Hilbert Problem Research Articles

Soliton resolution for the Ostrovsky–Vakhnenko equation

We consider the Cauchy problem of the Ostrovsky–Vakhnenko (OV) equation expressed in the new variables (y,τ)q3−q(logq)yτ−1=0 with Schwartz initial data q0(y)>0 which supports smooth and single-valued solitons. It is shown that the solution to the Cauchy problem for the OV equation can be characterized by a 3 × 3 matrix Riemann–Hilbert (RH) problem. Furthermore, by employing the ∂̄-steepest descent method to deform the RH problem into solvable models, we derive the soliton resolution for the OV equation across two space–time regions: y/τ>0 and y/τ<0. This result also implies that the N-soliton solutions of the OV equation in variables (y,τ) are asymptotically stable.

Riemann–Hilbert problem and soliton solutions with their asymptotic analysis for the focusing nonlocal Hirota equation with step-like initial data

The Riemann–Hilbert (RH) problem is developed to study the Cauchy problem of focusing space symmetric nonlocal Hirota (fsNH) equation with step-like initial data. Firstly, we analyze the characteristics of the Jost solutions and spectral functions, including analyticity, symmetry, and asymptotics as k→∞ and k→0. Then, we discuss three cases of zero point distributions of the spectral functions, and derive the expressions of spectral functions by constructing the special analytic functions and establishing the scalar RH problem. Furthermore, the constraint relations among the zeros of the spectral functions are analyzed. Next, the corresponding RH problem equipped with residual and singularity conditions is successfully derived. These singularity conditions are then transformed into a general residue condition, which reduces the problem of solving the RH problem with reflection-less potential to solving a system of algebraic equations. Ultimately, the corresponding soliton solutions of the fsNH equation and their asymptotics are derived from the solution of the RH problem.

Nonlocal combined nonlinear Schrödinger–Gerdjikov–Ivanov model: Integrability, Riemann–Hilbert problem with simple and double poles, Cauchy problem with step-like initial data

In this paper, we propose three new types of the integrable nonlocal combined nonlinear Schrödinger–Gerdjikov–Ivanov (NLS-GI) models. By the Riemann–Hilbert approach, we discuss the Cauchy problem of the reverse-space-time nonlocal combined NLS-GI model with step-like initial data: u(z, 0) = o(1) for z → −∞ and u(z, 0) = A + o(1) for z → +∞, where A is an arbitrary positive constant. First of all, we give an integrable nonlocal combined NLS-GI model and its Lax pair. Then, we consider the analytical and asymptotic behaviors, symmetries, and scattering matrix of the Jost solutions. Finally, we discuss the Cauchy problem for the nonlocal combined NLS-GI model with step-like initial data.

The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line

The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and this study addresses this issue by extending the application of the Fokas method to the higher-order Gerdjikov-Ivanov equation on the half line (−∞,0]. We have demonstrated the exclusive representation of the potential function u(z,t) in the higher-order Gerdjikov–Ivanov equation through the solution of a Riemann–Hilbert problem. The characteristic function is partitioned on the complex plane, and we obtain the jump matrix between each partition based on the positive and negative values of the partition as well as the spectral matrix determined by the initial data and boundary value data. The findings suggest that the spectral functions are not mutually independent; instead, they conform to a global relationship. The novel aspect of this study is the application of the Fokas method to a previously unexplored case, contributing to the theoretical and practical understanding of complex partial differential equations and filling a gap in the treatment of boundary conditions.

Integrability and Inverse Scattering Transform of the Modified Benjamin-Ono Equation

Abstract This paper presents a B{"a}cklund transformation, the Lax representation, and conserved quantities for the modified Benjamin--Ono equation. The initial problem of the modified Benjamin--Ono equation on the line was studied by the inverse scattering transform method, presenting a nonlocal Riemann--Hilbert problem in the inverse problem to reconstruct the explicit potential function. Further, the exact $N$-soliton solutions and long--time asymptotic behavior are provided, respectively. We also graphically show that the propagation of soliton solutions is consistent with the result of large-time asymptotic forms. As a revised version, the mBO equation incorporates the solution features of the BO equation, and its solutions are presented in logarithmic form, which seeks to describe related physical phenomena more accurately. Our results will contribute to further exploration of physical and mathematical problems related to the mBO equation.

