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.
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REFERENCES
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York (1953).
R. Terras and R. Swanson, J. Math. Phys., 21, 2140 (1980).
A. S. Fokas, Proc. Roy. Soc. London A, 53, 1411 (1997).
A. S. Fokas, J. Math. Phys., 41, 4188 (2000).
A. S. Fokas, Proc. Roy. Soc. London A, 457, 371 (2001).
A. S. Fokas and B. Pelloni, Math. Proc. Camb. Phil. Soc., 131, 521 (2001).
B. Pelloni, Theor. Math. Phys., 133, 1598 (2002).
A. Degasperis, S. V. Manakov, and P. M. Santini, “Initial-boundary value problems for linear PDEs: The analyticity approach,” nlin.SI/0210058 (2002).
A. S. Fokas, “Inverse scattering transform on the half line—the nonlinear analogue of the sine transform,” in: Inverse Problems: An Interdisciplinary Study (P. C. Sabatier, ed.), Acad. Press, London (1987), p. 409.
P. C. Sabatier, J. Math. Phys., 41, 414 (2000).
A. S. Fokas, A. R. Its, and L. Y. Sung, “The nonlinear Schrödinger equation on the half-line,” Preprint, Cambridge Univ. Press, Cambridge (2001).
A. S. Fokas, “Integrable nonlinear evolution equations on the half-line (October 2001),” Comm. Math. Phys. (to appear).
A. S. Fokas and A. R. Its, Phys. Rev. Lett., 68, 3117 (1992).
M. J. Ablowitz and H. Segur, J. Math. Phys., 16, 1054 (1975).
A. Degasperis, S. V. Manakov, and P. M. Santini, JETP Letters, 74, 481 (2001).
I. T. Khabibullin, Theor. Math. Phys., 130, 25 (2002).
J. Leon and A. V. Mikhailov, Phys. Lett. A, 253, 33 (1999).
J. Leon and A. Spire, J. Phys. A, 34, 7359 (2001); nlin.PS/0105066 (2001).
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981); V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Plenum, New York (1984).
A. C. Newell, “The inverse scattering transform,” in: Solitons (R. K. Bullough and P. J. Caudrey, eds.), Springer, Berlin (1980), p. 177.
F. Calogero and S. De Lillo, Inverse Problems, 4, L33 (1988); J. Math. Phys., 32, 99 (1991); M. J. Ablowitz and S. De Lillo, Phys. Lett. A, 156, 483 (1991).
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Degasperis, A., Manakov, S.V. & Santini, P.M. Initial-Boundary Value Problems for Linear and Soliton PDEs. Theoretical and Mathematical Physics 133, 1475–1489 (2002). https://doi.org/10.1023/A:1021138525261
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DOI: https://doi.org/10.1023/A:1021138525261