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On the Cauchy problem for the Kortewegde Vries equation with steplike finite-gap initial data II. Perturbations with finite moments

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Abstract

We solve the Cauchy problem for the Korteweg-de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of finite derivatives with finite moments.

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References

  1. T. Aktosun, On the Schrödinger equation with steplike potentials, J. Math. Phys. 40 (1999), 5289–5305.

    Article  MathSciNet  MATH  Google Scholar 

  2. V. B. Baranetskii and V. P. Kotlyarov, Asymptotic behavior in a back front domain of the solution of the KdV equation with a “step type” initial condition, Teoret. Mat. Fiz. 126 (2001), 214–227.

    MathSciNet  Google Scholar 

  3. E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994.

    MATH  Google Scholar 

  4. R. F. Bikbaev, Structure of a shock wave in the theory of the Korteweg-de Vries equation, Phys. Lett. A 141 (1989), 289–293.

    Article  MathSciNet  Google Scholar 

  5. R. F. Bikbaev, Time asymptotics of the solution of the nonlinear Schrödinger equation with boundary conditions of “step-like” type, Teoret. Mat. Fiz. 81 (1989), 3–11.

    MathSciNet  Google Scholar 

  6. R. F. Bikbaev and R. A. Sharipov, The asymptotic behavior as t → ∞of the solution of the Cauchy problem for the Korteweg-de Vries equation in a class of potentials with finite-gap behavior as x→±∞, Teoret. Mat. Fiz. 78 (1989), 345–356.

    MathSciNet  Google Scholar 

  7. I. N. Bondareva, The Korteweg-de Vries equation in classes of increasing functions with prescribed asymptotic behavior as |x| →∞, Math. USSR-Sb. 50 (1985), 125–135.

    Article  MATH  Google Scholar 

  8. I. Bondareva, and M. Shubin, Increasing asymptotic solutions of the Korteweg-de Vries equation and its higher analogues, Sov. Math. Dokl. 26:3 (1982), 716–719.

    MATH  Google Scholar 

  9. A. Boutet de Monvel, and I. Egorova, The Toda lattice with step-like initial data. Soliton asymptotics, Inverse Problems 16 (2000), 955–977.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Boutet de Monvel, I. Egorova, and G. Teschl, Inverse scattering theory for one-dimensional Schrödinger operators with steplike finite-gap potentials, J. Anal. Math. 106 (2008), 271–316.

    Article  MathSciNet  MATH  Google Scholar 

  11. V. S. Buslaev and V. N. Fomin, An inverse scattering problem for the one-dimensional Schrödinger equation on the entire axis, Vestnik Leningrad. Univ. 17 (1962), 56–64.

    MathSciNet  Google Scholar 

  12. A. Cohen, Solutions of the Korteweg-de Vries equation with steplike initial profile, Comm. Partial Differential Equations 9 (1984), 751–806.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Cohen and T. Kappeler, Scattering and inverse scattering for steplike potentials in the Schrödinger equation, Indiana Univ. Math. J. 34 (1985), 127–180.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Cohen and T. Kappeler, Solutions to the Korteweg-de Vries equation with initial profile in L 11 (R) ∩ L 1N (R +), SIAM J. Math. Anal. 18 (1987), 991–1025.

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Comm. Math. Phys. 63 (1978), 277–301.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Deift, S. Kamvissis, T. Kriecherbauer, and X. Zhou, The Toda rarefaction problem, Comm. Pure Appl. Math. 49 (1996), 35–83.

    Article  MathSciNet  MATH  Google Scholar 

  17. W. Eckhaus and A. Van Harten, The Inverse Scattering Transformation and Solitons: An Introduction, North-Holland, Amsterdam, 1984.

    Google Scholar 

  18. I. Egorova and G. Teschl, On the Cauchy problem for the modified Korteweg-de Vries equation with steplike finite-gap initial data, in Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, Contemp. Math. 526, Amer. Math. Soc., Providence, RI, 2010, pp. 151–158.

