Matrix Riemann-Hilbert Problem Research Articles

Initial–boundary problems for the vector modified Korteweg–de Vries equation via Fokas unified transform method

Initial–boundary problems for the vector modified Korteweg–de Vries equation on the half-line are investigated by Fokas unified transform method. Even though additional technical complications arise in the multi-component case compared with scalar ones, it is shown that the solution q(x,t) can be expressed in terms of the solution of a matrix Riemann–Hilbert problem. The Riemann–Hilbert problem involves a jump matrix, uniquely defined in terms of four matrix spectral functions denoted by {a(λ),b(λ),A(λ),B(λ)} that depend on the initial data and all boundary values, respectively. A key role is played by the so-called global relation which involves the known and unknown boundary values. By analyzing the global relation, an effective characterization of the latter two spectral functions is presented.

Explicit Solutions of the Coupled mKdV Equation by the Dressing Method via Local Riemann-Hilbert Problem

We study the coupled mKdV equation by the dressing method via local Riemann-Hilbert problem. With the help of the Lax pairs, we obtain the matrix Riemann-Hilbert problem with zeros. The explicit solutions for the coupled mKdV equation are derived with the aid of the regularization of the Riemann-Hilbert problem.

Diffraction by an impedance strip II. Solving Riemann–Hilbert problems by OE-equation method

The current paper is the second part of a series of two papers dedicated to 2D problem of diffraction of acoustic waves by a segment bearing impedance boundary conditions. In the first part some preliminary steps were made, namely, the problem was reduced to two matrix Riemann-Hilbert problem. Here the Riemann-Hilbert problems are solved with the help of a novel method of OE-equations. Each Riemann-Hilbert problem is embedded into a family of similar problems with the same coefficient and growth condition, but with some other cuts. The family is indexed by an artificial parameter. It is proven that the dependence of the solution on this parameter can be described by a simple ordinary differential equation (ODE1). The boundary conditions for this equation are known and the inverse problem of reconstruction of the coefficient of ODE1 from the boundary conditions is formulated. This problem is called the OE-equation. The OE-equation is solved by a simple numerical algorithm.

Diffraction by an impedance strip I. Reducing diffraction problem to Riemann–Hilbert problems

A 2D problem of acoustic wave scattering by a segment bearing impedance boundary conditions is considered. In the current paper (the first part of a series of two) some preliminary steps are made, namely, the diffraction problem is reduced to two matrix Riemann-Hilbert problems with exponential growth of unknown functions (for the symmetrical part and for the antisymmetrical part). For this, the Wiener--Hopf problems are formulated, they are reduced to auxiliary functional problems by applying the embedding formula, and finally the Riemann-Hilbert problems are formulated by applying the Hurd's method. In the second part the Riemann-Hilbert problems will be solved by a novel method of OE-equation.

Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line

Initial‐boundary value problems for the coupled nonlinear Schrödinger equation on the half‐line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k‐plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so‐called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.

Novel systems of resonant wave interactions

A matrix Riemann–Hilbert problem (RHP) is constructed using the dressing method starting from two uncoupled, one-directional linear wave equations; the RHP thus obtained is then used to derive a novel integrable matrix non-local system of equations describing resonant wave interactions, together with its Lax pair. This system is shown to be a matrix generalization of the equations for resonant three-wave interactions and stimulated Raman scattering. Several compatible reductions admitted by this system are also discussed.

A Riemann-Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation

We use the Fokas method to analyze the derivative nonlinear Schrödinger (DNLS) equation iqt(x,t)=−qxx(x,t)+(rq2)x on the interval [0, L]. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit (x,t) dependence, and it has jumps across {ξ ∈ℂ| Imξ4=0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a(ξ),b(ξ)},{A(ξ),B(ξ)}, and {A(ξ),B(ξ)}, which in turn are defined in terms of the initial data q0(x)=q(x,0), the boundary data g0(t)=q(0,t),g1(t)=qx(0,t), and another boundary values f0(t)=q(L,t),f1(t)=qx(L,t). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

Quantum Spectral Curve for PlanarN=4Super-Yang-Mills Theory

We present a new formalism, alternative to the old TBA-like approach, for solution of the spectral problem of planar N = 4 SYM. It takes a concise form of a non-linear matrix Riemann-Hilbert problem in terms of a few Q-functions. We demonstrate the formalism for two types of observables - local operators at weak coupling and cusped Wilson lines in a near BPS limit.

