Riemann-Hilbert Research Articles

Riemann–Hilbert approach and double-polesolutions for the third-order flow equation ofDNLS-type equation with nonzero boundaryconditions

Abstract In this paper, the Riemann-Hilbert approach is applied to study a third-order flow equation of derivative nonlinear Schrödinger-type equation with nonzero boundary conditions. By utilizing the analytical, symmetric, and asymptotic properties of eigenfunctions, a generalized Riemann-Hilbert problem is formulated for the third-order flow equation of derivative nonlinear Schrödinger-type equation with nonzero boundary conditions. The formulas of N-soliton solutions for cases of single pole and double poles are given. We present some kinds of soliton solutions of these two cases according to different distributions of spectral parameters to study the dynamical behavior of them.

On the dispersionless Davey-Stewartson hierarchy: the tau function, the Riemann-Hilbert problem and the Hamilton-Jacobi theory

The dDS (dispersionless Davey-Stewartson) hierarchy is constructed by two eigenfunctions of a special Hamiltonian vector field. This hierarchy consists of the infinite symmetries of the dDS system. Further, this paper explores the tau function, the Riemann-Hilbert problem and Hamilton-Jacobi theory related to dDS hierarchy.

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Utilizing the Riemann-Hilbert approach, we study the inverse scattering transformation, as well as multi-pole solitons and breathers, for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions at infinity. Beginning with the Lax pair, we introduce the uniformization variable to simplify both the direct and inverse problems on the two-sheeted Riemann surface. In the direct scattering problem, we systematically demonstrate the analyticity, asymptotic behaviors and symmetries of the Jost functions and scattering matrix. By solving the corresponding matrix Riemann-Hilbert problem, we work out the multi-pole solutions expressed as determinants for the reflectionless potential. Based on the parameter modulation, the dynamical properties of the simple-, double- and triple-pole solutions are investigated. In the defocusing cases, we show abundant simple-pole solitons including dark solitons, anti-dark-dark solitons, double-hump solitons, as well as double- and triple-pole solitons. In addition, the asymptotic expressions for the double-pole soliton solutions are presented. In the focusing cases, we illustrate the propagations of simple-pole, double-pole, and triple-pole breathers. Furthermore, the multi-pole breather solutions can be reduced to the bright soliton solutions for the focusing nonlocal Lakshmanan-Porsezian-Daniel equation.

The focusing coupled modified Korteweg–de Vries equation with nonzero boundary conditions: the Riemann–Hilbert problem and soliton classification

The focusing coupled modified Korteweg–de Vries equation with nonzero boundary conditions: the Riemann–Hilbert problem and soliton classification

Riemann-Hilbert hierarchies for hard edge planar orthogonal polynomials

abstract: We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $\varkappa$, the expansion holds with an $\mathrm{O}(N^{-\varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies---sequences of scalar Riemann-Hilbert problems---which allows us to express all higher order correction terms in closed form. Indeed, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $\partial\mathscr{D}$.

The Sasa-Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via mixed ∂-Riemann-Hilbert problem

Abstract In this paper, we investigate the Cauchy problem of the Sasa-Satsuma (SS) equation with initial data belonging to the Schwartz space. The SS equation is one of integrable higher-order extensions of the nonlinear Schrödinger equation and admits a 3 × 3 Lax representation. With the aid of the ∂-nonlinear steepest descent method of the mixed¯∂-Riemann-Hilbert problem, we give the soliton resolution and long-time asymptotics for the Cauchy problem of the Sasa-Satsuma equation with the existence of second-order discrete spectra in the space-time solitonic regions.

Riemann–Hilbert approach for the inhomogeneous discrete nonlinear Schrödinger equation with nonzero boundary conditions

In this paper, we systematically investigate the Riemann–Hilbert (RH) approach and obtain the soliton solutions for the inhomogeneous discrete nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs). Starting from the spectral problem and introducing the uniformization variable κ to avoid the complexity of double-valued function and Riemann surface, we deduce the analyticity, asymptotics and symmetries of the eigenfunctions and scattering coefficients, then the RH problem and reconstruction formula for the potential are successfully constructed. Under reflectionless condition and combining the time evolution of the scattering coefficients and eigenfunctions, we obtain various first-order soliton solutions with different direction of propagation caused by the change of the coefficients. Based on the analytic solution and the choice of special parameter values, we obtain the collision mechanism of two soliton solutions. Furthermore, the important advantage of the RH problem is that it can be further used to study the soliton resolution and the long-time asymptotic behavior of the solutions.

