arXiv subjects

Deng-Shan Wang

Publications and source records attributed to Deng-Shan Wang.

At least 19 recordsLinked to original sources

arXiv2026

Soliton gas for the derivative nonlinear Schrödinger equation: continuum dbar-problem, genus reduction and asymptotics

A soliton gas theory for the derivative nonlinear Schrödinger equation is developed, focusing on the Gerdjikov-Ivanov equation and using its gauge equivalence with the Kaup-Newell and Chen-Lee-Liu equations. Starting from the reflectionless inverse scattering problem, we formulate pure N-soliton solutions through a meromorphic Riemann-Hilbert problem and pass to the continuum limit as the discrete spectrum condenses on the planar spectral domains. Under a suitable scaling of the norming constants, the limit yields a compactly supported dbar-problem. For domains admitting a Schwarz function, in particular elliptic domains, this problem reduces to a Riemann-Hilbert problem on the associated mother body. For the elliptic soliton gas, the large-x and long-time asymptotic behaviors are established: the solution decays as x->+infty, approaches an elliptic finite-gap background as x->-infty, and exhibits stratified long-time sectors described by one-, two-, and three-phase Riemann theta functions. A distinctive feature of derivative nonlinear Schrodinger equation is the z->-z symmetry, which induces a quotient reduction of the effective Abelian geometry via lambda=z^2. We further derive a kinetic equation for the effective velocity of the elliptic soliton gas and an Its-Izergin-Korepin-Slavnov type Fredholm determinant representation of the continuum tau-function.

math.AP↗

arXiv2026

Riemann-Hilbert problem and long-time asymptotics of the Yajima-Oikawa equation

The Yajima-Oikawa equation is an integrable long wave-short wave resonance interaction model arising as a deformation of the Zakharov system for Langmuir waves coupled to ion-acoustic waves. In this work, a Riemann-Hilbert approach is developed for the Cauchy problem for the Yajima-Oikawa equation with rapidly decaying initial data. A main novelty is the formulation of a direct and inverse scattering theory adapted to its third-order spectral problem, including a detailed treatment of the singular spectral point $k=0$. The associated Riemann-Hilbert problem is expressed in terms of two reflection coefficients determined by the initial data, together with possible discrete eigenvalues and norming constants. We prove a vanishing lemma which ensures the unique solvability of the Riemann-Hilbert problem under suitable positivity assumptions, and hence obtain a rigorous reconstruction formula for the solution. We also classify the admissible discrete spectrum and derive exact pure soliton solutions from the reflectionless Riemann-Hilbert problem. In the solitonless case, we apply the Deift-Zhou nonlinear steepest descent method to obtain rigorous long-time asymptotic formulas in the different regions of the upper $(x,t)$-plane. The leading oscillatory behavior of the short-wave component is described explicitly in terms of the reflection coefficients evaluated at the stationary phase points, while the long-wave component is shown to be of lower order away from the transition region. These results provide, to the best of our knowledge, the first Riemann-Hilbert framework for the long-time asymptotic analysis of the Yajima-Oikawa equation in the presence of continuous spectrum.

nlin.SI↗

arXiv2026

Arbitrary-genus dark soliton gases in the defocusing nonlinear Schrödinger hydrodynamics

The defocusing nonlinear Schrödinger hydrodynamics supports exact dark solitons under finite density boundary conditions. However, the dark soliton gas, an interacting ensemble of dark solitons, has not yet been studied. In this work, we introduce an arbitrary-genus potential of dark soliton gases by considering the limit of the $\mathcal{N}$-dark soliton as $\mathcal{N}\to \infty$. The large-space asymptotics and long-time evolution of this dark soliton gas potential are analytically investigated through Deift-Zhou nonlinear steepest descent approach. The genus-$N$ dark soliton gas potential approaches the genus-$N$ finite-gap solution as $x \to -\infty$ and the background $1$ as $x \to +\infty$. In the long-time evolution, as the self-similar variable $ξ=x/t$ increases, the gas configuration exhibits a cascade of behaviours, passing from unmodulated and modulated genus-$N$ regions and progressively reducing the genus down to the planar region (unmodulated genus-$0$ region). Notably, the evolution of lower-genus soliton gases can be embedded within that of higher-genus gases, exhibiting identical dynamics within specific regimes. This phenomenon is encoded by the underlying spectra. We also include numerical validations, in perfect agreement with the theoretical predictions.