Riemann–Hilbert approach and soliton solutions for the Lakshmanan–Porsezian–Daniel equation with nonzero boundary conditions

We construct the Riemann–Hilbert problem of the Lakshmanan–Porsezian–Daniel equation with nonzero boundary conditions, and use the Laurent expansion and Taylor series expansion to obtain the exact formulas of the soliton solutions in the case of a higher-order pole and multiple higher-order poles. The dynamic behaviors of a simple pole, a second-order pole and a simple pole plus a second-order pole are demonstrated.

Multi-geometric discrete spectral problem with several pairs of zeros for Sasa–Satsuma equation

Sasa–Satsuma equation is proposed to model the propagation and interaction of the sub-picosecond or femtosecond pulses in a monomode optical fiber. Different from several integrable equations in the Ablowitz–Kaup–Newell–Segur system, the higher-order zeros of Riemann–Hilbert problem for the Sasa–Satsuma appear in quadruples. A new approach to study the multi-geometric discrete spectral problem with several pairs of zeros for the Sasa–Satsuma equation is proposed. Thus, the complete soliton solutions corresponding to the higher-order zeros with arbitrary geometric and algebraic multiplicities are derived. Moreover, the inelastic interactions between or among the solitons corresponding to the higher-order non-elementary zeros exhibit the shape-changing phenomena.

A Riemann–Hilbert approach to the existence results for the Benjamin–Ono equation on a half-line

The main problem addressed in this paper is to study the local existence in time of solutions to the non-homogeneous Neumann initial boundary value problem for the Benjamin–Ono equation on a half-line. In this result, we observe the influence of the boundary data on the behavior of solutions. In order to obtain the characterization of the solution it is essential to use the theory concerning the Riemann–Hilbert problem. We prove local existence in time of the solutions.

Exploring Limit Cycles of Differential Equations through Information Geometry Unveils the Solution to Hilbert's 16th Problem.

The detection of limit cycles of differential equations poses a challenge due to the type of the nonlinear system, the regime of interest, and the broader context of applicable models. Consequently, attempts to solve Hilbert's sixteenth problem on the maximum number of limit cycles of polynomial differential equations have been uniformly unsuccessful due to failing results and their lack of consistency. Here, the answer to this problem is finally obtained through information geometry, in which the Riemannian metrical structure of the parameter space of differential equations is investigated with the aid of the Fisher information metric and its scalar curvature R. We find that the total number of divergences of |R| to infinity provides the maximum number of limit cycles of differential equations. Additionally, we demonstrate that real polynomial systems of degree n≥2 have the maximum number of 2(n-1)(4(n-1)-2) limit cycles. The research findings highlight the effectiveness of geometric methods in analyzing complex systems and offer valuable insights across information theory, applied mathematics, and nonlinear dynamics. These insights may pave the way for advancements in differential equations, presenting exciting opportunities for future developments.

Optical solitons and their propagations in a time-varying coefficient modified nonlinear Schrödinger equation with non-zero boundary conditions via Riemann–Hilbert method

