Riemann-Hilbert Research Articles

A free boundary problem of elasticity in an angle and a vector Riemann–Hilbert problem on a torus

An inverse problem of elasticity of reconstructing the shape of an inclusion placed in a wedge, when the whole inclusion is subjected to a prescribed antiplane uniform stress, is considered. The doubly connected physical domain is conformally mapped onto a parametric plane cut along two finite segments on the real axis. Determination of such a map requires solving a vector Riemann–Hilbert problem of the theory of holomorphic functions on a symmetric genus-1 Riemann surface. The main steps of the method include factorization of a discontinuous function on a torus, solution of the associated Jacobi inversion problem, representation of the unknown vector-functions in terms of Weierstrass integrals, and reduction of the vector Riemann–Hilbert problem to a single integral kernel of the second kind with a discontinuous kernel on the elliptic surface. An integral representation of the conformal map in terms of the solution to the integral equation is given. In addition to four dimensionless problem parameters, the conformal map possesses five free parameters. Numerical results, which reconstruct the inclusion shape for sample sets of the parameters, are reported and discussed.

Mathematical Structures of Non-perturbative Topological String Theory: From GW to DT Invariants

We study the Borel summation of the Gromov–Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson–Thomas (DT) invariants of the resolved conifold, having a direct relation to the Riemann–Hilbert problem formulated by Bridgeland (Invent Math 216(1), 69–124, 2019). There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann–Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by Jafferis and Moore (Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909, 2008).

Riemann-Hilbert approach for the Kundu equation with non-vanishing boundary conditions: Simple and double poles

The inverse scattering transform for the Kundu equation with non-vanishing boundary conditions at infinity is studied via the Riemann-Hilbert approach. A detailed spectral analysis for analyticity, symmetry properties, and asymptotic behavior of the modified Jost eigenfunctions and scattering matrix is discussed. And an appropriate matrix Riemann-Hilbert problem is formulated. A closed system is solved for the explicit N-soliton solutions to the Kundu equation in the case of the reflectionless potential. Besides, the trace formula for the analytic scattering coefficients and the so-called theta condition for the phase difference between the boundary values are derived. A consistent framework is constructed to obtain solutions associated with double zeros to the analytic scattering coefficients. The solitons in the simple and double poles case of the Chen-Lee-Liu equation as a reduced example are studied, including any combination of dark, bright and breather solitons, and the dynamical properties of these solutions are displayed and analyzed.

Long-time asymptotic behavior of the coupled dispersive AB system in low regularity spaces

In this paper, we mainly investigate the long-time asymptotic behavior of the solution for coupled dispersive AB systems with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method. Based on the spectral analysis of Lax pairs, the Cauchy problem of coupled dispersive AB systems is transformed into a Riemann–Hilbert problem, and the existence and uniqueness of its solution is proved by the vanishing lemma. The stationary phase points play an important role in determining the long-time asymptotic behavior of these solutions. We demonstrate that in any fixed time cone Cx1,x2,v1,v2=(x,t)∈R2∣x=x0+vt,x0∈x1,x2,v∈v1,v2, the long-time asymptotic behavior of the solution for coupled dispersive AB systems can be expressed by N(I) solitons on the discrete spectrum, the leading order term O(t−1/2) on the continuous spectrum, and the allowable residual O(t−3/4).

Multi-soliton solutions for the three types of nonlocal Hirota equations via Riemann–Hilbert approach

The purpose of the paper is to formulate multi-soliton solutions for the nonlocal Hirota equations via the Riemann–Hilbert (RH) approach. The RH problems are constructed and the zero structures are studied via performing spectral analysis of the Lax pair. Then we consider three types of nonlocal Hirota equations by discussing different symmetry reductions of the potential matrix. On the basis of the resulting matrix RH problem under the restriction of the reflectionless case, we successfully obtain the multi-soliton solutions of the nonlocal Hirota equations.

The nonlinear Schrödinger equationon the half-line with homogeneous Robin boundary conditions.

We consider the nonlinear Schrödinger equationon the half-line with a Robin boundary condition at and with initial data in the weighted Sobolev space . We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann-Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution. In particular, by restricting our asymptotic result to solutions whose initial data are close to the initial profile of the stationary one-soliton, we obtain results on the asymptotic stability of the stationary one-soliton under any small perturbation in . In the focusing case, such a result was already established by Deift and Park using different methods, and our work provides an alternative approach to obtain such results. We treat both the focusing and the defocusing versions of theequation.

