Hilbert Problem Research Articles

Jeffery’s paradox for the rotation of a single ‘stick–slip’ cylinder

We consider the problem of determining the two-dimensional fluid velocity due to the rotation of an infinitely-long ‘stick–slip’ cylinder in an otherwise quiescent Stokes flow. Stick–slip boundary conditions are introduced as a model of a rough superhydrophobic surface, via a distribution of alternating solid–liquid (stick) and gas–liquid (slip) interfaces. This leads to a mixed boundary-value problem for Stokes flow. Complex variable techniques are employed to transform the flow problem into a Hilbert problem, which involves finding a function analytic in a plane region assuming that on some portions of the boundary its real part is known, while on others its imaginary part is given. We solve the Hilbert problem to obtain semi-analytic expressions for all the pertinent fluid-dynamic quantities. We find that in the general aperiodic case there is no solution in which the velocity of the fluid vanishes at infinity. This is a form of Jeffery’s paradox, typically associated with viscous flow due to the counter-rotation of two equal rigid cylinders. Our work provides the first example of Jeffery’s paradox due to the rotation of a single cylinder.

Asymptotics of discrete q-Freud orthogonal polynomials from the q-Riemann Hilbert problem

We investigate a Riemann–Hilbert problem (RHP), whose solution corresponds to a group of q-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the polynomials approach infinity. The RHP formulation also enables us to obtain further properties. In particular, we consider how the class of polynomials and their asymptotic behaviours change under translations of the q-discrete lattice and determine the asymptotics of a related q-Painlevé equation.

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

AbstractThe two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so‐called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark‐dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark‐bright, bright‐bright, breather‐breather, M(‐type)‐W(‐type) solitons, and their interactions.

Painlevé asymptotics for the coupled Sasa-Satsuma equation

We compute the long-time asymptotics of the solution to the Cauchy problem for coupled Sasa-Satsuma equation on the line with decaying initial data. By performing a nonlinear steepest descent arguments for an associated 5 × 5 5\times 5 matrix Riemann–Hilbert problem, it turns out that in the sector | x / t 1 / 3 | ≤ N |x/t^{1/3}|\leq N , N N constant, the asymptotics can be expressed in terms of the solution of a coupled modified Painlevé II equation, which is related to a 5 × 5 5\times 5 matrix Riemann–Hilbert problem.

The Riemann–Hilbert approach for the higher-order Gerdjikov–Ivanov equation, soliton interactions and position shift

In this paper, we are concerned with the Riemann–Hilbert approach for the higher-order Gerdjikov–Ivanov equation with nonzero boundary conditions. A uniformization variable will be introduced to make it forms a one-to-one mapping with the spectral parameter. The analyticity, symmetric and asymptotic behaviors of the eigenfunctions and scattering datas will be calculated in detail and the Riemann–Hilbert problem will be constructed to obtain the expression for the N-soliton solutions. In addition, the case of Jost solutions and scattering coefficients near the branch points will also be analyzed. On the other hand, when the boundary conditions turn to zero, the formula for the position shifts and phase shifts of solitons resulting from soliton interactions will be solved.

Constructing basises in solution space of the system of equations for the Lauricella Function FD (N)

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function , depending on N complex variables and having complex parameters , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points and in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons.

Why and whence the Hilbert space in quantum theory?

We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, of scalar product, of orthogonality, and of the linear functional are derivable from the statistical description of quantum micro-events and from Hilbertian sum of squares $|\mathfrak{a}_1|^2+|\mathfrak{a}_2|^2+\cdots$. The latter leads (non-axiomatically) to the standard writing of the Born formula $\mathtt{f} = |\langle\psi|\varphi\rangle|^2$. As a corollary, the status of Pythagorean theorem, the concept of a length, and the 6-th Hilbert problem undergo a quantum `revision'. An issue of deriving the norm topology may no have a short-length solution (too many abstract math-axioms) but is likely solvable in the affirmative; the problem is reformulated as a mathematical one.