In this article, a time-varying coefficient modified nonlinear Schrödinger equation (tvcmNLSE) with non-zero boundary conditions (NZBCs) at infinity, phase term of which contains time-varying coefficients, is proposed to describe the nonlinear propagation behavior of optical pulses in non-uniform optical fibre. Meanwhile, the Riemann–Hilbert method (RHM) for the tvcmNLSE with NZBCs is presented for the first time. Specifically, a special transformation is constructed to convert the tvcmNLSE with NZBCs into the case with constant NZBCs, and the associated asymptotic spectral problem (ASP) is obtained. By introducing a suitable two-piece Riemann surface to map the original spectral parameter into a single-valued one, and analyzing the ASP, we construct the corresponding Jost solutions and spectral matrix, and sequentially analyze their analytical, symmetric, and asymptotic properties. Then, a generalized Riemann–Hilbert problem (RHP) is established that relates to the converted tvcmNLSE with constant NZBCs. Using the reconstruction formula, we further derive N-soliton solution with NZBCs in the absence of reflection potential, and analyze the dynamic characteristics of single and double solitons. Most notably, by modulating discrete spectral points (DSPs) and time-varying coefficients, we discover some previously unreported new waveforms, such as parabolic and periodic solitons with NZBCs. These waveforms have important implications in both theory and practice. In addition, the influences of different time-varying coefficients, NZBCs, and DSPs on soliton solutions are also discussed and analyzed. This study shows that the selection of time-varying coefficients, distribution of DSPs, and constraints from NZBCs all have significant impacts on the properties of soliton solutions.

Initial Boundary Value Problem for the Coupled Kundu Equations on the Half-Line

In this article, the coupled Kundu equations are analyzed using the Fokas unified method on the half-line. We resolve a Riemann–Hilbert (RH) problem in order to illustrate the representation of the potential function in the coupled Kundu equations. The jump matrix is obtained from the spectral matrix, which is determined according to the initial value data and the boundary value data. The findings indicate that these spectral functions exhibit interdependence rather than being mutually independent, and adhere to a global relation while being connected by a compatibility condition.

On the crossing limit cycles created by a discontinuous piecewise differential system formed by three linear Hamiltonian saddles

We study planar discontinuous piecewise differential systems formed by three linear Hamiltonian saddles separated by the non-regular line Σ = { ( x , y ) ∈ R 2 : ( y = 0 ) ∨ ( x = 0 ∧ y ≥ 0 ) } . We prove that when the linear Hamiltonian saddles are homogeneous they have no limit cycles, and when they are non homogeneous they can have at most three limit cycles having exactly one point on each branch of Σ, and at most one limit cycle having four intersection points on Σ. Moreover we show that they can have at most one limit cycle having four intersection points on Σ and three limit cycles having three intersection points on Σ, simultaneously. Thus we have solved the extension of the 16th Hilbert problem to this class of piecewise differential systems. Furthermore we show that for the three types of combinations of the limit cycles here studied in two of them the upper bound is sharp by providing examples of these systems with the maximum number of possible limit cycles. For the remaining type of limit cycles the upper bound is four and we have examples with three limit cycles. So at this moment it is an open problem to know if the sharp upper bound is either four or three.

Soliton resolution for the modified short pulse equation with the weighted Sobolev initial data

In this paper, the Cauchy problem of the modified short pulse (mSP) equation with initial conditions in weighted Sobolev space is studied by using [Formula: see text]-steepest descent method. Based on the spectral analysis of Lax pair and the transformations of field variables and independent variables, a basic Riemann–Hilbert problem (RHP) is established, and the solution of Cauchy problem for the mSP equation is transformed into the corresponding RHP. The asymptotic expansion of the solution of mSP equation is derived in a fixed space-time cone [Formula: see text] According to this, the soliton resolution conjecture of the mSP equation is given, in which the leading term is characterized by an [Formula: see text]-soliton on the discrete spectrum, the second term comes from the soliton–radiation interactions on the continuum spectrum, and the error term [Formula: see text] is generated by the corresponding [Formula: see text]-problem.

The revised Riemann–Hilbert approach to the Kaup–Newell equation with a non-vanishing boundary condition: Simple poles and higher-order poles

With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.