The Soliton Solutions for Nonlocal Multi-Component Higher-Order Gerdjikov–Ivanov Equation via Riemann–Hilbert Problem

The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert method. Compared to the local HOGI equation, the solutions of nonlocal equations not only depend on the local spatial and time variables, but also the nonlocal variables. To exhibit the dynamic behavior, we consider the reverse-spacetime multi-component HOGI equation and its Riemann–Hilbert problem. When the Riemann–Hilbert problem is regular, the integral form solution can be given. Conversely, the exact solutions can be obtained explicitly. Finally, as concrete examples, the periodic solutions of the two-component nonlocal HOGI equation are given, which is different from the local equation.

On behavior analysis of solutions for the coupled higher-order WKI equation

A coupled higher-order WKI equation is proposed and its 4 × 4 matrix Lax pair is derived. On the basis of spectral analysis of the 4 × 4 matrix spectral problems related to the coupled higher-order WKI equation, the basic Riemann–Hilbert problem is established by using the inverse scattering transformation. Resorting to the nonlinear steepest descent method, the solution of the basic Riemann–Hilbert problem is transformed to the solution of the model Riemann–Hilbert problem after several contour deformation. Finally, we explicitly obtain the long-time asymptotics of the solution to the Cauchy problem of the coupled higher-order WKI equation in the solitonless sector region with the aid of the parabolic cylinder function.

Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems

We consider the Riemann–Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation. An RH problem for the solution of the finite-band problem has been recently derived via the Fokas method (Deconinck et al. 2021 Lett. Math. Phys. 111 , 1–18. ( doi:10.1007/s11005-021-01356-7 ); Fokas & Lenells. 2021 Proc. R. Soc. A 477 , 20200605. ( doi:10.1007/s11005-021-01356-7 )) Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.

Generating new gravitational solutions by matrix multiplication

Explicit solutions to the nonlinear field equations of some gravitational theories can be obtained, by means of a Riemann–Hilbert approach, from a canonical Wiener–Hopf factorization of certain matrix functions called monodromy matrices. In this paper, we describe other types of factorization from which solutions can be constructed in a similar way. Our approach is based on an invariance problem, which does not constitute a Riemann–Hilbert problem and allows to construct solutions that could not have been obtained by Wiener–Hopf factorization of a monodromy matrix. It gives rise to a novel solution generating method based on matrix multiplications. We show, in particular, that new solutions can be obtained by multiplicative deformation of the canonical Wiener–Hopf factorization, provided the latter exists, and that one can superpose such solutions. Examples of applications include Kasner, Einstein–Rosen wave and gravitational pulse wave solutions.

Robust inverse scattering analysis of discrete high-order nonlinear Schrödinger equation

Abstract 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.

Spectral structures and soliton dynamical behaviors of two shifted nonlocal NLS equations via a novel Riemann–Hilbert approach: A reverse-time NLS equation and a reverse-spacetime NLS equation

By extending the traditional Riemann–Hilbert (RH) approach of soliton equations to a novel version, this paper is devoted to studying two shifted nonlocal NLS equations: a reverse-time NLS equation with a temporal parameter t0, and a reverse-spacetime NLS equation with the spatial and temporal parameters x0,t0. It is shown that the defocusing cases of the two shifted nonlocal NLS equations are equivalent to their focusing cases under linear transformations, only the focusing cases of the two shifted nonlocal NLS equations need to be treated in this work which differs from the classical local NLS equation whose focusing and defocusing cases should be treated separately. Firstly, the spectral analysis of the Lax pair of the two shifted nonlocal NLS equations are performed, from which spectral structures of the two shifted equations are revealed in detail, respectively. Secondly, based on the obtained symmetry relations of the scattering data, the soliton solutions of the two shifted nonlocal NLS equations are rigorously obtained by introducing a novel reduction-proof technique which is different but more direct than the traditional RH approach. It is demonstrated that, due to the existence of the shifted parameters, the spectral analysis for deriving the symmetry relations of the scattering data and thus the calculations of the soliton solutions for the two shifted nonlocal NLS equations involve more tedious and ingenious computations than those unshifted ones without shifted parameters. Finally, the dynamical behaviors underlying the obtained soliton solutions are theoretically explored and graphically illustrated by highlighting the roles that the shifted parameters play, which manifest the peculiar soliton characteristics that may have potential applications of the shifted nonlocal reverse-time and reverse-spacetime NLS equations.

