Generation of asymptotic solitons of the nonlinear Schrödinger equation by boundary data

Journal of Mathematical Physics

Volumn 44, Issue 8, 2003, Pages 3185-3215

  Boutet de Monvel, Anne ^a   Kotlyarov, Vladimir ^b  

^a INSTITUT DE MATHÉMATIQUES DE JUSSIEU   (France)

^b B VERKIN INSTITUTE FOR LOW TEMPERATURE PHYSICS AND ENGINEERING   (Ukraine)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0043020561     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1588465     Document Type: Article

References (44)

1. Ablowitz, M. J. and Segur, H., Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981).
2. Ablowitz, M. J. and Segur, H., "The inverse scattering transform: Semi-infinite interval," J. Math. Phys. 16, 1054-1056 (1975).
3. Belokolos, E. D., "General Formulas for Solutions of Initial and Boundary-Value Problem for the sine-Gordon Equation," Teor. Mat. Fiz. 103, 358-367 (1995) [English translation: Theor. Math. Phys. 103, 613-620 (1995)].
4. Berezanskii, Yu. M., "Integration of nonlinear difference equations by the method of the inverse spectral problem," Dokl. Akad. Nauk SSSR 281, 16-19 (1985) [English translation: Sov. Math. Dokl. 31, 264-267 (1985)].
5. Berezanskii, Yu. M. and Shmoish, M., "Nonisospectral flows of the semi-infinite Jacobi matrices," J. Nonlinear Math. Phys. 1, 116-146 (1994).
6. Bikbaev, R. F. and Its, A. R., "Algebraic-Geometric solutions of a Boundary Value Problem for the Nonlinear Schrödinger Equation," Mat. Zametki 45, 3-9 (1989) [English translation: Math. Notes 45, 349-354 (1989)].
7. Bikbaev, R. F. and Tarasov, V. O., "Initial Boundary Value Problem for the Nonlinear Schrödinger Equation," J. Phys. A 24, 2507-2516 (1991).
8. Bikbaev, R. F. and Tarasov, V. O., "An inhomogeneous boundary value problem on the semi-axis and on a segment for the sine-Gordon equation," Algebra i Analiz 3, 78-92 (1991) [English translation: St. Petersbg. Math. J. 3, 775-789 (1992)].
9. Bikbaev, R. F. and Tarasov, V. O., "An inhomogeneous boundary value problem on the semi-axis and on a segment for the sine-Gordon equation," Algebra i Analiz 3, 78-92 (1991) [English translation: St. Petersbg. Math. J. 3, 775-789 (1992)].
10. Boutet de Monvel, A. and Kotlyarov, V. P., "Scattering problem for the Zakharov-Shabat equations on the semi-axis," Inverse Probl. 16, 1813-1837 (2000).
11. Boutet de Monvel, A. and Kotlyarov, V. P., "Nonlinear Schrödinger equation on the half-axis. Generation of asymptotic solitons by boundary data," preprint (BiBoS, Bielefeld, 2001).
12. Calogero, F. and De Lillo, S., "The Burgers Equation on the semi-infinite and finite intervals," Nonlinearity 2, 37-43 (1989).
13. Caroll, R. and Bu, Q., "Solution of the forced nonlinear Schrödinger equation using PDE techniques," Appl. Anal. 41, 33-51 (1991).
14. Degasperis, A., Manakov, S. V., and Santini, P. M., "On the Initial-Boundary Value Problems for Soliton Equations," JETP Lett. 74, 481-485 (2001).
15. Degasperis, A., Manakov, S. V., and Santini, P. M., "Initial-Boundary Value Problems for Linear and Soliton PDEs," Theor. Math. Phys. 133, 1475-1489 (2002).
16. Dubrovin, B. A., "Theta functions and nonlinear equations," Usp. Mat. Nauk 36, 11-80 (1981) [English translation: Russ. Math. Surveys 36, 11-92 (1982)].
17. Faddeev, L. D. and Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons (Nauka, Moscow, 1986) [English translation: Springer Series in Soviet Mathematics (Springer, Berlin, 1987)].
18. Faddeev, L. D. and Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons (Nauka, Moscow, 1986) [English translation: Springer Series in Soviet Mathematics (Springer, Berlin, 1987)].
19. Fokas, A. S., "An Initial Boundary Value Problem for the Nonlinear Schrödinger Equation," Physica D 35, 167-185 (1989).
20. Fokas, A. S., "Integrable Nonlinear Evolution Equations on the Half-Line," Commun. Math. Phys. 230, 1-39 (2002).