    Google Scholar 

  19. I. Egorova, K. Grunert, and G. Teschl, On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data I. Schwartz-type perturbations, Nonlinearity 22 (2009), 1431–1457.

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Egorova, J. Michor, and G. Teschl, Inverse scattering transform for the Toda hierarchy with quasi-periodic background, Proc. Amer. Math. Soc. 135 (2007), 1817–1827.

    Article  MathSciNet  MATH  Google Scholar 

  21. V. D. Ermakova, The asymptotics of the solution of the Cauchy problem for the Korteweg-de Vries equation with nondecreasing initial data of special type, Dokl. Akad. Nauk Ukrain. SSR Ser. A 7 (1982), 3–6.

    MathSciNet  Google Scholar 

  22. V. D. Ermakova, The inverse scattering problem on the whole axis for the Schrödinger equation with nondecreasing potential of special form, Vestnik Khar’kov Gos. Univ. 230 (1982), 50–60.

    Google Scholar 

  23. L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, 1987.

    MATH  Google Scholar 

  24. N. E. Firsova, An inverse scattering problem for the perturbed Hill operator, Mat. Zametki 18 (1975), 831–843.

    MathSciNet  MATH  Google Scholar 

  25. N. E. Firsova, A direct and inverse scattering problem for a one-dimensional perturbed Hill operator, Mat. Sb. (N. S.) 130(172) (1986), 349–385.

    MathSciNet  Google Scholar 

  26. N. E. Firsova, The Riemann surface of a quasimomentum, and scattering theory for a perturbed Hill operator, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 51(1975), 183–196.

    MathSciNet  MATH  Google Scholar 

  27. N. E. Firsova, Solution of the Cauchy problem for the Korteweg-de Vries equation with initial data that are the sum of a periodic and a rapidly decreasing function, Math. USSR-Sb. 63 (1989), 257–265.

    Article  MathSciNet  MATH  Google Scholar 

  28. C. S. Gardner, J. M. Green, M. D. Kruskal, and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095–1097.

    Article  MATH  Google Scholar 

  29. F. Gesztesy, Scattering theory for one-dimensional systems with nontrivial spatial asymptotics in Schrödinger Operators, Springer, Berlin, 1986, pp. 93–122.

    Chapter  Google Scholar 

  30. F. Gesztesy and H. Holden, Soliton Equations and their Algebro-Geometric Solutions. Volume I: (1 + 1)-Dimensional Continuous Models, Cambridge University Press, Cambridge, 2003.

    MATH  Google Scholar 

  31. F. Gesztesy, R. Nowell, and W. Pötz, One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics, Differential Integral Equations 10 (1997), 521–546.

    MathSciNet  MATH  Google Scholar 

  32. F. Gesztesy, R. Ratnaseelan, and G. Teschl, The KdV hierarchy and associated trace formulas, in Proceedings of the International Conference on Applications of Operator Theory, Birkhäuser, Basel, 1996, pp. 125–163.

    Google Scholar 

  33. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007.

    MATH  Google Scholar 

  34. A. R. Its and A. F. Ustinov, Time asymptotics of the solution of the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of finite density type, Dokl. Akad. Nauk SSSR 291 (1986), 91–95.

    MathSciNet  Google Scholar 

  35. S. Kamvissis, On the Toda shock problem, Physica D 65 (1993), 242–266.

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Kamvissis and G. Teschl, Stability of periodic soliton equations under short range perturbations, Phys. Lett. A 364 (2007), 480–483.

    Article  MathSciNet  MATH  Google Scholar 

  37. S. Kamvissis and G. Teschl, Stability of the periodic Toda lattice under short range perturbations, arxiv: 0705.0346v5.

  38. T. Kappeler, Solutions of the Korteweg-de Vries equation with steplike initial data, J. Differential Equations 63 (1986), 306–331.

