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Seiberg-Witten/Whitham Equations and Instanton Corrections in N = 2 Supersymmetric Yang-Mills Theory

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Communications in Theoretical Physics 2018年第5期69卷 558-568页

作者： Jia-Liang Dai en-gui fan School of Mathematical Science Fudan University Shanghai 200433 China

We obtain the instanton correction recursion relations for the low energy effective prepotential in pure Ν = 2SU(n) supersymmetric Yang-Mills gauge theory from Whitham hierarchy and Seiberg-Witten/Whitham equations. These formulae provide us a powerful tool to calculate arbitrary order instanton corrections coefficients from the perturbative contributions of the effective prepotential in Seiberg-Witten gauge theory. We apply this idea to evaluate one-and two-order instanton corrections coefficients explicitly in SU(n) case in detail through the dynamical scale parameter expressed in terms of Riemann's theta-function.

关键词： Whitham hierarchy Seiberg-Witten/Whitham equations instanton corrections recursion relations

Propagation and modulational instability of Rossby waves in stratified fluids

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Chinese Physics B 2022年第7期31卷 84-96页

作者： Xiao-Qian Yang en-gui fan Ning Zhang College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590China Department of Fundamental Course Shandong University of Science and TechnologyTai’an 271019China School of Mathematical Sciences Fudan UniversityShanghai 200433China

Perturbation analysis and scale expansion are used to derive the(2+1)-dimensional coupled nonlinear Schr¨odinger(CNLS)equations that can describe interactions of two Rossby waves propagating in stratified ***(2+1)-dimensional equations can reflect and describe the wave propagation more intuitively and *** properties of the two waves in the process of propagation can be analyzed by the solution obtained from the equations using the Hirota bilinear method,and the influence factors of modulational instability are *** results suggest that,when two Rossby waves with slightly different wave numbers propagate in the stratified fluids,the intensity of bright soliton decreases with the increases of dark soliton *** addition,the size of modulational instable area is related to the amplitude and wave number in y direction.

关键词： Rossby waves Hirota bilinear method modulational instability stratified fluids

Long-Time Asymptotics for the Nonlocal MKdV Equation

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Communications in Theoretical Physics 2019年第5期71卷 475-488页

作者： Feng-Jing He en-gui fan Jian Xu School of Mathematical Sciences Fudan UniversityShanghai200433China College of Science University of Shanghai for Science and TechnologyShanghai200093China

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation(nonlocal mKdV) qt(x, t)+qxxx(x, t)-6 q(x, t)q(-x,-t)qx(x, t) = 0, which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal m KdV equation. In contrast with the classical mKdV equation,we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results.

关键词： nonlocal mKdV equation Riemann-Hilbert problem Deift-Zhou nonlinear steepest-descent long-time asymptotics

A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrfdinger Equation

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Communications in Theoretical Physics 2001年第10期36卷 401-404页

作者： fan en-gui Institute of Mathematics Fudan University Shanghai 200433 China

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.

关键词： Schrodinger equation, N-fold Darboux transformation, N-soliton solutions

Bi-Hamiltonian Structure and Liouville Integrablity for a Gerdjikov-Ivanov Equation Hierarchy

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Chinese Physics Letters 2001年第1期18卷 1-3页

作者： fan en-gui Institute of Mathematics Fudan UniversityShanghai 200433

A new spectral problem is introduced and the associated hierarchy of nonlinear evolution equations is derived with the help of the zero curvature equation as well as recursive *** is found that the Gerdjikov-Ivanov(GI)equation,which is one of three important derivative nonlinear Schrodinger equations,exactly belongs to the hierarchy as a special reduction.A powerful tool of the trace identity is used to establish the bi-Harniltonian structure for the whole GI ***,it is shown that GI hierarchy admits an infinite common set of conserved quantities which are in involution in pairs under Poisson's *** indicates that the whole GI hierarchy is completely integrable in Liouville's sense.

关键词： Liouville hierarchy equation

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

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Communications in Theoretical Physics 2007年第2X期48卷 205-210页

作者： LUO Lin fan en-gui Department of Mathematics Xiaogan University Xiaogan 432100 China School of Mathematics Fudan University Shanghai 200433 China

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems （FDHS） in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.

关键词： nonlinear equations Hamiltonian system symmetry constraint Darboux transformation

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

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Communications in Theoretical Physics 2008年第6期49卷 1399-1402页

作者： LUO Lin fan en-gui Department of Mathematics Xiaogan University Xiaogan 432100 China School of Mathematics Fudan University Shanghai 200433 China

Starting from a new discrete spectral problem, the corresponding hierarchy of nonlinear lattice equations is proposed. It is shown that the lattice soliton hierarchy possesses the bi-Hamiltonian structures and infinit... 详细信息

关键词： Hamiltonian system infinite conservation laws discrete spectral problem

Characteristic Numbers of Matrix Lie Algebras

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Communications in Theoretical Physics 2008年第4期49卷 845-850页

作者： ZHANG Yu-Feng fan en-gui Mathematical School Liaoning Normal University Dalian 116029 China Institute of Mathematics Fudan University Shanghai 200433 China

A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.

关键词： characteristic number Lie algebra integrable couplings

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

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Communications in Theoretical Physics 2006年第6X期46卷 1091-1100页

作者： ZHANG Wei-Guo DONG Chun-Yan fan en-gui College of Science University of Shanghai for Science and Technology Shanghai 200093 China Institute of Mathematics Fudan University Shanghai 200433 China

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.

关键词： generalized compound KdV equation generalized compound KdV-Burgers equation