math-ph↗

arXiv2026

Long-time asymptotics of the Newell equation on the line

In 1978, A. C. Newell [SIAM J. Appl. Math. 35(4) (1978) 650-664] proposed an exactly solvable model called Newell equation, which simulates the investigation of significant interaction mechanism between long and short waves. Nearly fifty years have passed, yet the long-time asymptotics of the Newell equation remains an open problem to date, with no results reported. In this work, the long-time asymptotic behaviors of the solutions to this model under Schwartz class initial conditions are studied by using the Riemann-Hilbert formulation. Through direct and inverse scattering analysis, the corresponding Riemann-Hilbert problem is formulated, and its relationship with the solution to the initial-value problem of the Newell equation is established. The existence and uniqueness of the solution to the Riemann-Hilbert problem is proved by vanishing lemma. Subsequently, the asymptotic expressions of the solution to the initial-value problem in the dispersive wave region are obtained by using the Deift-Zhou nonlinear steepest descent method. This work extends Newell's original results, providing a rigorous proof for the findings presented in Section 4 of his paper, along with explicit expressions. Furthermore, the comparison between direct numerical simulations and the theoretical results obtained in this paper demonstrates the reliability of the asymptotic expressions.

math-ph↗

arXiv2026

Long-time asymptotics of the Tzitzéica equation on the line

In this paper, the renowned Riemann-Hilbert method is employed to investigate the initial value problem of Tzitzéica equation on the line. Initially, our analysis focuses on elucidating the properties of two reflection coefficients, which are determined by the initial values. Subsequently, leveraging these reflection coefficients, we construct a Riemann-Hilbert problem that is a powerful tool to articulate the solution of the Tzitzéica equation. Finally, the nonlinear steepest descent method is applied to the oscillatory Riemann-Hilbert problem, which enables us to delineate the long-time asymptotic behaviors of solutions to the Tzitzéica equation across various regions. Moreover, it is shown that the leading-order terms of asymptotic formulas match well with direct numerical simulations.

math-ph↗

arXiv2026

Genus two KdV soliton gases and their long-time asymptotics

This paper employs the Riemann-Hilbert problem to provide a comprehensive analysis of the asymptotic behavior of the high-genus Korteweg-de Vries soliton gases. It is demonstrated that the two-genus soliton gas is related to the two-phase Riemann-Theta function as $x \to +\infty$, and approaches to zero as $x \to -\infty$. Additionally, the long-time asymptotic behavior of this two-genus soliton gas can be categorized into five distinct regions in the $x$-$t$ plane, which from left to right are rapidly decay, modulated one-phase wave, unmodulated one-phase wave, modulated two-phase wave, and unmodulated two-phase wave. Moreover, an innovative method is introduced to solve the model problem associated with the high-genus Riemann surface, leading to the determination of the leading terms, which is also related with the multi-phase Riemann-Theta function. A general discussion on the case of arbitrary $N$-genus soliton gas is also presented.

nlin.SI↗

arXiv2026

Long-time asymptotics of the Sawada-Kotera equation on the line

The Sawada-Kotera (SK) equation is an integrable system characterized by a third-order Lax operator and is related to the modified Sawada-Kotera (mSK) equation through a Miura transformation. This work formulates the Riemann-Hilbert problem associated with the SK and mSK equations by using direct and inverse scattering transforms. The long-time asymptotic behaviors of the solutions to these equations are then analyzed via the Deift-Zhou steepest descent method for Riemann-Hilbert problems. It is shown that the asymptotic solutions of the SK and mSK equations are categorized into four distinct regions: the decay region, the dispersive wave region, the Painlevé region, and the rapid decay region. Notably, the Painlevé region is governed by the F-XVIII equation in the Painlevé classification of fourth-order ordinary differential equations, a fourth-order analogue of the Painlevé transcendents. This connection is established through the Riemann-Hilbert formulation in this work. Similar to the KdV equation, the SK equation exhibits a transition region between the dispersive wave and Painlevé regions, arising from the special values of the reflection coefficients at the origin. Finally, numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.

nlin.SI↗

arXiv2025

Long-time asymptotics of the good Boussinesq equation and its modified version: Painlevé region

