Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 001, 17 pages      arXiv:1405.3500

Initial Value Problems for Integrable Systems on a Semi-Strip

Alexander L. Sakhnovich
Vienna University of Technology, Institute of Analysis and Scientific Computing, Wiedner Hauptstr. 8, A-1040 Vienna, Austria

Received September 01, 2015, in final form December 28, 2015; Published online January 03, 2016

Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix solutions of the defocusing nonlinear Schrödinger equation with quasi-analytic boundary conditions is dealt with. (The result is new even for a scalar nonlinear Schrödinger equation.) Next, a special case of the nonlinear optics ($N$-wave) equation is considered.

Key words: Weyl-Titchmarsh function; initial condition; quasi-analytic functions; system on a semi-strip; nonlinear Schrödinger equation; nonlinear optics equation.

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  1. Ablowitz M.J., Haberman R., Resonantly coupled nonlinear evolution equations, J. Math. Phys. 16 (1975), 2301-2305.
  2. Ablowitz M.J., Prinari B., Trubatch A.D., Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Note Series, Vol. 302, Cambridge University Press, Cambridge, 2004.
  3. Ashton A.C.L., On the rigorous foundations of the Fokas method for linear elliptic partial differential equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2012), 1325-1331.
  4. Bang T., On quasi-analytic functions, in C. R. Dixième Congrès Math. Scandinaves 1946, Jul. Gjellerups Forlag, Copenhagen, 1947, 249-254.
  5. Beals R., Coifman R.R., Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984), 39-90.
  6. Berezanskii Yu.M., Integration of non-linear difference equations by means of inverse problem technique, Dokl. Akad. Nauk SSSR 281 (1985), 16-19.
  7. Berezanskii Yu.M., Gekhtman M.I., Inverse problem of spectral analysis and nonabelian chains of nonlinear equations, Ukrain. Math. J. 42 (1990), 645-658.
  8. Beurling A., The collected works of Arne Beurling. Vol. 1. Complex analysis, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1989.
  9. Bikbaev R.F., Tarasov V.O., Initial-boundary value problem for the nonlinear Schrödinger equation, J. Phys. A: Math. Gen. 24 (1991), 2507-2516.
  10. Bona J., Winther R., The Korteweg-de Vries equation, posed in a quarter-plane, SIAM J. Math. Anal. 14 (1983), 1056-1106.
  11. Bona J.L., Fokas A.S., Initial-boundary-value problems for linear and integrable nonlinear dispersive partial differential equations, Nonlinearity 21 (2008), T195-T203.
  12. Carroll R., Bu Q., Solution of the forced nonlinear Schrödinger (NLS) equation using PDE techniques, Appl. Anal. 41 (1991), 33-51.
  13. Chu C.K., Xiang L.W., Baransky Y., Solitary waves induced by boundary motion, Comm. Pure Appl. Math. 36 (1983), 495-504.
  14. Clark S., Gesztesy F., Weyl-Titchmarsh $M$-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators, Trans. Amer. Math. Soc. 354 (2002), 3475-3534, math.SP/0102040.
  15. Damanik D., Killip R., Simon B., Perturbations of orthogonal polynomials with periodic recursion coefficients, Ann. of Math. 171 (2010), 1931-2010, math.SP/0702388.
  16. Degasperis A., Manakov S.V., Santini P.M., Mixed problems for linear and soliton partial differential equations, Theoret. and Math. Phys. 133 (2002), 1475-1489.
  17. Fokas A.S., Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys. 230 (2002), 1-39.
  18. Fokas A.S., A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 78, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008.
  19. Fokas A.S., Pelloni B. (Editors), Unified transform for boundary value problems. Applications and advances, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015.
  20. Fritzsche B., Kirstein B., Roitberg I.Ya., Sakhnovich A.L., Weyl theory and explicit solutions of direct and inverse problems for Dirac system with a rectangular matrix potential, Oper. Matrices 7 (2013), 183-196, arXiv:1105.2013.
  21. Fritzsche B., Kirstein B., Sakhnovich A.L., Weyl functions of generalized Dirac systems: integral representation, the inverse problem and discrete interpolation, J. Anal. Math. 116 (2012), 17-51, arXiv:1007.4304.
  22. Gesztesy F., Mitrea M., Zinchenko M., On Dirichlet-to-Neumann maps and some applications to modified Fredholm determinants, in Methods of Spectral Analysis in Mathematical Physics, Oper. Theory Adv. Appl., Vol. 186, Birkhäuser Verlag, Basel, 2009, 191-215, arXiv:1002.0390.
  23. Gesztesy F., Weikard R., Zinchenko M., Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials, Oper. Matrices 7 (2013), 241-283, arXiv:1109.1613.
  24. Gohberg I., Kaashoek M.A., Sakhnovich A.L., Scattering problems for a canonical system with a pseudo-exponential potential, Asymptot. Anal. 29 (2002), 1-38.
  25. Habibullin I.T., Backlund transformation and integrable boundary-initial value problems, in Nonlinear World, Vol. 1 (Kiev, 1989), World Sci. Publ., River Edge, NJ, 1990, 130-138.
  26. Harris B.J., The asymptotic form of the Titchmarsh-Weyl $m$-function associated with a Dirac system, J. London Math. Soc. 31 (1985), 321-330.