A family of generalized Liénard differential systems with an explicit limit cycle

One of the important problems in the qualitative theory of real planar differential systems is limit cycles i.e., isolated periodic solutions in the set of all periodic this concept was defined at the end of the 19th century by Poincaré, and with the works of van der Pol and Liénard. The significance of these isolated solutions lies in their key role in understanding the dynamics of a certain differential system. Afterward, Hilbert specified a list of 23 problems for the advancement of mathematical science, and from then it started intensive research on these problems throughout the 20th century. Of the 23 problems, only the so-called 16th Hilbert’s problem and the Riemann conjecture remain open until now. The second part of the 16th Hilbert problem has two parts and asks for an upper bound on the number of possible limit cycles and their positions for a planar polynomial differential system of a given degree. The theory of the limit cycle operating as an indispensable mathematical tool has found broad and important applications in modern physics, chemical, biological, ,and other fields. Then the progress of these fields, in turn, promotes limit cycle research. In this paper, we prove the existence and uniqueness of the limit cycle for a family of generalized polynomial Liénard differential systems, the explicit expression of this limit cycle is given. An example illustrating our results is additionally presented.

Riemann–Hilbert method to the Ablowitz–Ladik equation: Higher‐order cases

AbstractWe focus on the Ablowitz–Ladik equation on the zero background, specifically considering the scenario of pairs of multiple poles. Our first goal was to establish a mapping between the initial data and the scattering data, which allowed us to introduce a direct problem by analyzing the discrete spectrum associated with pairs of higher‐order zeros. Next, we constructed another mapping from the scattering data to a matrix Riemann–Hilbert (RH) problem equipped with several residue conditions set at pairs of multiple poles. By characterizing the inverse problem on the basis of this RH problem, we are able to derive higher‐order soliton solutions in the reflectionless case.

On the convergence of Fourier spectral methods involving noncompact operators

Abstract Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann–Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.

Hilbert's tenth problem for lacunary entire functions of finite order

AbstractIn the context of Hilbert's tenth problem, an outstanding open case is that of complex entire functions in one variable. A negative solution is known for polynomials (by Denef) and for exponential polynomials of finite order (by Chompitaki, Garcia‐Fritz, Pasten, Pheidas, and Vidaux), but no other case is known for rings of complex entire functions in one variable. We prove a negative solution to the analogue of Hilbert's tenth problem for rings of complex entire functions of finite order having lacunary power series expansion at the origin.

Riemann–Hilbert problem for a (3+1)-dimensional nonlinear evolution equation

This paper concentrates on a (3+1)-dimensional nonlinear evolution equation. By introducing a transformation, the (3+1)-dimensional nonlinear evolution equation is decomposed into three integrable (1+1)-dimensional models. On the basis of a quartet Lax pair, we build the associated matrix Riemann–Hilbert problem. As a consequence, solving the obtained matrix Riemann–Hilbert problem with the identity jump matrix, corresponding to the reflectionless, the soliton solution to the (3+1)-dimensional nonlinear evolution equation is acquired. Specially, the one-soliton solutions are worked out and analyzed graphically.

Riemann–Hilbert problem for the defocusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

Abstract In this work, Riemann-Hilbert approach is demonstrated to investigate the defocusing Lakshmanan-Porsezian-Daniel equation with fully asymmetric non-zero boundary conditions. In contrast to the symmetry case, this paper focuses at the branch points related to the scattering problem rather than using the Riemann surfaces. For the direct problem, we analyse the Jost solution of lax pairs and some properties of scattering matrix, including two kinds of symmetries. The inverse problem at branch points can be presented, corresponding to the associated Riemann-Hilbert. Moreover, we investigate the time evolution problem and estimate the value of solving the solutions by Jost function. For the inverse problem, we construct it as a Riemann-Hilbert problem and formulate the reconstruction formula for the defocusing Lakshmanan-PorsezianDaniel equation. The solutions of the Riemann-Hilbert problem can be constructed by estimating the solutions. Finally, we work out the solutions with fully asymmetric non-zero boundary conditions precisely via utilizing Sokhotski-Plemelj formula and the square of the negative column transformation with the assistance of Riemann surfaces. These results are valuable for understanding physical phenomena and developing further applications of optical problems.