Riemann–Hilbert problem for the Fokas–Lenells equation in the presence of high-order discrete spectrum with non-vanishing boundary conditions

We extend the Riemann–Hilbert (RH) method to study the Fokas–Lenells (FL) equation with nonzero boundary conditions at infinity and successfully find its multiple soliton solutions with one high-order pole and N high-order poles. The mathematical structures of the FL equation are constructed, including global conservation laws and local conservation laws. Then, the conditions (analytic, symmetric, and asymptotic properties) needed to construct the RH problem are obtained by analyzing the spectral problem. The reflection coefficient r(z) with two cases appearing in the RH problem is considered, including one high-order pole and N high-order poles. In order to overcome the difficulty of establishing the residue expressions corresponding to high-order poles, we introduce the generalized residue formula. Finally, the expression of exact soliton solutions with reflectionless potential is further derived by a closed algebraic system.

Explicit Nth order solutions of Fokas–Lenells equation based on revised Riemann–Hilbert approach

We develop a revised Riemann–Hilbert problem (RHP) to the Fokas–Lenells (FL) equation with a zero boundary condition, satisfying the normalization condition, and the potential of the FL equation is recovered from the asymptotic behavior of RHP when the spectral parameter goes to zero. Under the reflection-less situation, we consider the RHP with 2N simple poles and two Nth order poles, respectively, and obtain the explicit formulas of Nth order soliton and positon solutions. As applications, the first-order soliton, the second-order soliton, and positon are displayed. Additionally, the collisions of N solitons are studied, and the phase shift and space shift are displayed.

A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation

AbstractIn the Korteweg–de Vries equation (KdV) context, we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann–Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step‐type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context, our method offers same benefits as the classical binary Darboux transformation does.

Soliton hierarchies and soliton solutions of type ([formula omitted], [formula omitted] ) reduced nonlocal nonlinear Schrödinger equations of arbitrary even order

We present mixed-type reduced soliton hierarchies of nonlocal integrable nonlinear Schrödinger equations of arbitrary even order by conducting two nonlocal group reductions for the Ablowitz–Kaup–Newell–Segur matrix spectral problems. Based on specific distributions of eigenvalues and adjoint eigenvalues, we construct soliton solutions by solving the corresponding reflectionless generalized Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues.

Laguerre unitary ensembles with jump discontinuities, PDEs and the coupled Painlevé V system

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at tk,k=1,…,m. By employing the ladder operator approach to establish Riccati equations, we show that σn(t1,…,tm), the logarithmic derivative of the n-dimensional Hankel determinant, satisfies an m-variable generalization of the σ-form of a Painlevé V equation. Through investigating the Riemann–Hilbert problem for the associated orthogonal polynomials and via the Lax pair, we express σn in terms of solutions of a coupled Painlevé V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and the Lax pair. In addition, when each tk tends to the hard edge of the spectrum and n goes to infinity, the scaled σn is shown to satisfy a generalized σ-form of a Painlevé III equation.

On long-time asymptotics to the nonlocal short pulse equation with the Schwartz-type initial data: Without solitons

In this work, the initial value problem of the nonlocal short pulse (NSP) equation is studied with the Schwartz-type initial data. Our aim is to adequately study the long-time asymptotic behavior of the solution of the NSP equation in view of the initial value condition. Starting with the Lax pair of the NSP equation, we give some discussion about the NSP equation, including infinite number of conservation laws and finite-dimensional Hamiltonian function. The relevant spectral analysis is discussed, among which it is important to note that we discuss it separately due to the singularity of the spectrum. According to these results, a suitable Riemann–Hilbert (RH) problem which is used to express the solution of the NSP equation is established. By using the nonlinear steepest descent method, the original RH problem is transformed into a model RH problem, which is given by a parabolic cylinder function. The long-time asymptotic solution of the NSP equation is therefore given, especially the error term is quite different from the local situation.

