# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
 [1] A. Boutet de Monvel, A.S. Fokas, D. Shepelsky, The mKdV equation on the half-line, J. Inst. Math. Jussieu, in press · Zbl 1057.35050 [2] A. Boutet de Monvel, D. Shepelsky, Initial boundary value problem for the modified KdV equation on a finite interval, Preprint · Zbl 1137.35419 [3] Fokas, A.S., A unified transform method for solving linear and certain nonlinear pdes, Proc. roy. soc. London ser. A, 453, 1411-1443, (1997) · Zbl 0876.35102 [4] Fokas, A.S., On the integrability of linear and nonlinear partial differential equations, J. math. phys., 41, 4188-4237, (2000) · Zbl 0994.37036 [5] Fokas, A.S., Integrable nonlinear evolution equations on the half-line, Comm. math. phys., 230, 1-39, (2002) · Zbl 1010.35089 [6] A.S. Fokas, A.R. Its, The nonlinear Schrödinger equation on the interval, Preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.