Formula omitted]-Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz–Ladik systems

In this paper, we establish the l2-Sobolev space bijectivity of the inverse scattering transform related to the defocusing Ablowitz–Ladik system. On the one hand, based on the spectral problem, we establish the reflection coefficient and the corresponding Riemann–Hilbert problem for the direct scattering problem. And we prove that if the potential belongs to l2,k space, then the reflection coefficient belongs to Hθk(Σ). On the other hand, based on the Riemann–Hilbert problem, we obtain the corresponding reconstruction formula and recover the potential from the reflection coefficient. Furthermore, we confirm that if the reflection coefficient is in Hθk(Σ), then we show that the corresponding potential also belongs to l2,k for the inverse scattering problem. This study also confirms that for the initial-value problem of the defocusing Ablowitz–Ladik system, if the initial potential qn(0) belongs to l2,k and satisfies ∥qn∥∞<1, then the solution qn(t) for t≠0 also belongs to l2,k.

Long-time asymptotic behavior for the matrix modified Korteweg–de Vries equation

The long-time asymptotics of solution to the integrable matrix modified Korteweg–de Vries equation with a 4 × 4 Lax pair on the line in the case of decaying initial data is established. The study makes crucial use of the inverse scattering transform in the form of an associated 4 × 4 matrix Riemann–Hilbert problem, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. It was shown that the (x,t)-plane mainly decomposes asymptotically in time into three types of regions: a left oscillating region, where to leading order the solution exhibits decaying (of the order O(t−1/2)) modulated oscillations, a central Painlevé region where the asymptotic behavior is expressed in terms of the solution of a coupled Painlevé II equation which is related to a 4 × 4 matrix Riemann–Hilbert problem, and a right fast decaying region.

Application of the Riemann–Hilbert approach to the derivative nonlinear Schrödinger hierarchy

In our paper, we mainly study the [Formula: see text]-soliton solutions for the derivative nonlinear Schrödinger (DNLS) hierarchy. From the spectral problem of the DNLS hierarchy, the associated Riemann–Hilbert problem is constructed. Based on the scattering relationship and solving the Riemann–Hilbert problem, the [Formula: see text]-soliton solutions for the DNLS are given explicitly.

The Hermitian symmetric space Fokas–Lenells equation: spectral analysis and long-time asymptotics

Abstract Based on the inverse scattering transformation, we carry out spectral analysis of the $4\times 4$ matrix spectral problems related to the Hermitian symmetric space Fokas–Lenells (FL) equation, by which the solution of the Cauchy problem of the Hermitian symmetric space FL equation is transformed into the solution of a Riemann–Hilbert problem. The nonlinear steepest descent method is extended to study the Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems and strict error estimates, we obtain explicitly the long-time asymptotics of the Cauchy problem of the Hermitian symmetric space FL equation with the aid of the parabolic cylinder function.

Inverse scattering for the derivative nonlinear Schrödinger equation with nonzero boundary conditions and triple poles

The inverse scattering transform approach is used to investigate the derivative nonlinear Schrödinger equation (DNLSE) with nonzero boundary conditions (NZBCs) at infinity and triple zeros of analytical scattering coefficients. Based on the analytical, symmetric and asymptotic properties of eigenfunctions, a matrix Riemann–Hilbert problem (RHP) associated with DNLSE with NZBCs is constructed. Then, the reconstruction formula for the potential is found by solving the RHP. In particular, the reflectionless potential with triple poles for the NZBCs is carried out explicitly by means of determinants.

Non-degenerate high-order solitons of the coupled nonlinear Schrödinger equation

In this paper, we establish the Riemann–Hilbert problem for the non-degenerate high-order solitons of the coupled nonlinear Schrödinger equation. The dynamics of these two components q1 and q2 behave quite differently. By choosing some proper parameters, one component will have a double-hump soliton but the other one will not. Especially, this kind of high-order solitons will not satisfy the scalar nonlinear Schrödinger equation under the SU(2) symmetry. Moreover, we also give the asymptotic analysis of large t for these different components and check the asymptotic solutions and the exact solutions numerically.