On Determinants of Integrable Operators with Shifts

Integrable integral operator can be studied by means of a matrix Riemann–Hilbert problem. However, in the case of so-called integrable operators with shifts, the associated Riemann–Hilbert problem becomes operator-valued and this complicates strongly the analysis. In this note, we show how to circumvent, in a very simple way, the use of such a setting while still being able to characterize the large-x asymptotic behavior of the determinant associated with the operator.

Complete linearization of a mixed problem to the Maxwell–Bloch equations by matrix Riemann–Hilbert problems

Considered in this paper, the Maxwell–Bloch (MB) equations became known after Lamb (1967 Phys. Lett. A 25 181–2; 1971 Rev. Mod. Phys. 43 99–144; 1973 Phys. Rev. Lett. 31 196–199; 1974 Phys. Rev. A 9 422–30). Ablowitz, Kaup and Newell (1974 J. Math. Phys. 15 1852–58) proposed the inverse scattering transform (IST) to the MB equations for studying a physical phenomenon known as the self-induced transparency. A description of general solutions to the MB equations and their classification was done by Gabitov, Zakharov and Mikhailov (1985 Teor. Mat. Fiz. 63 11–31). In particular, they gave an approximate solution of the mixed problem to the MB equations in the domain x, t ∈ (0, L) × (0, ∞) and, on this basis, a description of the phenomenon of superfluorescence. It was emphasized in Gabitov et al (1985 Teor. Mat. Fiz. 63 11–31) that the IST method is non-adopted for the mixed problem. Authors of the mentioned papers have developed the IST method in the form of the Marchenko integral equations. We propose another approach for solving the mixed problem to the MB equations in the quarter plane. We use a simultaneous spectral analysis of both the Lax operators and matrix Riemann–Hilbert (RH) problems. Firstly, we introduce appropriate compatible solutions of the corresponding Ablowitz–Kaup–Newell–Segur equations and then we suggest such a matrix RH problem which corresponds to the mixed problem for MB equations. Secondly, we generalize this matrix RH problem, prove a unique solvability of the new RH problem and show that the RH problem (after a specialization of jump matrix) generates the MB equations. As a result we obtain solutions defined on the whole line and studied in Ablowits (1974 J. Math. Phys. 15 1852–58) and Gabitov et al (1985 Teor. Mat. Fiz. 63 11–31), solutions to the mixed problem studied below in this paper and solutions with periodic (finite-gap) boundary conditions. The type of solution is defined by specialization of the conjugation contour and jump matrix. Suggested matrix RH problems will be useful for studying the long time/long distance () asymptotic behavior of solutions to the MB equations using the Deift–Zhou method of steepest decent.

The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem uniquely determined in terms of the initial datum q(x,0), as well as of the boundary values {q(0, t),qx(0, t),qxx(0, t)}. For the Dirichlet problem, it is necessary to characterize the unknown boundary values qx(0, t) and qxx(0, t) in terms of the given data q(x, 0) and q(0, t). It is shown here that in the particular case of a vanishing initial datum and of a sine wave as Dirichlet datum, qx(0, t) and qxx(0, t) can be computed explicitly at least up to third order in a perturbative expansion and that at least up to this order, these functions are asymptotically periodic for large t.

Counting free fermions on a line: a Fisher–Hartwig asymptotic expansion for the Toeplitz determinant in the double-scaling limit

We derive an asymptotic expansion for a Wiener–Hopf determinant arising in the problem of counting one-dimensional free fermions on a line segment at zero temperature. This expansion is an extension of the result in the theory of Toeplitz and Wiener–Hopf determinants known as the generalized Fisher–Hartwig conjecture. The coefficients of this expansion are conjectured to obey certain periodicity relations, which renders the expansion explicitly periodic in the ‘counting parameter’. We present two methods to calculate these coefficients and verify the periodicity relations order by order: the matrix Riemann–Hilbert problem and the Painlevé V equation. We show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients.

Boundary Value Problems for the Elliptic Sine-Gordon Equation in a Semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.

Exact solutions for a class of matrix Riemann-Hilbert problems

In contrast to scalar Riemann–Hilbert problems, a general matrix Riemann–Hilbert problem cannot be solved in terms of Sokhotskyi–Plemelj integrals. As far as the authors know, the exact solutions are known for a class of homogeneous matrix Riemann–Hilbert problems with commutative and factorable kernels. This article considered matrix Riemann–Hilbert problems in which all the partial indices are zero and the logarithms of the components of the kernels and their non-homogeneous vectors are exponential-type (equivalently, band-limited) functions. Then, it develops exact solutions for such matrix Riemann–Hilbert problems. Applications in a class of spectral factorizations and a class of the Wiener–Hopf system of integrations are given.

