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• W2140505510 abstract "Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n−N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂ n x q. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving q t +q xxx and q t −q xxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively. Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II equations. We first show that the unknown boundary values at each end (for example, q x (0,t) and q x (L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution." @default.
• W2140505510 created "2016-06-24" @default.
• W2140505510 creator A5004114262 @default.
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• W2140505510 date "2006-01-26" @default.
• W2140505510 modified "2023-10-05" @default.
• W2140505510 title "Integrable Nonlinear Evolution Equations on a Finite Interval" @default.
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• W2140505510 doi "https://doi.org/10.1007/s00220-005-1495-2" @default.
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