Geodesic
Well-Posed Boundary Value Problems for Integrable Evolution Equations on a~Finite Interval
Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 2, pp. 327-336

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We consider boundary value problems posed on an interval $[0,L]$ for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order $n$. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing $N$ conditions at $x=0$ and $n-N$ conditions at $x=L$, where $N$ depends on $n$ and on the sign of the highest-degree coefficient $n$ in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.
Keywords: boundary value problems, Riemann–Hilbert problem, spectral analysis. 
@article{TMF_2002_133_2_a16,
     author = {B. Pelloni},
     title = {Well-Posed {Boundary} {Value} {Problems} for {Integrable} {Evolution} {Equations} on {a~Finite} {Interval}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {327--336},
     publisher = {mathdoc},
     volume = {133},
     number = {2},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_133_2_a16/}
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B. Pelloni. Well-Posed Boundary Value Problems for Integrable Evolution Equations on a~Finite Interval. Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 2, pp. 327-336. http://geodesic.mathdoc.fr/item/TMF_2002_133_2_a16/