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Communications in Theoretical Physics 2024年第1期76卷 1-6页

作者： engui fan Yanxi Zhang School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Science Fudan UniversityShanghai200433China Tianjin Xinhua High School No.99 machang RoadTianjin300204China

In this paper,we address interesting soliton resolution,asymptotic stability of N-soliton solutions and the Painleve asymptotics for the Kundu-Eckhaus(KE)equation with nonzero boundary conditions iq_(t)+q_(xx)-2(l|q|^(2)-1)q+4β^(2)(lql^(4)-1)q+4iβ(lql^(2))_(x)q=0,q(x,0)=q_(0)(x)-±1,x→±∞.The key to proving these results is to establish the formulation of a Riemann-Hilbert(RH)problem associated with the above Cauchy problem and find its connection with the RH problem of the defocusing NLS *** the■-steepest descent method and the results of the defocusing NLS equation,we find complete leading order approximation formulas for the defocusing KE equation on the whole(x,t)half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region,Zakharov-Shabat asymptotics in a solitonless region and the Painlevéasymptotics in two transition regions.

关键词： defocusing Kundu-Eckhaus equation Riemann-Hilbert problems steepest descent method soliton resolution asymptotic stability Painleve transcendents

Existence of Global Solutions to the Nonlocal mKdV Equation on the Line

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Chinese Annals of Mathematics,Series B 2024年第4期45卷 497-528页

作者： Anran LIU engui fan School of Mathematical Sciences Fudan University

In this paper, the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries（nonlocal mKdV for short） equation under the initial data u0 ∈ H3（R） ∩H1,1（R） with the L1（R） small-norm assumption. A Lipschitz L2-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem. The map from initial potential to reflection coefficient is obtained in direct scattering transform. The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator. The bijective relation naturally yields the existence of global solutions in a Sobolev space H3（R） ∩ H1,1（R） to the Cauchy problem.

关键词： Global solutions Lipschitz continuous Nonlocal mKdV equation Plemelj projection operator Riemann-Hilbert problem

Long-time asymptotic behavior for the nonlocal nonlinear Schro¨dinger equation in the solitonic region

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Science China Mathematics 2024年

作者： Gaozhan Li Yiling Yang engui fan School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences Fudan University

In this paper, we extend the ■-steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schr?dinger（NNLS） equation with weighted Sobolev initial data iqt+qxx+2σ■2（x, t）q（-x, t） = 0,q（x, 0） = q0（x）,where q0∈ H1,1（R）. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem is expressed in terms of the solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. We further obtain the asymptotic expansion of the solution to the Cauchy problem for the NNLS equation in the solitonic region. The leading term is soliton solutions, the second term is the interaction between solitons and dispersion, and the error term comes from a corresponding?ˉ-*** with the asymptotic results on the classical NLS equation, the major difference is the second and third terms of the asymptotic expansion for the NNLS equation, which were affected by a function depending on the scattering data and the stationary phase point.

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Long Time Asymptotics Behavior of the Focusing Nonlinear Kundu-Eckhaus Equation

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Chinese Annals of Mathematics,Series B 2023年第2期44卷 235-264页

作者： Ruihong MA engui fan School of Mathematical Sciences Fudan UniversityShanghai 200433China

The authors study the Cauchy problem for the focusing nonlinear KunduEckhaus(KE for short)equation and construct the long time asymptotic expansion of its solution in fixed space-time cone with C(x_(1),x_(2),v_(1),v_(2))={(x,t)∈R^(2):x=x_(0)+vt,x_(0)∈[x_(1),x_(2)],v∈[v_(1),v_(2)]}.By using the inverse scattering transform,Riemann-Hilbert approach and δ steepest descent method,they obtain the lone time asymptotic behavior of the solution,at the same time,they obtain the solitons in the cone compare with the all N-soliton the residual error up to order O(t-^(3/4)).

关键词： Focusing Kundu-Eckhaus equation Riemann-Hilbert problem δsteepest descent method

Long-time Asymptotic Behavior for the Derivative Schrödinger Equation with Finite Density Type Initial Data

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Chinese Annals of Mathematics,Series B 2022年第6期43卷 893-948页

作者： Yiling YANG engui fan School of Mathematical Sciences Fudan UniversityShanghai 200433China

In this paper,the authors apply■steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data iqt+qxx+1(lq|^(2)q)_(x)=0,q(x,0)=q0(x),where lim x→±∞ qo(x)=g0(x)=q±and|q±|=*** on the spectral analysis of the Lax pair,they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert *** compute the long time asymptotic expansion of the solution in differeit space-time *** the regionζ=x/t with|ζ+2|1,the long time asymptotic is given by q(x,t)=t(∞)^(-2)q^(r)Λ(x,t)-t^(-1/2)if11+O(t^(-3/4)) in which the leading term is N(I)solitons,the second t^(-1/2)order term is soliton-radiation interactions and the third term is a residual error from a■*** results are verification of the soliton resolution conjectuore for the derivative Schrödinger *** their case of finite density type initial data,the phase functionθ(z)is more complicated that in finite mass initial ***,two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis,respectively.