  27. Holmer J., The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations 18 (2005), 647-668, math.AP/0602152.
  28. Kac M., van Moerbeke P., A complete solution of the periodic Toda problem, Proc. Nat. Acad. Sci. USA 72 (1975), 2879-2880.
  29. Kaikina E.I., Inhomogeneous Neumann initial-boundary value problem for the nonlinear Schrödinger equation, J. Differential Equations 255 (2013), 3338-3356.
  30. Kamvissis S., Shepelsky D., Zielinski L., Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation, J. Nonlinear Math. Phys. 22 (2015), 448-473, arXiv:1412.7636.
  31. Kaup D.J., The forced Toda lattice: an example of an almost integrable system, J. Math. Phys. 25 (1984), 277-281.
  32. Kaup D.J., Newell A.C., The Goursat and Cauchy problems for the sine-Gordon equation, SIAM J. Appl. Math. 34 (1978), 37-54.
  33. Kaup D.J., Steudel H., Recent results on second harmonic generation, in Recent Developments in Integrable Systems and Riemann-Hilbert Problems (Birmingham, AL, 2000), Contemp. Math., Vol. 326, Amer. Math. Soc., Providence, RI, 2003, 33-48.
  34. Khryptun V.G., Expansion of functions of quasi-analytic classes in series in polynomials, Ukrain. Math. J. 41 (1989), 569-574.
  35. Kostenko A., Sakhnovich A., Teschl G., Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not. 2012 (2012), 1699-1747.
  36. Kreiss H.-O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277-298.
  37. Krichever I.M., An analogue of the d'Alembert formula for the equations of a principal chiral field and the sine-Gordon equation, Dokl. Akad. Nauk SSSR 253 (1980), 288-292.
  38. Paley R.E.A.C., Wiener N., Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, Vol. 19, Amer. Math. Soc., Providence, RI, 1987.
  39. Sabatier P.C., Elbow scattering and inverse scattering applications to LKdV and KdV, J. Math. Phys. 41 (2000), 414-436.
  40. Sabatier P.C., Generalized inverse scattering transform applied to linear partial differential equations, Inverse Problems 22 (2006), 209-228.
  41. Sakhnovich A.L., The $N$-wave problem on the half-line, Russ. Math. Surv. 46 (1991), no. 4, 198-200.
  42. Sakhnovich A.L., The Goursat problem for the sine-Gordon equation, and an inverse spectral problem, Russ. Math. Iz. VUZ (1992), no. 11, 42-52.
  43. Sakhnovich A.L., Second harmonic generation: Goursat problem on the semi-strip, Weyl functions and explicit solutions, Inverse Problems 21 (2005), 703-716, nlin.SI/0402055.
  44. Sakhnovich A.L., Weyl functions, the inverse problem and special solutions for the system auxiliary to the nonlinear optics equation, Inverse Problems 24 (2008), 025026, 23 pages, arXiv:0708.1112.
  45. Sakhnovich A.L., On the compatibility condition for linear systems and a factorization formula for wave functions, J. Differential Equations 252 (2012), 3658-3667.
  46. Sakhnovich A.L., Inverse problem for Dirac systems with locally square-summable potentials and rectangular Weyl functions, J. Spectr. Theory 5 (2015), 547-569, arXiv:1401.3605.
  47. Sakhnovich A.L., Nonlinear Schrödinger 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 (2015), 746-757.
  48. Sakhnovich A.L., Sakhnovich L.A., Roitberg I.Ya., Inverse problems and nonlinear evolution equations. Solutions, Darboux matrices and Weyl-Titchmarsh functions, De Gruyter Studies in Mathematics, Vol. 47, De Gruyter, Berlin, 2013.
  49. Sakhnovich L.A., Evolution of spectral data, and nonlinear equations, Ukrain. Math. J. 40 (1988), 459-461.
  50. Sakhnovich L.A., Integrable nonlinear equations on the semi-axis, Ukrain. Math. J. 43 (1991), 1470-1476.
  51. Sakhnovich L.A., The method of operator identities and problems in analysis, St. Petersburg Math. J. 5 (1994), 1-69.
  52. Sakhnovich L.A., Spectral theory of canonical differential systems. Method of operator identities, Operator Theory: Advances and Applications, Vol. 107, Birkhäuser Verlag, Basel, 1999.
  53. Seeley R.T., Classroom notes: Fubini implies Leibniz implies $F_{yx} = F_{xy}$, Amer. Math. Monthly 68 (1961), 56-57.
  54. Simon B., A new approach to inverse spectral theory. I. Fundamental formalism, Ann. of Math. 150 (1999), 1029-1057, math.SP/9906118.
  55. Sklyanin E.K., Boundary conditions for integrable equations, Funct. Anal. Appl. 21 (1987), 164-166.
  56. Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, Vol. 72, Amer. Math. Soc., Providence, RI, 2000.
  57. Ton B.A., Initial boundary value problems for the Korteweg-de Vries equation, J. Differential Equations 25 (1977), 288-309.
  58. Zakharov V.E., Manakov S.V., The theory of resonance interaction of wave packets in nonlinear media, Soviet Phys. JETP 69 (1975), 1654-1673.
  59. Zakharov V.E., Shabat A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Phys. JETP 61 (1971), 62-69.
  60. Zakharov V.E., Shabat A.B., Integration of nonlinear equations of mathematical physics by the method of the inverse scattering problem. II, Funct. Anal. Appl. 13 (1979), 166-174.

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