    Article  MathSciNet  MATH  Google Scholar 

  39. T. Kappeler, P. Perry, M. Shubin, and P. Topalov, Solutions of mKdV in classes of functions unbounded at infinity, J. Geom. Anal. 18 (2008), 443–477.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. B. Khasanov and G. U. Urazboev, Solution of the general KdV equation in the class of step functions, J. Math. Sciences 136 (2006), 3625–3640.

    Article  MathSciNet  Google Scholar 

  41. E. Ya. Khruslov, Asymptotics of the solution of the Cauchy problem for the Korteweg-de Vries equation with initial data of step type, Mat. Sb (N.S.) 99(141) (1976), 261–281.

    MathSciNet  Google Scholar 

  42. E. Ya. Khruslov and V. P. Kotlyarov, Soliton asymptotics of nondecreasing solutions of nonlinear completely integrable evolution equations, in Spectral Operator Theory and Related Topics, Amer. Math. Soc., Providence, RI, 1994, pp. 129–181.

    Google Scholar 

  43. E. Ya. Khruslov and V. P. Kotlyarov, Solitons of the nonlinear Schrödinger equation, which are generated by the continuous spectrum, Teoret. Mat. Fiz. 68 1986, 172–186.

    MathSciNet  Google Scholar 

  44. E. Ya. Khruslov and V. P. Kotlyarov, Time asymptotics of the solution of the Cauchy problem for the modified Korteweg-de Vries equation with nondecreasing initial data, Dokl. Akad. Nauk Ukrain. SSR Ser. A 10 (1986), 61–64.

    MathSciNet  Google Scholar 

  45. E. Ya. Khruslov and H. Stephan, Splitting of some nonlocalized solutions of the Korteweg-de Vries equation into solitons, Mat. Fiz. Anal. Geom. 5 (1998), 49–67.

    MathSciNet  MATH  Google Scholar 

  46. H. Krüger and G. Teschl, Stability of the periodic Toda lattice in the soliton region, Int. Math. Res. Not. IMRN 2009, 3996–4031.

  47. E. A. Kuznetsov and A. V. Mikhailov, Stability of stationary waves in nonlinear weakly dispersive media, Soviet Physics JETP 40 (1974), 855–859.

    MathSciNet  Google Scholar 

  48. V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986.

    MATH  Google Scholar 

  49. A. Mikikits-Leitner and G. Teschl, Trace formulas for Schrödinger operators in connection with scattering theory for finite-gap backgrounds, in Spectral Theory and Analysis, Birkhäuser, Basel, 2011, pp. 107–124.

    Chapter  Google Scholar 

  50. S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Springer, Berlin, 1984.

    Google Scholar 

  51. V. Yu. Novokshenov, Time asymptotics for soliton equations in problems with step initial conditions, J. Math. Sci. 125 (2005), 717–749.

    Article  MathSciNet  Google Scholar 

  52. A. Rybkin, Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half-line, Nonlinearity 23 (2010), 1143–1167.

    Article  MathSciNet  MATH  Google Scholar 

  53. S. Venakides, P. Deift, and R. Oba, The Toda shock problem, Comm. Pure Appl. Math. 44 (1991), 1171–1242.

    Article  MathSciNet  MATH  Google Scholar 

  54. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1927.

    MATH  Google Scholar 

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Authors and Affiliations

  1. B. Verkin Institute for Low Temperature Physics, 47 Lenin Avenue, 61103, Kharkiv, Ukraine

    Iryna Egorova

  2. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Wien, Austria

    Gerald Teschl

  3. International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090, Wien, Austria

    Gerald Teschl

Authors
  1. Iryna Egorova
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  2. Gerald Teschl
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Correspondence to Iryna Egorova.

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Egorova, I., Teschl, G. On the Cauchy problem for the Kortewegde Vries equation with steplike finite-gap initial data II. Perturbations with finite moments. JAMA 115, 71–101 (2011). https://doi.org/10.1007/s11854-011-0024-9

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  • DOI: https://doi.org/10.1007/s11854-011-0024-9

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