This work investigates the long-time asymptotic behaviors of initial value problem for the good Boussinesq equation and the modified Boussinesq equation in Painlevé region. The Deift-Zhou steepest descent method is used to deform the associated $3 \times 3$ Riemann-Hilbert problem to the Painlevé IV model. Then asymptotic formulas for the modified Boussinesq equation in both the Painlevé region and the Painlevé transition region are derived, characterized by the Clarkson-McLeod solution of the Painlevé IV equation. Additionally, the leading-order term of the good Boussinesq equation in Painlevé region is obtained via the Miura transformation. The theoretical asymptotic solutions are validated against direct numerical simulations, confirming the accuracy of the asymptotic analysis.

math.AP↗

arXiv2025

Long-time behaviors of the two-component nonlinear Klein-Gordon equation: higher-order asymptotics

This work investigates the long-time asymptotic behaviors of solutions to the initial value problem of the two-component nonlinear Klein-Gordon equation by inverse scattering transform and Riemann-Hilbert formulism. Two reflection coefficients are defined and their properties are analyzed in detail. The Riemann-Hilbert problem associated with the initial value problem is constructed in term of the two reflection coefficients. The Deift-Zhou nonlinear steepest descent method is then employed to analyze the Riemann-Hilbert problem, yielding the long-time asymptotics of the solution in different regions. Specifically, a higher-order asymptotic expansion of the solution inside the light cone is provided, and the leading term of this asymptotic solution is compared with results from direct numerical simulations, showing excellent agreement. This work not only provides a comprehensive analysis of the long-time behaviors of the two-component nonlinear Klein-Gordon equation but also offers a robust framework for future studies on similar nonlinear systems with third-order Lax pair.

nlin.SI↗

arXiv2025

Long-time asymptotics of the defocusing mKdV equation with step initial data

This work investigates the long-time asymptotics of solution to defocusing modified Korteweg-de Vries equation with a class of step initial data. A rigorous asymptotic analysis is conducted on the associated Riemann-Hilbert problem by applying Deift-Zhou nonlinear steepest descent method. In this process, the construction of odd-symmetry g-function is generalized and the method of genus reduction on the Riemann-theta function is proposed via conformal transformation and symmetries. It is revealed that for sufficiently large time, the solution manifests a tripartite spatiotemporal structure, i.e., in the left plane-wave region, the solution decays to a modulated plane wave with oscillatory correction; in the central dispersive shock wave region, the solution is governed by a modulated elliptic periodic wave; in the right plane wave region, the solution converges exponentially to a constant. The results from the long-time asymptotic analysis have been shown to match remarkably well with that obtained by direct numerical simulations.

math.AP↗

arXiv2025

Long-time asymptotics of the KdV equation with delta function initial profile

This work investigates the long-time asymptotic behaviors of the solution to the KdV equation with delta function initial profiles in different regions, employing the Riemann-Hilbert formulation and Deift-Zhou nonlinear steepest descent method. When the initial value is a delta potential well, the asymptotic solution is predominantly dominated by a single soliton in certain region for $x>0$, while in other regions, the dispersive tails including self-similar region, collisionless shock region and dispersive wave region, play a more significant role. Conversely, when the initial value is a delta potential barrier, the soliton region is absent, although the dispersive tails still persist. Moreover, the general delta function initial profile with $L$-spikes is also studied and it is proved that one to $L$ solitons will be generated in soliton region, which depends on the sizes of the distance and height of the spikes. The leading-order terms of the solution in each region are derived, highlighting the efficacy of the Riemann-Hilbert formulation in elucidating the long-time behaviors of integrable systems.

math.AP↗

arXiv2024

The generalized Darboux matrices with the same poles and their applications

Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$ Lax pair, in which the coefficient matrices are polynomials of spectral parameter. The first-order monic Darboux matrix is constructed explicitly and its classification theorem is presented. Then by using the solutions of the corresponding adjoint Lax pair, the $n$-order monic Darboux matrix and its inverse, both sharing the same unique pole, are derived explicitly. Further, a theorem is proposed to describe the invariance of Darboux matrix regarding pole distributions in Darboux matrix and its inverse. Finally, a unified theorem is offered to construct formal Darboux transformation in general form. All Darboux matrices expressible as the product of $n$ first-order monic Darboux matrices can be constructed in this way. The nonlocal focusing NLS equation, the focusing NLS equation and the Kaup-Boussinesq equation are taken as examples to illustrate the application of these Darboux transformations.

nlin.SI↗

arXiv2024

Interactions of soliton and mean field in KdV equation with well type initial data

