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A Riemann–Hilbert Approach to Complex Sharma–Tasso–Olver Equation on Half Line

Communications in Theoretical Physics 2017年第11期67卷 580-594页

作者：张宁夏铁成胡贝贝 Department of Mathematics Shanghai University Department of Basical Courses Shandong University of Science and Technology School of Mathematicas and Finance Chuzhou University

In this paper, the fokas unified method is used to analyze the initial-boundary value problem of a complex Sharma–Tasso–Olver(c STO) equation on the half line. We show that the solution can be expressed in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly given in terms of the matrix-value spectral functions spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which depending on initial data u_0(x) = u(x, 0) and boundary data g_0(y) = u(0, y), g_1(y) = ux(0, y), g_2(y) = u_(xx)(0, y). These spectral functions are not independent, they satisfy a global relation.

关键词： the cSTO equation initial-boundary value problem Riemann–Hilbert problem jump matrix fokas unified method

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A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

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Acta Mathematicae Applicatae Sinica 2018年第3期34卷 493-515页

作者： Ning ZHANG Tie-cheng XIA En-gui FAN Department of Basical Courses Shandong University of Science and Technology Taian 271019 China Department of Mathematics Shanghai University Shanghai 200444 China School of Mathematical Sciences Fudan University Shanghai 200433 China

In this paper, the fokas unified method is used to analyze the initial-boundary value for the ChenLee-Liu equation i？tu ＋？xxu-i｜u2｜？xu = 0 on the half line（-∞, 0] with decaying initial value. Assuming that the solution u（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit（x, t） dependence and is given in terms of the spectral functions{a（λ）, b（λ）}and{A（λ）, B（λ）}, which are obtained from the initial data u0（x） = u（x, 0） and the boundary data g0（t） = u（0, t）, g1（t） = ux（0, t）, respectively. The spectral functions are not independent,but satisfy a so-called global relation.

关键词： Chen-Lee-Liu equation initial-value problem RiemannIHilbert problem fokas unified method jump matrix

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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：肖羽徐建范恩贵 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)problem.The 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,respectively.What'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

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