New integrable multi-Lévy-index and mixed fractional nonlinear soliton hierarchies

In this letter, we present a simple and new idea to generate two types of novel integrable multi-Lévy-index and mix-Lévy-index (mixed) fractional nonlinear soliton hierarchies, containing multi-index and mixed fractional higher-order nonlinear Schrödinger (NLS) hierarchy, fractional complex modified Korteweg–de Vries (cmKdV) hierarchy, and fractional mKdV hierarchy. Their explicit forms can be given using the completeness of squared eigenfunctions. Moreover, we present their anomalous dispersion relations via their linearizations, and fractional multi-soliton solutions via the inverse scattering transform with matrix Riemann–Hilbert problems. These obtained fractional multi-soliton solutions may be useful to understand the related super-dispersion transports of nonlinear waves in multi-index fractional nonlinear media.

A Riemann–Hilbert approach to the modified Camassa–Holm equation with step-like boundary conditions

The paper aims at developing the Riemann–Hilbert (RH) approach for the modified Camassa–Holm (mCH) equation on the line with non-zero boundary conditions, in the case when the solution is assumed to approach two different constants at different sides of the line. We present detailed properties of spectral functions associated with the initial data for the Cauchy problem for the mCH equation and obtain a representation for the solution of this problem in terms of the solution of an associated RH problem.

Large-Degree Asymptotics of Rational Painlevé-IV Solutions by the Isomonodromy Method

The Painlevé-IV equation has two families of rational solutions generated, respectively, by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann–Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painlevé-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painlevé-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement.

Riemann–Hilbert approach of the complex Sharma–Tasso–Olver equation and its N-soliton solutions

We study the complex Sharma–Tasso–Olver equation using the Riemann–Hilbert approach. The associated Riemann–Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair. Subsequently, in the case that the Riemann–Hilbert problem is irregular, the N-soliton solutions of the equation can be deduced. In addition, the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.

Explicit solitons of Kundu equation derived by Riemann-Hilbert problem

We apply Riemann-Hilbert problem (RHP) to Kundu equation with zero boundary condition. By solving the revised RHP related to N pairs of simple pole points, a pair of differential equations are solved from the asymptotic behaviors, and the explicit formula of N-th order soliton is obtained. Unlike the known representation of N-th order soliton in many literatures, our formula does not include unsatisfactory integral due to using revised RHP. Based on this new formula, the explicit first order soliton is obtained directly, and the explicit second order bound-state soliton (BSS) is also obtained by applying limit technique. Additionally, higher-order soliton and bound-state soliton (BSS) solutions are discussed, and the approximate trajectories of second order BSS solution are studied.

On the Poincare Problem for the Stokes-Bitsadze Equation with a Supersingular Point in Minor Term Coefficients

Riemann-Hilbert and Poincaré type problems are studied for the Bitsadze equation with additional minor terms consisting of first-order partial derivatives and with coefficients containing a supersingular point. It is shown that with certain constraints on the minor term coefficients, the Bitsadze equation and the Riemann-Hilbert type problem are reduced to the equivalent Poincaré problem. Matters concerned with the solution uniqueness of the problems under consideration and their explicit representation are addressed.

On the spectral properties of the Hilbert transform operator on multi-intervals

Let J,E\subset\mathbb{R} be two multi-intervals with non-intersecting interiors. Consider the operator A\colon L^2( J )\to L^2(E),\quad (Af)(x) = \frac 1\pi\int_J \frac {f(y) d y}{{y-x}}, and let A^\dagger be its adjoint. We introduce a self-adjoint operator \mathscr K acting on L^2(E)\oplus L^2(J) , whose off-diagonal blocks consist of A and A^\dagger . In this paper we study the spectral properties of \mathscr K and the operators A^\dagger A and A A^\dagger . Our main tool is to obtain the resolvent of \mathscr K , which is denoted by \mathscr R , using an appropriate Riemann–Hilbert problem, and then compute the jump and poles of \mathscr R in the spectral parameter \lambda . We show that the spectrum of \mathscr K has an absolutely continuous component [0,1] if and only if J and E have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of \mathscr K consists only of eigenvalues and 0 . If there are common endpoints, then \mathscr K may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at 0 . In all cases, \mathscr K does not have a singular continuous spectrum. The spectral properties of A^\dagger A and A A^\dagger , which are very similar to those of \mathscr K , are obtained as well.

The explicit bound-state soliton of Kundu equation derived by Riemann–Hilbert problem

With zero boundary condition, a revised Riemann–Hilbert problem (RHP) for Kundu equation is provided. Assuming the refection coefficient having one pair of Nth order poles, the RHP is solved, the expression of integral factor involved in the solution is also solved, and the explicit formula of Nth order bound-state soliton of Kundu equation is presented.