Riemann-Hilbert Research Articles

Riemann–Hilbert problem and dynamics of soliton solutions of the fifth-order nonlinear Schrödinger equation

The theory of inverse scattering is developed to investigate the initial-value problem for the fifth-order nonlinear Schrödinger (foNLS) equation under the zero boundary conditions at infinity. The spectral analysis is performed in the direct scattering process, including the establishment of the analytical, asymptotic and symmetric properties of the scattering matrix and the Jost functions. In the inverse scattering process, a suitable Riemann–Hilbert (RH) problem is successfully established by using the modified eigenfunctions and scattering data, and the relationship between the potential function and the solution of the RH problem is successfully established. In order to further analyze the propagation behavior of the solutions of the foNLS equation, we present some new phenomena of studying three-soliton solutions corresponding to simple zeros and two-soliton solutions corresponding to double zeros in scattered data.

On the monodromy of the deformed cubic oscillator

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlevé equation. We use the generalised monodromy map for this equation to give solutions to the Riemann-Hilbert problems of (Bridgeland in Invent Math 216(1):69–124, 2019) arising from the Donaldson-Thomas theory of the A_2 quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map.

The "good" Boussinesq equation: a Riemann-Hilbert approach

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.

SOLVING CONVOLUTION SINGULAR INTEGRAL EQUATIONS WITH REFLECTION AND TRANSLATION SHIFTS UTILIZING RIEMANN-HILBERT APPROACH

<abstract> In this paper, method of solution for some kinds of convolution singular integral equations with reflection will be discussed in class {0}. By means of the theory of Fourier analysis and the theory of boundary value problems of analytic functions, such equations can be transformed into Riemann boundary value problems (i.e., Riemann-Hilbert problems) with nodes and reflection, or a system of linear algebraic equations. In spite of the classical method for solution, we are to give a new method, by which analytic solutions and conditions of Noether solvability are obtained respectively. At the end of this paper, we propose two kinds of convolution singular integral equations with reflections and a finite set of translation shifts. </abstract>

Explicit Solitons of Kundu Equation Derived by Riemann-Hilbert Problem

Explicit Solitons of Kundu Equation Derived by Riemann-Hilbert Problem

Long-time asymptotics for the modified complex short pulse equation

&lt;p style='text-indent:20px;'&gt;Based on the spectral analysis and the inverse scattering method, by introducing some spectral function transformations and variable transformations, the initial value problem for the modified complex short pulse (mCSP) equation is transformed into a &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ 2\times2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; matrix Riemann-Hilbert problem. It is proved that the solution of the initial value problem for the mCSP equation has a parametric expression related to the solution of the matrix Riemann-Hilbert problem. Various Deift-Zhou contour deformations and the motivation behind them are given. Through several appropriate transformations and strict error estimates, the original matrix Riemann-Hilbert problem can be reduced to the model Riemann-Hilbert problem, whose solution can be solved explicitly in terms of the parabolic cylinder functions. Finally, the long-time asymptotics of the solution of the initial value problem for the mCSP equation is obtained by using the nonlinear steepest decent method.&lt;/p&gt;

Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

&lt;p style='text-indent:20px;'&gt;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. 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 &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ J = \overline{J} = 1,2,3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 4 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for example, we obtain some interesting phenomenon such as breather-type solitons, arc solitons, three soliton and four soliton. Furthermore, the influence of parameter &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \delta $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; on these solutions is further considered via the graphical analysis. Finally, the eigenvalues and conserved quantities are investigated under a few special initial conditions.&lt;/p&gt;

ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS

For the fifth Painlevé transcendents, an asymptotic representation by the Jacobi sn-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrized by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the sn-function and the ϑ-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the sn-function satisfying the Boutroux equations and to construct deductively the elliptic representation.

The general fifth-order nonlinear Schrödinger equation with nonzero boundary conditions: Inverse scattering transform and multisoliton solutions

Under investigation in this work is the inverse scattering transform of the general fifth-order nonlinear Schr\"{o}dinger equation with nonzero boundary conditions (NZBCs), which can be reduced to several integrable equations. Firstly, a matrix Riemann-Hilbert problem for the equation with NZBCs at infinity is systematically investigated.Then the inverse problems are solved through the investigation of the matrix Riemann-Hilbert problem. Thus, the general solutions for the potentials, and explicit expressions for the reflection-less potentials are well constructed. Furthermore, the trace formulae and theta conditions are also presented. In particular, we analyze the simple-pole and double-pole solutions for the equation with NZBCs. Finally, the dynamics of the obtained solutions are graphically discussed. These results provided in this work can be useful to enrich and explain the related nonlinear wave phenomena in nonlinear fields.

Nonlinear evolution equations related to Kac-Moody algebras $A_r^{(1)}$: spectral aspects

We analyze three types of integrable nonlinear evolution equations (NLEE) related to the Kac-Moody algebras $A_r^{(1)}$. These are $\mathbb{Z}_h$-reduced derivative NLS equations (DNLS), multicomponent mKdV equations and 2-dimensional Toda field theories (2dTFT). We outline the basic tools of this analysis: i) the gradings of the simple Lie algebras using their Coxeter automorphisms; ii) the construction of the relevant Lax representations; and iii) the spectral properties of the Lax operators and their reduction to Riemann-Hilbert problems. We also formulate the minimal set of scattering data which allow one to recover the asymptotics of the fundamental analytic solutions to $L$ and its potential.