关键词： Derivative Schrödinger equation Riemann-Hilbert problem ■steepest descent method Long-time asymptotics Soliton resolution Asymptotic stability

A Riemann-Hilbert Approach to the Harry-Dym Equation on the Line

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Chinese Annals of Mathematics,Series B 2016年第3期37卷 373-384页

作者： Yu XIAO engui fan Department of Mathematics Fudan University

In this paper, the authors consider the Harry-Dym equation on the line with decaying initial value. They construct the solution of the Harry-Dym equation via the solution of a 2 × 2 matrix Riemann-Hilbert problem in ... 详细信息

关键词： Harry-Dym equation Riemann-Hilbert problem Initial-value problem One-cusp soliton solution

The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

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Science China Mathematics 2009年第1期52卷 147-159页

作者： LUO Lin fan engui Department of Mathematics Shanghai Second Polytechnic UniversityShanghai 201209China School of Mathematical Sciences Fudan UniversityShanghai 200433China Department of Mathematics Xiaogan UniversityXiaogan 432100China

An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ-... 详细信息

关键词： discrete zero curvature equation integrable couplings τ-symmetry algebra 35Q51

N-SOLITON SOLUTION OF THE KUNDU-TYPE EQUATION VIA RIEMANN-HILBERT APPROACH

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Acta Mathematica Scientia 2020年第1期40卷 113-126页

作者： Lili WEN Ning ZHANG engui fan School of Mathematical Sciences Fudan UniversityShanghai 200433China Department of Basic Courses Shandong University of Science and TechnologyTaian 266510China

In this article, we focus on investigating the Kundu-type equation with zero boundary condition at infinity. Based on the analytical and symmetric properties of eigenfunctions and spectral matrix of its Lax pair, a Riemann-Hilbert problem for the initial value problem of the Kundu-type equation is constructed. Further through solving the regular and nonregular Riemann-Hilbert problem, a kind of general N-soliton solution of the Kundu-type equation are presented. As special cases of this result, the N-soliton solution of the Kaup-Newell equation, Chen-Lee-Liu equation, and Gerjikov-Ivanov equation can be obtained respectively by choosing different parameters.

关键词： the Kundu-type equation Lax pair Riemann-Hilbert problem soliton solution

THE GLM REPRESENTATION OF THE TWO-COMPONENT NONLINEAR SCHR?DINGER EQUATION ON THE HALF-LINE

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Acta Mathematica Scientia 2018年第6期38卷 1846-1860页

作者： Qiaozhen ZHU engui fan Jian XU School of Mathematical Sciences Fudan UniversityShanghai 200433China College of Science University of Shanghai for Science and TechnologyShanghai 200093China

The Gelfand-Levitan-Marchenko representation is used to analyze the initial-boundary value problem of two-component nonlinear SchrSdinger equation on the *** has shown that the global relation can be effectively analyzed by the Geffand-Levitan-Marchenko representation, we also derive expressions for the Dirichlet-to-Neumann map tocharacterize the unknown boundary values.

关键词： Gelfand-Levitan-Marchenko representation initial-boundary value problem two-component nonlinear Schr?dinger equation

THE INITIAL BOUNDARY VALUE PROBLEMS FOR A NONLINEAR INTEGRABLE EQUATION WITH 3×3 LAX PAIR ON THE FINITE INTERVAL

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Acta Mathematica Scientia 2021年第5期41卷 1733-1748页

作者： Yu XIAO Jian XU engui fan College of Information and Management Science Henan Agricultural UniversityZhengzhou 450046China College of Science University of Shanghai for Science and TechnologyShanghai 200093China School of Mathematical Science Fudan UniversityShanghai 200433China

In this paper,we apply Fokas unified method to study the initial boundary value(IBV)problems for nonlinear integrable equation with 3×3 Lax pair on the finite interval[0,L].The solution can be expressed by the solution of a 3×3 Riemann-Hilbert(RH)*** relevant jump matrices are written in terms of matrix-value spectral functions s(k),S(k),S_(l)(k),which are determined by initial data at t=0,boundary values at x=0 and boundary values at x=L,***'s more,since the eigenvalues of 3×3 coefficient matrix of k spectral parameter in Lax pair are three different values,search for the path of analytic functions in RH problem becomes a very interesting thing.

关键词： integral equation initial boundary value problems Fokas unified method Riemann-Hilbert problem