Riemann–Hilbert problems and the mKdV equation with step initial data: short-time behavior of solutions and the nonlinear Gibbs-type phenomenon

We produce a family of matrix Riemann–Hilbert (RH) problems parametrized by a given scalar function r(k). The jump matrix of the RH problem depends on r(k) and external real parameters x and t. We prove the unique solvability of such a problem. A solution M(x, t, k) of this matrix RH problem gives a smooth solution q(x, t) of the modified Korteweg–de Vries (mKdV) equation qt + 6q2qx + qxxx = 0. For a special choice of the function r(k) we show that the corresponding initial data q(x, 0) is a Heaviside-type function. We consider the small-time limit of the solution q(x, t). The leading-order term is explicitly calculated as an integral of the Airy functionThe first summand is an exact solution to the linear KdV equation pt + pxxx = 0 with initial dataThus this specific solution q(x, t) to the mKdV equation, being smooth with respect to and t > 0, takes pointwise its limit values and exhibits a Gibbs-type phenomenon with rapid oscillations near x = 0 as t → +0.

The unified method: I. Nonlinearizable problems on the half-line

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving four scalar functions of k, called the spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant and on the computation of the large k asymptotics of the eigenfunctions defining the relevant spectral functions.

The unified method: II. NLS on the half-line with t-periodic boundary conditions

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving four scalar functions of k, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved by simply using algebraic manipulations. Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand–Levitan–Marchenko representations of the eigenfunctions defining the spectral functions. We then concentrate on the physically significant case of t-periodic Dirichlet boundary data. After presenting certain heuristic arguments which suggest that the Neumann boundary values become periodic as t → ∞, we show that for the case of the NLS with a sine-wave as Dirichlet data, the asymptotics of the Neumann boundary values can be computed explicitly at least up to third order in a perturbative expansion and indeed at least up to this order are asymptotically periodic.

The unified method: III. Nonlinearizable problems on the interval

Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and extensively used in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving six scalar functions of k, called the spectral functions. Two of these functions depend on the initial data, whereas the other four depend on all boundary values. The most difficult step of the new method is the characterization of the latter four spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data. We present two different characterizations of this problem. One is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant and on the computation of the large k asymptotics of the eigenfunctions defining the relevant spectral functions. The other is based on the analysis of the global relation and on the introduction of the so-called Gelfand–Levitan–Marchenko representations of the eigenfunctions defining the relevant spectral functions. We also show that these two different characterizations are equivalent and that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained recently for the case of boundary value problems formulated on the half-line.

Gradient Catastrophe and Fermi-Edge Resonances in Fermi Gas

Any smooth spatial disturbance of a degenerate Fermi gas inevitably becomes sharp. This phenomenon, called the gradient catastrophe, causes the breakdown of a Fermi sea to multiconnected components characterized by multiple Fermi points. We argue that the gradient catastrophe can be probed through a Fermi-edge singularity measurement. In the regime of the gradient catastrophe the Fermi-edge singularity problem becomes a nonequilibrium and nonstationary phenomenon. We show that the gradient catastrophe transforms the single-peaked Fermi-edge singularity of the tunneling (or absorption) spectrum to a sequence of multiple asymmetric singular resonances. An extension of the bosonic representation of the electronic operator to nonequilibrium states captures the singular behavior of the resonances.

Long-time behavior of the solution to the mKdV equation with step-like initial data

We consider the modified Korteweg–de Vries equation on the line. The initial data is the pure step function, i.e. q(x, 0) = cr for x ⩾ 0 and q(x, 0) = cl for x < 0, where cl > cr > 0 are arbitrary real numbers. Long-time behavior of the solution to the mKdV equation in the case cl > cr = 0 and t → ∞ was studied recently in Kotlyarov and Minakov (2010 J. Math. Phys. 51 093506) where the structure of the compression wave was obtained in the form of a modulated elliptic wave. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t → −∞, i.e. we study the long-time dynamics of the rarefaction wave. Using the steepest descent method and the so-called g-function mechanism we deform the original oscillatory matrix Riemann–Hilbert problem to the explicitly solvable model forms and show that the solution of the initial-value problem has a different asymptotic behavior in different regions of the xt-plane. In the regions x < 6c2lt and x > 6c2rt the main term of asymptotics of the solution is equal to cl and cr, respectively. In the region the asymptotics of the solution tends to . An influence of the dispersion is also studied: the second term of the asymptotics is obtained in the region x > −6c2rt, where the background constant cr is perturbed by the self-similar vanishing wave.