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The Korteweg–de Vries equation on the interval

### Abstract

The initial-boundary value problem for the Korteweg–de Vries equation posed on a finite interval of the spatial variable is considered. Using the method of simultaneous spectral analysis of the associated Lax pair, this problem is mapped into a Riemann–Hilbert problem formulated in the complex plane of the spectral parameter, but with explicit dependence on the space-time variables appearing in the Korteweg–de Vries equation. It is shown that, under certain conditions, the solution of this Riemann–Hilbert is uniquely determined by certain functions of the spectral parameter, which are defined by the initial and boundary data of the original problem. In turn, the solution of the Riemann–Hilbert problem provides the solution of the initial-boundary value problem for the Korteweg–de Vries equation, for which an integral representation is derived.

© 2010 American Institute of Physics

Received 08 May 2009
Accepted 09 July 2010
Published online 26 August 2010

Acknowledgments:
This work was partially supported by a University of Patras *Caratheodory* grant (Grant No. 2785), the Greek State Scholarships Foundation, and a *Pythagoras* research grant (Grant No. B.365.015) of the European Social Fund (EPEAEK II). The first author (I.H.) thanks Professors A. S. Fokas and V. Papageorgiou for the illuminating discussions.

Article outline:

I. INTRODUCTION
II. FROM THE LAX PAIR TO THE RIEMANN–HILBERT PROBLEM
A. Properties of the eigenfunctions
1. Boundedness and holomorphicity
2. The spectral matrices
3. Behavior at infinity
4. Unit determinants
5. Symmetry properties
6. Behavior at zero
B. Properties of the spectral functions
1. Relation to ordinary differential equations
2. Boundedness and holomorphicity
3. Unit determinants
4. Behavior at infinity
5. Behavior at zero
C. Toward a Riemann–Hilbert formulation
III. FROM THE RH TO THE INITIAL-BOUNDARY VALUE PROBLEM
A. The inverse spectral maps
1. *The inverse spectral map*
2. The inverse spectral map
3. *The inverse spectral map*
B. The Global relation
C. The Riemann–Hilbert problem
D. On the unique solvability of the Riemann–Hilbert problem
E. Integral representation of the solution

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2010-08-26

2016-09-28

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