In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can...In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ.The jump matrix L(x,t,θ)has an explicit representation dependent on x,t and it can be represented exactly by the two pairs of spectral functions y(θ),z(θ)(obtained from the initial value u0(x))and Y(θ),Z(θ)(obtained from the boundary conditions v0(t),{vk(t)}_(1)^(4)).Furthermore,the two pairs of spectral functions y(θ),z(θ)and Y(θ),Z(θ)are not independent of each other,but are related to the compatibility condition,the so-called global relation.展开更多
基金supported by the National Natural Science Foundation of China under Grant Nos.12147115 and 11835011the Natural Science Foundation of Anhui Province under Grant No.2108085QA09+3 种基金the University Natural Science Research Project of Anhui Province under Grant No.KJ2021A1094China Postdoctoral Science Foundation under Grant No.2022M712833the Program for Science and Technology Innovation Talents in Universities of Henan Province under Grant No.22HASTIT019the Natural Science Foundation of Henan Province under Grant No.202300410524
文摘In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ.The jump matrix L(x,t,θ)has an explicit representation dependent on x,t and it can be represented exactly by the two pairs of spectral functions y(θ),z(θ)(obtained from the initial value u0(x))and Y(θ),Z(θ)(obtained from the boundary conditions v0(t),{vk(t)}_(1)^(4)).Furthermore,the two pairs of spectral functions y(θ),z(θ)and Y(θ),Z(θ)are not independent of each other,but are related to the compatibility condition,the so-called global relation.
