A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line
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摘要
In this paper, the Fokas unified method is used to analyze the initial-boundary value for the ChenLee-Liu equation i?_tu + ?_(xx)u-i|u~2|?_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.
In this paper, the Fokas unified method is used to analyze the initial-boundary value for the ChenLee-Liu equation i?_tu + ?_(xx)u-i|u~2|?_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.
In this paper, the Fokas unified method is used to analyze the initial-boundary value for the ChenLee-Liu equation i?_tu + ?_(xx)u-i|u~2|?_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.
引文
[1]Adler,V.E.,Khabibullin,I.T.Boundary conditions for integrable chains.(Russian)Funktsional.Anal.i Prilozhen.31(2):1–14(1997),translation in Funct.Anal.Appl.,31(2):75–85(1997)
[2]Chen,F.,Lu,C.N.,Yang,H.W.Time-space fractional(2+1)dimensional nonlinear Schr¨odinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions.Adv.Diff.Equs.,56:(2018)
[3]Chen,H.H.,Lee,Y.C.,Liu,C.S.Integrability of nonlinear Hamiltonian systems by inverse scattering method.Phys.Scr.,20:490–492(1979)
[4]Cui,Y.J.Existence of Solutions For Coupled Integral Boundary Value Problem At Resonance.Publ.Mathe-Debrecen,89:73–88(2016)
[5]Cui,Y.J.Uniqueness of solution for boundary value problems for fractional differential Equations.Appl.Math.Lett.,51:48–54(2016)
[6]Cui,Y.J.,Zou,Y.M.An Existence And Uniqueness Theorem for a Second Order Nonlinear System with Coupled Integral Boundary Value Conditions.Appl.Math.Comput.,256:438–444(2015)
[7]Deift,P.,Zhou,X.Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space.Dedicated to the memory of Ju¨rgen K.Moser.Comm.Pure Appl.Math.,56(8):1029–1077(2003)
[8]Ding,S.F.,Huang,H.J.,Xu,X.Z.,Wang,J.Polynomial Smooth Twin Support Vector Machines.Appl.Math.Inf.Sci.,8,4:2063–2071(2014)
[9]Dong,H.H.,Chen,T.T.,Chen,L.F.,Zhang,Y.A new integrable symplectic map and the lie point symmetry associated with nonlinear lattice equations.J.Nonlinear Sci.Appl.,9:5107–5118(2016)
[10]Dong,H.H.,Guo,B.Y.,Yin,B.S.Generalized Fractional Supertrace Identity for Hamiltonian Structure of Nls-Mkdv Hierarchy With Self-consistent Sources.Anal.Math.Phys.,6,2:199–209(2016)
[11]Fan,E.G.A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations.J.Math.Phys.,42:4327–4344(2001)
[12]Fan,E.G.Integrable systems of derivative nonlinear Schr¨odinger type and their multi-Hamiltonian structure.J.Phys.A:Math.Gen.,34:513–519(2001)
[13]Fang,Y.,Dong,H.H.,Hou,Y.J.,Kong,Y.Frobenius Integrable Decompositions Of Nonlinear Evolution Equations With Modified Term.Appl.Math.Comput.,226:435–440(2014)
[14]Fokas,A.S.On a class of physically important integrable equations.Phys.D.,87:145–150(1995)
[15]Fokas,A.S.A unified transform method for solving linear and certain nonlinear PDEs.Proc.R.Soc.Lond.,A 453:1411–1443(1997)
[16]Fokas,A.S.Two dimensional linear PDEs in a convex ploygon.Proc.R.Soc.Lond.A.,457:371–393(2001)
[17]Fokas,A.S.,Its,A.R.The nonlinear Schro¨dinger equation on the interval.J.Phys.A.,37:6091–6114(2004)
[18]Gerdjikov,V.S.,Ivanov,I.A quadratic pencil of general type and nonlinear evolution equations:I,Hierarchies of Hamiltonian structures.J.Phys.Bulg.,10:130–143(1983)