Riemann–Hilbert Problems for Axially Symmetric Monogenic Functions in $${\mathbb {R}}^{n+1}$$

We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems $$\big ($$ for short RHBVPs $$\big )$$ for axially monogenic functions on Euclidean space $${\mathbb {R}}^{n+1},n\in {\mathbb {N}}$$ . With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to $$\big ({\mathcal {D}}-\alpha \big )\phi =0,\alpha \in {\mathbb {R}}$$ .

Topological Recursion and Uncoupled BPS Structures II: Voros Symbols and the tau -Function

We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland’s “BPS Riemann–Hilbert problem”. In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices Omega in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland’s tau -function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function Z_textrm{TR}, up to a simple factor.

Riemann–Hilbert problems and soliton solutions for the reverse space-time nonlocal Sasa–Satsuma equation

The main work of this paper is to study the soliton solutions and asymptotic behavior of the integrable reverse space-time nonlocal Sasa–Satsuma equation, which is derived from the coupled two-component Sasa–Satsuma system with a specific constraint. The soliton solutions of the nonlocal Sasa–Satsuma equation are constructed through solving the inverse scattering problems by Riemann–Hilbert method. Compared with local systems, discrete eigenvalues and eigenvectors of the reverse space-time nonlocal Sasa–Satsuma equation have novel symmetries and constraints. On the basis of these symmetry relations of eigenvalues and eigenvectors, the one-soliton and two-soliton solutions are obtained and the dynamic properties of these solitons are shown graphically. Furthermore, the asymptotic behaviors of two-soliton solutions are analyzed. All these results about physical features and mathematical properties may be helpful to comprehend nonlocal nonlinear system better.

Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions

We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in GL(N,{mathbb {C}}) can be written in terms of a Fredholm determinant of Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of SL(2,{mathbb {C}}) this combinatorial expression takes the form of a dual Nekrasov–Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results we also propose a definition of the tau function of the Riemann–Hilbert problem on a torus with generic jump on the A-cycle.

Riemann–Hilbert problems and N-soliton solutions of the nonlocal reverse space-time Chen–Lee–Liu equation

In this paper, the N-soliton solutions to the nonlocal reverse space-time Chen–Lee–Liu equation have been derived. Under the nonlocal symmetry reduction to the matrix spectral problem, the nonlocal reverse space-time Chen–Lee–Liu equation can be obtained. Based on the spectral problem, the specific matrix Riemann–Hilbert problem is constructed for this nonlocal equation. Through solving this associated Riemann–Hilbert problem, the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.

Inverse scattering transforms for the nonlocal Hirota–Maxwell–Bloch system

The Hirota and Maxwell–Bloch (H-MB) system is a mathematical model that can be used to describe the wave propagation in an erbium-doped nonlinear fiber with higher-order dispersion. By extending the parameter ω in the H-MB system to the complex domain, all the nonlocal forms of the H-MB system, including the reverse-space-time, complex reverse-space, complex reverse-time, complex reverse-space-time H-MB systems, are found. Then the Riemann–Hilbert problem of the nonlocal H-MB system is established to analyze the corresponding inverse scattering problem and construct the corresponding soliton solutions. When N = 1, the singularity of the one-soliton solutions of each nonlocal H-MB system is analyzed. When N = 2, we take the nonlocal reverse-space-time H-MB system and the nonlocal complex reverse-time H-MB system as examples, showing that a two-soliton can also be regarded as a superposition of two single-soliton in the nonlocal cases as .

A generalized Sasa–Satsuma equation on the half line: From Dirichlet to Neumann map

In this paper, we study the initial-boundary value (IBV) problem for a generalized Sasa–Satsuma equation with [Formula: see text] Lax pair by Fokas unified method on the half line. Based on the analyticity and asymptotics of the eigenfunctions, the IBV problem is formulated as a Riemann–Hilbert (RH) problem. Further, the global relation among IBVs is established and the map from the Dirichlet boundary value to Neumann boundary value is obtained.