Inverse scattering and soliton dynamics for the mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation

Under investigation in this letter is a mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation which can be considered as the simplest model to approximate the dynamics of weakly nonlinear and dispersive waves, taking into account the self-steepening effect. The inverse scattering transform under the zero boundary conditions and analytical scattering coefficients with an arbitrary number of simple and double zeros is detailedly discussed. Particularly, the inverse problem is solved by the study of a matrix Riemann–Hilbert problem. As a consequence, we present the general solution for the potential, and explicit expression for the reflectionless potential.

Long-time asymptotics for the integrable nonlocal Lakshmanan–Porsezian–Daniel equation with decaying initial value data

In this work, we study the Cauchy problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation with rapid attenuation of initial data. The basic Riemann–Hilbert problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is constructed from Lax pair. Using Deift-Zhou nonlinear steepest descent method, the explicit long-time asymptotic formula of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is derived, which is different from the local model. Besides, compared to the nonlocal nonlinear Schrödinger equation, since the increase of real stationary phase points, the long-time asymptotic formula for nonlocal Lakshmanan-Porsezian-Daniel equation becomes more complex.

Soliton resolution for the complex short-pulse positive flow with weighted Sobolev initial data in the space-time soliton regions

The Riemann Hilbert approach and ∂‾ steepest descent method are used to study the Cauchy problem of a new complex short-pulse positive flow, from which the long-time asymptotics and soliton resolution conjecture for the complex short-pulse positive flow with initial conditions is obtained in weighted Sobolev spaces. Resorting to the spectral analysis of Lax pair, the introduced transformations of field variables and transformations of independent variables, we establish the basic Riemann-Hilbert problem, and convert the solution of the Cauchy problem of the complex short-pulse positive flow into the corresponding solution of the Riemann Hilbert problem. The jump matrix then expands and deforms continuously over the initial contour. We finally obtain the long-time asymptotics and soliton resolution for the complex short-pulse positive flow in the soliton region by using the ∂‾ steepest descent method. The results also show that N-soliton solutions of the complex short-pulse positive flow are asymptotically stable.

Whitham approach to certain large fluctuation problems in statistical mechanics

We show the relationship between the strongly non-linear limit (also termed the dispersionless or the Whitham limit) of the macroscopic fluctuation theory of certain statistical models and the inverse scattering method. We show that in the strongly non-linear limit the inverse scattering problem can be solved using the steepest descent method of the associated Riemann–Hilbert problem. The importance of establishing this connection, is that the equations in the strongly non-linear limit can often be solved exactly by simple means, the connection then provides a limit in which one can solve the inverse scattering problem, thus aiding potentially the exact solution of a particular large deviation problem.

Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data

We characterize the long-time asymptotic behavior of the solution of the initial value problem for the coupled complex short-pulse equation associated with the 4×4 matrix spectral problem. The spectral analysis of the 4×4 matrix spectral problem is very difficult because of the existence of energy-dependent potentials and the WKI type. The method we adopted is a combination of the inverse scattering transform and Deift-Zhou nonlinear steepest descent method. Starting from the Lax pair associated with the coupled complex short-pulse equation, we derive a basic Riemann-Hilbert problem by introducing some appropriate spectral function transformations, and reconstruct the potential parameterized from the solution of the basic Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at k→0. We finally obtain the leading order asymptotic behavior of the solution of the coupled complex short-pulse equation through a series of Deift-Zhou contour deformations.

Stable vector bundles and the Riemann-Hilbert problem on a Riemann surface

Stable vector bundles and the Riemann-Hilbert problem on a Riemann surface

Soliton resolution and asymptotic stability of N-solutions for the defocusing Kundu–Eckhaus equation with nonzero boundary conditions

In this paper, we address interesting soliton resolution, asymptotic stability of N-soliton solutions and the Painlevé asymptotics for the Kundu-Eckhaus (KE) equation with nonzero boundary conditions iqt+qxx−2(∣q∣2−1)q+4β2(∣q∣4−1)q+4iβ∣q∣2xq=0,q(x,0)=q0(x)∼±1,x→±∞. The key to proving these results is to establish the formulation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem and find its connection with the RH problem of the defocusing NLS equation. With the ∂¯ -steepest descent method and the results of the defocusing NLS equation, we find complete leading order approximation formulas for the defocusing KE equation on the whole (x,t) half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region, Zakharov-Shabat asymptotics in a solitonless region and the Painlevé asymptotics in two transition regions.