[19]Guo M.,Zhang,Y.,Wang,M.,Chen,Y.D.,Yang,H.W.A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism.Comput.Math.Appl.,75:3589 C 3603(2018)
[20]Kaup,D.J.,Newell,A.C.An exact solution for a derivative nonlinear Schro¨dinger equation.J.Math.Phys.,19:798–801(1978)
[21]Kawata,T.,Kobayashi,N.,Inoue,H.Soliton solution of the derivative nonlinear Schro¨dinger equation.J.Phys.Soc.Jpn.,46:1008–1015(1979)
[22]Kitaev,A.V.,Vartanian,A.H.Higher order asymptotics of the modified non-linear Schr¨odinger equation.Comm.Partial Dif.Equations,25:1043–1098(2000)
[23]Kundu,A.,Strampp,W.,Oevel,W.Gauge transformations of constrained KP flows:new integrable hierarchies.J.Math.Phys.,36:2972–2984(1995)
[24]Lenells,J.The derivative nonlinear Schr¨odinger equation on the half-line.Physica.D.,237:3008–3019(2008)
[25]Lenells,J.,Fokas,A.S.On a novel integrable generalization of the nonlinear Schr¨odinger equation.Nonlinearity,22:709–722(2009)
[26]Lenells,J.,Fokas,A.S.An integrable generalization of the nonlinear Schr¨odinger equation on the half-line and solitons.Inverse Probl.,25,115006:1–32(2009)
[27]Wang,D.S.,Wang,X.L.Long-time asymptotics and the bright N-soliton solutions of the Kundu-Eckhaus equation via the Riemann-Hilbert approach.Nonlinear.Anal-Real.41:334–361(2018)
[28]Wang,D.S.,Wei,X.Q.Integrability and exact solutions of a two-component Korteweg-de Vries system,Appl.Math.Lett.51:60-67(2016)
[29]Xu,X.X.An Integrable Coupling Hierarchy of the Mkdv-integrable Systems,Its Hamiltonian Structure and Corresponding Nonisospectral Integrable Hierarchy.Appl.Math.and Comput.,216,1:344–353(2010)
[30]Zakharov,V.E.,Shabat,A.A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem,I and II.Funct.Anal.Appl.8:226–235(1974)
[31]Zhang,N.A Riemann-Hilbert approach to complex Sharma-Tasso-Olver equation on half line.Commun.Theor.Phys.,68:580–594(2017)
[32]Zhang,T.Q.,Meng,X.Z.,Zhang,T.H.,Song,Y.Global Dynamics for a New High-dimensional Sir Model with Distributed Delay.Appl.Math.Comput.,218,24:11806–11819(2012)
[2]Chen,F.,Lu,C.N.,Yang,H.W.Time-space fractional(2+1)dimensional nonlinear Schr¨odinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions.Adv.Diff.Equs.,56:(2018)
[3]Chen,H.H.,Lee,Y.C.,Liu,C.S.Integrability of nonlinear Hamiltonian systems by inverse scattering method.Phys.Scr.,20:490–492(1979)
[4]Cui,Y.J.Existence of Solutions For Coupled Integral Boundary Value Problem At Resonance.Publ.Mathe-Debrecen,89:73–88(2016)
[5]Cui,Y.J.Uniqueness of solution for boundary value problems for fractional differential Equations.Appl.Math.Lett.,51:48–54(2016)
[6]Cui,Y.J.,Zou,Y.M.An Existence And Uniqueness Theorem for a Second Order Nonlinear System with Coupled Integral Boundary Value Conditions.Appl.Math.Comput.,256:438–444(2015)
[7]Deift,P.,Zhou,X.Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space.Dedicated to the memory of Ju¨rgen K.Moser.Comm.Pure Appl.Math.,56(8):1029–1077(2003)
[8]Ding,S.F.,Huang,H.J.,Xu,X.Z.,Wang,J.Polynomial Smooth Twin Support Vector Machines.Appl.Math.Inf.Sci.,8,4:2063–2071(2014)
[9]Dong,H.H.,Chen,T.T.,Chen,L.F.,Zhang,Y.A new integrable symplectic map and the lie point symmetry associated with nonlinear lattice equations.J.Nonlinear Sci.Appl.,9:5107–5118(2016)
