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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

Full text: PDF file (290 kB)
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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
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\paper Initial-Boundary Value Problems for Linear and Soliton PDEs
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\vol 133
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\pages 184--201
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\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}
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. I. T. Habibullin, E. V. Gudkova, “Boundary Conditions for Multidimensional Integrable Equations”, Funct. Anal. Appl., 38:2 (2004), 138–148  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. E. V. Gudkova, I. T. Habibullin, “Kadomtsev–Petviashvili Equation on the Half-Plane”, Theoret. and Math. Phys., 140:2 (2004), 1086–1094  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. A. N. Vil'danov, “Integrable Boundary Value Problem for the Boussinesq Equation”, Theoret. and Math. Phys., 141:2 (2004), 1494–1508  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    5. Vu PL, “Some problems for cubic nonlinear equations on a half-line”, Acta Applicandae Mathematicae, 84:1 (2004), 97–120  crossref  mathscinet  zmath  isi  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. De Lillo, S, “Neumann problem on the semi-line for the Eckhaus equation”, Nonlinearity, 18:5 (2005), 2365  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    8. Degasperis, A, “Integrable and nonintegrable initial boundary value problems for soliton equations”, Journal of Nonlinear Mathematical Physics, 12 (2005), 228  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. Chevriaux, D, “Bistable transmitting nonlinear directional couplers”, Modern Physics Letters B, 20:10 (2006), 515  crossref  zmath  adsnasa  isi  elib  scopus  scopus
    10. de Monvel, AB, “Integrable nonlinear evolution equations on a finite interval”, Communications in Mathematical Physics, 263:1 (2006), 133  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    11. Gentile, G, “Conservation of resonant periodic solutions for the one-dimensional nonlinear Schrodinger equation”, Communications in Mathematical Physics, 262:3 (2006), 533  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. Gurses, M, “Integrable boundary value problems for elliptic type Toda lattice in a disk”, Journal of Mathematical Physics, 48:10 (2007), 102702  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    13. Escher, J, “Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation”, Indiana University Mathematics Journal, 56:1 (2007), 87  crossref  mathscinet  zmath  isi  scopus  scopus
    14. Biondini G., Wang D., “Initial-boundary-value problems for discrete linear evolution equations”, IMA J Appl Math, 75:6 (2010), 968–997  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    15. De Lillo S., Sommacal M., “Neumann problem on the semi-line for the Burgers equation”, Bound Value Probl, 2011, 1–10  mathscinet  isi
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    19. Alexander L. Sakhnovich, “Initial Value Problems for Integrable Systems on a Semi-Strip”, SIGMA, 12 (2016), 001, 17 pp.  mathnet  crossref
    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  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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