21. Fokas, A. S. and Gelfand, I. M., "Integrability of Linear and Nonlinear Evolution Equations and the Associated Nonlinear Fourier Transform," Lett. Math. Phys. 32, 189-210 (1992).
22. Fokas, A. S. and Its, A. R., "An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates," Teor. Mat. Fiz. 92, 387-403 (1992) [English translation: Theor. Math. Phys. 92, 964-978 (1992)].
23. Fokas, A. S. and Its, A. R., "An initial-boundary value problem for the Korteweg-de Vries equation," Math. Comput. Simul. 37, 293-321 (1994).
24. Fokas, A. S. and Its, A. R., "The Linearization of the Initial Boundary Value Problem of the Nonlinear Schrödinger Equation," SIAM J. Math. Anal. 27, 738-764 (1996).
25. Gattobigio, M., Liguori, A., and Mintchev, M., "The Nonlinear Schrödinger equation on the half line," J. Math. Phys. 40, 2949-2970 (1999).
26. Habibullin, I. T., "Bäcklund Transformation and Integrable Boundary-Initial Value Problems," Nonlinear world, Vol. 1. Kiev NTPP-89 (World Scientific, River Edge, 1990), pp. 130-138.
27. Khabibullin, I. T., "The KdV equation on a half-line with a zero boundary condition," Teor. Mat. Fiz. 119 397-404 (1999) [English translation: Theor. Math. Phys. 119, 712-718 (1999)].
28. Kotlyarov, V. P., "The effect of a doubly continuous spectrum of the Dirac operator on asymptotic solitons of the nonlinear Schrödinger equation," Mat. Zametki 49, 84-94 (1991) [English translation: Math. Notes 49, 172-180 (1992)].
29. Kotlyarov, V. P., "Asymptotic analysis of the Marchenko integral equation and soliton asymptotics of a solution of the nonlinear Schrödinger equation," Physica D 87, 176-185 (1995).
30. Krichever, I. M., "Integration of non-linear equations," Funkc. Anal. Priloz. 11, 15-31 (1977).
31. Sabatier, P. C., "New direct linearizations for KdV and solutions of the other Cauchy problem," J. Math. Phys. 40, 2983-3020 (1999).
32. Sabatier, P. C., "Elbow scattering and inverse scattering applications to LKdV and KdV," J. Math. Phys. 41, 414-436 (2000).
33. Sabatier, P. C., "Past and future of inverse problems," J. Math. Phys. 41, 4082-4124 (2000).
34. Sabatier, P. C., "Elbow scattering and boundary value problems of NLPDE," J. Nonlinear Math. Phys. 8, suppl. 249-253 (2001).
35. Sabatier, P. C., "Should we study sophisticated inverse problems?" Inverse Probl. 17, 1219-1223 (2001).
36. Sakhnovich, L. A., "Explicit Formulas for Spectral Characteristics and Solution of the sinh-Gordon Equation," Ukr. Mat. Zh. 42, 1517-1523 (1990) [English translation: Ukr. Mat. J. 42, 1359-1365 (1990)].
37. Sakhnovich, L. A., "Integrable Nonlinear equations on the semi-axis," Ukr. Mat. Zh. 43, 1578-1584 (1991) [English translation: Ukr. Mat. J. 43, 1470-1476 (1991)].
38. Sakhnovich, L. A., "The Goursat Problem for the sine-Gordon Equation and an Inverse Spectral Problem," Izv. Vyssh. Uchebn. Zaved. Mat. 36, 44-54 (1992) [English translation: Russian Math. (Iz. VUZ) 36, 42-52 (1992)].
39. Sakhnovich, L. A., "Sine-Gordon Equation in Laboratory coordinates and Inverse Problem on the Semi-Axis," Algebraic and Geometrical Methods in Mathematical Physics, Math. Phys. Stud. Vol 19 (Kluwer, Dordrecht, 1996). pp. 443-447.
40. Sklyanin, E. K., "Boundary Conditions for Integrable Equations," Funkc. Anal. Priloz. 21, 86-87 (1987) [English translation: Funct. Anal. Appl. 21, 164-166 (1997)].
41. Sung, L. Y., "Solution of the Initial Boundary Problem of the nonlinear Schrödinger equation using PDE techniques," preprint (Clarkson University, Potsdam, NY, 1993).
42. Tarasov, V. O., "The Boundary Value Problem for the Nonlinear Schrödinger Equation," Zap. Nauchn. Semin. LOMI 169, 151-165 (1988).
43. Zakharov, V. E. and Shabat, A. B., "An exact theory of two-dimensional self-focusing and one-dimensional automodulation of waves in a nonlinear medium," Sov. Phys. JETP 34, 62-78 (1972).
44. Zakharov, V. E. and Shabat, A. B., "A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I," Funkc. Anal. Priloz. 6, 43-53 (1974) [English translation: Funct. Anal. Appl. 8, 226-235 (1974)].