Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle’s corners—this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann–Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.