For the KdV equation with well-type initial value, the interaction between the trial soliton and the mean field is studied. The well initial value will lead to the appearance of rarefaction wave and dispersion shock wave, and there will be a linear wave region after a long time. The interaction between trial soliton and mean field is described within the framework of Whitham modulation theory, and the trajectory of soliton is given. The predicted soliton amplitude and phase changes are numerically confirmed, verifying the correctness of the theoretical analysis.

math-ph↗

arXiv2024

Modulation theory of soliton-mean flow in KdV equation with box type initial data

For the KdV equation with box type initial data, the interaction between a trial soliton and large-scale dispersive mean flow is studied theoretically and numerically. The pure box initial value can cause rarefaction wave and dispersive shock wave, and can create an area of soliton train. The key to the interaction of soliton and mean flow is that the dynamic evolutions of the mean flow and the local soliton can be described by the same modulation system. The soliton modulation system is derived from the degenerations of the two-genus Whitham modulation system. Considering the influence of rarefaction wave, dispersive shock wave and soliton train on the trial soliton, in the framework of Whitham modulation theory, the equation describing the soliton trajectory and the changes in amplitude and phase shift are given explicitly. The predicted results are compared with the numerical simulations, which verifies the corrections of the theoretical analysis. The exotic interaction phenomena between soliton and mean flow found in this work have broad applications to shallow water soliton propagations and real soliton experiments in fluid dynamics.

nlin.PS↗

arXiv2024

Miura transformations and large-time behaviors of the Hirota-Satsuma equation

The good Boussinesq equation has several modified versions such as the modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma equation. This work builds the full relations among these equations by Miura transformation and invertible linear transformations and draws a pyramid diagram to demonstrate such relations. The direct and inverse spectral analysis shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation has simple pole at origin, the solution of Riemann-Hilbert problem for the good Boussinesq equation has double pole at origin, while the solution of Riemann-Hilbert problem for the modified Boussinesq equation and Mikhailov-Lenells equation doesn't have singularity at origin. Further, the large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz class initial value is studied by Deift-Zhou nonlinear steepest descent analysis. In such initial condition, the asymptotic expressions of the Hirota-Satsuma equation and good Boussinesq equation away from the origin are derived and it is shown that the leading term of asymptotic formulas matches well the direct numerical simulations.

nlin.SI↗

arXiv2023

Long-time asymptotics of the Sawada-Kotera equation and Kaup-Kupershmidt equation on the line

Both Sawada-Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are integrable systems with third-order Lax operator. Moreover, they are related with the same modified nonlinear equation (called modified SK-KK equation) by Miura transformations. This work first constructs the Riemann-Hilbert problem associated with the SK equation, KK equation and modified SK-KK equation by direct and inverse scattering transforms. Then the long-time asymptotics of these equations are studied based on Deift-Zhou steepest-descent method for Riemann-Hilbert problem. Finally, it is shown that the asymptotic solutions match very well with the results of direct numerical simulations.

nlin.SI↗

arXiv2023

Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics

The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work, the long-time asymptotics for the defocusing nonlinear Schrödinger equation with general step-like initial data is investigated by the Whitham modulation theory and Riemann-Hilbert formulation. The Whitham modulation theory shows that there are six cases for the initial discontinuity problem according to the orders of the Riemann invariants. The leading-order terms and the corresponding error estimates for each region of the six cases are formulated by the Deift-Zhou nonlinear steepest descent method for oscillatory Riemann-Hilbert problems. It is demonstrated that the long-time asymptotic solutions match very well with the results from Whitham modulation theory and the numerical simulations.

math.AP↗

arXiv2022

Normalized solution to the nonlinear p-Laplacian equation with an L^2 constrain: mass supercritical case

In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\geq3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -Δ_{p}u+{\vert u\vert}^{p-2}u=f(u)-μu \quad \text{ in } \mathbb{R}^N,\\ {\Vert u\Vert}^2_{L^2(\mathbb{R}^N)}=m,\\ u\in W^{1,p}(\mathbb{R}^N)\cap L^2(\mathbb{R}^N), \end{cases} \end{equation*} where $-Δ_{p}u=div\left( {\vert\nabla u\vert}^{p-2}\nabla u \right)$, $2\leq p 0$, $μ\in\mathbb{R}$ will appear as a Lagrange multiplier and the continuous nonlinearity $f$ satisfies mass supercritical conditions. We mainly study the behavior of ground state energy $E_m$ with $m>0$ changing within a certain range and aim at extending nonlinear scalar field equation when $p=2$ and reducing the constraint condition of nonlinearity $f$.

math.AP↗