[10]Dong,H.H.,Guo,B.Y.,Yin,B.S.Generalized Fractional Supertrace Identity for Hamiltonian Structure of Nls-Mkdv Hierarchy With Self-consistent Sources.Anal.Math.Phys.,6,2:199–209(2016)
[11]Fan,E.G.A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations.J.Math.Phys.,42:4327–4344(2001)
[12]Fan,E.G.Integrable systems of derivative nonlinear Schr¨odinger type and their multi-Hamiltonian structure.J.Phys.A:Math.Gen.,34:513–519(2001)
[13]Fang,Y.,Dong,H.H.,Hou,Y.J.,Kong,Y.Frobenius Integrable Decompositions Of Nonlinear Evolution Equations With Modified Term.Appl.Math.Comput.,226:435–440(2014)
[14]Fokas,A.S.On a class of physically important integrable equations.Phys.D.,87:145–150(1995)
[15]Fokas,A.S.A unified transform method for solving linear and certain nonlinear PDEs.Proc.R.Soc.Lond.,A 453:1411–1443(1997)
[16]Fokas,A.S.Two dimensional linear PDEs in a convex ploygon.Proc.R.Soc.Lond.A.,457:371–393(2001)
[17]Fokas,A.S.,Its,A.R.The nonlinear Schro¨dinger equation on the interval.J.Phys.A.,37:6091–6114(2004)
[18]Gerdjikov,V.S.,Ivanov,I.A quadratic pencil of general type and nonlinear evolution equations:I,Hierarchies of Hamiltonian structures.J.Phys.Bulg.,10:130–143(1983)
[19]Guo M.,Zhang,Y.,Wang,M.,Chen,Y.D.,Yang,H.W.A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism.Comput.Math.Appl.,75:3589 C 3603(2018)
[20]Kaup,D.J.,Newell,A.C.An exact solution for a derivative nonlinear Schro¨dinger equation.J.Math.Phys.,19:798–801(1978)
[21]Kawata,T.,Kobayashi,N.,Inoue,H.Soliton solution of the derivative nonlinear Schro¨dinger equation.J.Phys.Soc.Jpn.,46:1008–1015(1979)
[22]Kitaev,A.V.,Vartanian,A.H.Higher order asymptotics of the modified non-linear Schr¨odinger equation.Comm.Partial Dif.Equations,25:1043–1098(2000)
[23]Kundu,A.,Strampp,W.,Oevel,W.Gauge transformations of constrained KP flows:new integrable hierarchies.J.Math.Phys.,36:2972–2984(1995)
[24]Lenells,J.The derivative nonlinear Schr¨odinger equation on the half-line.Physica.D.,237:3008–3019(2008)
[25]Lenells,J.,Fokas,A.S.On a novel integrable generalization of the nonlinear Schr¨odinger equation.Nonlinearity,22:709–722(2009)
[26]Lenells,J.,Fokas,A.S.An integrable generalization of the nonlinear Schr¨odinger equation on the half-line and solitons.Inverse Probl.,25,115006:1–32(2009)
[27]Wang,D.S.,Wang,X.L.Long-time asymptotics and the bright N-soliton solutions of the Kundu-Eckhaus equation via the Riemann-Hilbert approach.Nonlinear.Anal-Real.41:334–361(2018)
[28]Wang,D.S.,Wei,X.Q.Integrability and exact solutions of a two-component Korteweg-de Vries system,Appl.Math.Lett.51:60-67(2016)
[29]Xu,X.X.An Integrable Coupling Hierarchy of the Mkdv-integrable Systems,Its Hamiltonian Structure and Corresponding Nonisospectral Integrable Hierarchy.Appl.Math.and Comput.,216,1:344–353(2010)
[30]Zakharov,V.E.,Shabat,A.A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem,I and II.Funct.Anal.Appl.8:226–235(1974)
[31]Zhang,N.A Riemann-Hilbert approach to complex Sharma-Tasso-Olver equation on half line.Commun.Theor.Phys.,68:580–594(2017)
[32]Zhang,T.Q.,Meng,X.Z.,Zhang,T.H.,Song,Y.Global Dynamics for a New High-dimensional Sir Model with Distributed Delay.Appl.Math.Comput.,218,24:11806–11819(2012)
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