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 TMF, 2002, Volume 133, Number 2, Pages 184–201 (Mi tmf389)

This article is cited in 20 scientific papers (total in 20 papers)

Initial-Boundary Value Problems for Linear and Soliton PDEs

A. Degasperisa, S. V. Manakovb, P. M. Santinia

a University of Rome "La Sapienza"
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We consider evolution PDEs for dispersive waves in both linear and nonlinear integrable cases and formulate the associated initial-boundary value problems in the spectral space. We propose a solution method based on eliminating the unknown boundary values by proper restrictions of the functional space and of the spectral variable complex domain. Illustrative examples include the linear Schrödinger equation on compact and semicompact n-dimensional domains and the nonlinear Schrödinger equation on the semiline.

Keywords: solitons, integrability, boundary conditions

DOI: https://doi.org/10.4213/tmf389

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English version:
Theoretical and Mathematical Physics, 2002, 133:2, 1475–1489

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Citation: A. Degasperis, S. V. Manakov, P. M. Santini, “Initial-Boundary Value Problems for Linear and Soliton PDEs”, TMF, 133:2 (2002), 184–201; Theoret. and Math. Phys., 133:2 (2002), 1475–1489

Citation in format AMSBIB
\Bibitem{DegManSan02} \by A.~Degasperis, S.~V.~Manakov, P.~M.~Santini \paper Initial-Boundary Value Problems for Linear and Soliton PDEs \jour TMF \yr 2002 \vol 133 \issue 2 \pages 184--201 \mathnet{http://mi.mathnet.ru/tmf389} \crossref{https://doi.org/10.4213/tmf389} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2001532} \transl \jour Theoret. and Math. Phys. \yr 2002 \vol 133 \issue 2 \pages 1475--1489 \crossref{https://doi.org/10.1023/A:1021138525261} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000180061400005} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. de Monvel, AB, “Generation of asymptotic solitons of the nonlinear Schrodinger equation by boundary data”, Journal of Mathematical Physics, 44:8 (2003), 3185
2. I. T. Habibullin, E. V. Gudkova, “Boundary Conditions for Multidimensional Integrable Equations”, Funct. Anal. Appl., 38:2 (2004), 138–148
3. E. V. Gudkova, I. T. Habibullin, “Kadomtsev–Petviashvili Equation on the Half-Plane”, Theoret. and Math. Phys., 140:2 (2004), 1086–1094
4. A. N. Vil'danov, “Integrable Boundary Value Problem for the Boussinesq Equation”, Theoret. and Math. Phys., 141:2 (2004), 1494–1508
5. Vu PL, “Some problems for cubic nonlinear equations on a half-line”, Acta Applicandae Mathematicae, 84:1 (2004), 97–120
6. de Monvel AB, Kotlyarov V, “Characteristic properties of the scattering data for the mKdV equation on the half-line”, Communications in Mathematical Physics, 253:1 (2005), 51–79
7. De Lillo, S, “Neumann problem on the semi-line for the Eckhaus equation”, Nonlinearity, 18:5 (2005), 2365
8. Degasperis, A, “Integrable and nonintegrable initial boundary value problems for soliton equations”, Journal of Nonlinear Mathematical Physics, 12 (2005), 228
9. Chevriaux, D, “Bistable transmitting nonlinear directional couplers”, Modern Physics Letters B, 20:10 (2006), 515
10. de Monvel, AB, “Integrable nonlinear evolution equations on a finite interval”, Communications in Mathematical Physics, 263:1 (2006), 133
11. Gentile, G, “Conservation of resonant periodic solutions for the one-dimensional nonlinear Schrodinger equation”, Communications in Mathematical Physics, 262:3 (2006), 533
12. Gurses, M, “Integrable boundary value problems for elliptic type Toda lattice in a disk”, Journal of Mathematical Physics, 48:10 (2007), 102702
13. Escher, J, “Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation”, Indiana University Mathematics Journal, 56:1 (2007), 87
14. Biondini G., Wang D., “Initial-boundary-value problems for discrete linear evolution equations”, IMA J Appl Math, 75:6 (2010), 968–997
15. De Lillo S., Sommacal M., “Neumann problem on the semi-line for the Burgers equation”, Bound Value Probl, 2011, 1–10
16. Biondini G., Bui A., “On the Nonlinear Schrodinger Equation on the Half Line with Homogeneous Robin Boundary Conditions”, Stud. Appl. Math., 129:3 (2012), 249–271
17. Sakhnovich A., “Nonlinear Schrodinger Equation in a Semi-Strip: Evolution of the Weyl-Titchmarsh Function and Recovery of the Initial Condition and Rectangular Matrix Solutions From the Boundary Conditions”, J. Math. Anal. Appl., 423:1 (2015), 746–757
18. Geng X., Liu H., Zhu J., “Initial-Boundary Value Problems For the Coupled Nonlinear Schrodinger Equation on the Half-Line”, Stud. Appl. Math., 135:3 (2015), 310–346
19. Alexander L. Sakhnovich, “Initial Value Problems for Integrable Systems on a Semi-Strip”, SIGMA, 12 (2016), 001, 17 pp.
20. Mi Y., Liu Yu., Guo B., Luo T., “The Cauchy Problem For a Generalized Camassa-Holm Equation”, J. Differ. Equ., 266:10 (2019), 6739–6770
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