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The modified KdV equation on a finite interval. (English. Abridged French version) Zbl 1044.35080
Summary: We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex \(k\)-plane. This Riemann-Hilbert problem has explicit \((x,t)\)-dependence and it involves certain functions of \(k\) referred to as “spectral functions”. Some of these functions are defined in terms of the initial condition \(q(x,0)=q_0(x)\), while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic “global relation” that characterize the boundary values in spectral terms.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q15 Riemann-Hilbert problems in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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