A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

In this paper, we consider the Muttalib-Borodin ensemble of Laguerre type, a determinantal point process on [0,∞) which depends on the varying weights xαe−nV(x), α>−1, and a parameter θ. For θ being a positive integer, we derive asymptotics of the associated biorthogonal polynomials near the origin for a large class of potential functions V as n→∞. This further allows us to establish the hard edge scaling limit of the correlation kernel, which is previously only known in special cases and conjectured to be universal. Our proof is based on a Deift/Zhou nonlinear steepest descent analysis of two 1×2 vector-valued Riemann-Hilbert problems that characterize the biorthogonal polynomials and the explicit construction of (θ+1)×(θ+1)-dimensional local parametrices near the origin in terms of Meijer G-functions.

Inverse scattering transform of the general three-component nonlinear Schrödinger equation and its multisoliton solutions

The work investigates the integrable general three-component nonlinear Schrödinger equation, which can be reduced to several integrable equations. A 4 × 4 matrix spectral problem is performed for the equation, from which a Riemann–Hilbert problem is strictly formulated. Moreover, the multi-soliton solutions to this equation are obtained through a particular Riemann–Hilbert formulation.

Gap probability for products of random matrices in the critical regime

The singular values of a product of M independent Ginibre matrices of size N×N form a determinantal point process. Near the soft edge, as both M and N go to infinity in such a way that M/N→α, α>0, a scaling limit emerges. We consider a gap probability for the corresponding limiting determinantal process, namely, the probability that there are no particles in the interval (a,+∞). We derive a Tracy–Widom-like formula in terms of the unique solution of a certain matrix Riemann–Hilbert problem of size 2 × 2. The right-tail asymptotics for this solution is obtained by the Deift–Zhou non-linear steepest descent analysis.

Role of crack length, crack spacing and layer thickness ratio in the electric potential and temperature of thermoelectric bi-materials systems

The influence of dual collinear interface cracks on the electric potential and temperature of thermoelectric bi-materials system under the electric load and thermal load is reported. The nonlinear governing equations of thermoelectric materials are transformed into the Laplace equations with the help of the driving force of the electric current density and the energy flux introduced. Firstly, based on the boundary conditions about the electric potential and the temperature, the interface electric current density and the interface energy flux the boundary of crack-free system are obtained. Then, the problem of the dual interface cracks is transformed into a Riemann-Hilbert problem, and analytical solutions of the electric current density and the energy flux are achieved. Furthermore, we provide the intensity factors of electric current density and energy flux around the crack tip. Finally, we unveil the crucial role of crack length, crack spacing and layer thickness ratio in the electric potential and temperature of thermoelectric bi-materials by numerical experiments.

Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-time

We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equationuxt+αβ2u−2iαβux−αuxx−iαβ2|u|2ux=0 with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at k=0. (ii) Four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial coneC(x1,x2,v1,v2)={(x,t)∈R2|x=x0+vt,x0∈[x1,x2],v∈[v1,v2]}, the long-time asymptotic behavior of the solution u(x,t) for the focusing FL equation can be characterized with an N(I)-soliton on discrete spectrums and a leading order term O(t−1/2) on continuous spectrum up to a residual error order O(t−3/4). The main tool is a ∂‾-generalization of the Deift-Zhou nonlinear steepest descent method.

Nonlocal PT-symmetric integrable equations and related Riemann–Hilbert problems

We aim to discuss about how to construct and classify nonlocal PT-symmetric integrable equations via nonlocal group reductions of matrix spectral problems. The nonlocalities considered are reverse-space, reverse-time and reverse-spacetime, each of which can involve either the transpose or the Hermitian transpose. The associated spectral problems are used to formulate a kind of Riemann–Hilbert problems and thus inverse scattering transforms. Soliton solutions are generated from specific Riemann–Hilbert problems with the identity jump matrix. We focus on two expository examples: nonlocal PT-symmetric matrix nonlinear Schrödinger and modified Korteweg–de Vries equations.

Inverse scattering transforms for non-local reverse-space matrix non-linear Schrödinger equations

The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.

A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup–Newell equation

Due to the fact that the higher-order Kaup–Newell (KN) system has more complex and diverse solutions than the classical second-order flow KN system, the research on it has attracted much attention. In this paper, we consider a higher-order KN equation with third-order dispersion and fifth-order nonlinearity. Based on the theory of the inverse scattering, the matrix Riemann–Hilbert problem is established. Through the dressing method, the solution matrix with simple zeros without reflection is constructed. In particular, a new form of solution is given which is more direct and simpler than previous methods. In addition, through the determinant solution matrix, the vivid diagrams and dynamic analysis of single-soliton solution and two-soliton solution are given in detail. Finally, by using the technique of limit, we construct the general solution matrix in the case of multiple zeros, and the examples of solutions for the cases of double zeros, triple zeros, single–double zeros, and double–double zeros are especially shown.

Conformal TBA for Resolved Conifolds

We revisit the Riemann–Hilbert problem determined by Donaldson–Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2–D0-brane bound states. We explore various prescriptions to make the sum well defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the $$\tau $$ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz integral representation for Faddeev’s quantum dilogarithm.

Riemann–Hilbert approach for discrete sine‐Gordon equation with simple and double poles

AbstractIn this paper, we present the inverse scattering transformation (IST) for discrete sine‐Gordon equation via the Riemann–Hilbert approach. In the direct scattering part, we establish analyticity, symmetries, and asymptotic properties of Jost solutions and reflection coefficients. In the inverse part, we solve the discrete sine‐Gordon equation by establishing a Riemann–Hilbert problem with simple and double poles, respectively. N‐soliton solutions are obtained via the reconstruction formula correspondent to the Riemann–Hilbert problem. As examples of N‐solitons, we present some explicit solutions such as breathers, degenerate solitons for discrete sine‐Gordon equation, and the dynamical properties of these solutions are further analyzed.