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Inverse Problems for the Higher Order Nonlinear Schrödinger Equation

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The results on the well-posedness of inverse problems with integral overdetermination on a bounded interval for the higher order nonlinear Schrödinger equation are established. Either the right-hand side of the equation, or the boundary data, or both are chosen as controls. Assumptions on the smallness of the input data or a time interval are required.

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References

  1. G. Fibich, “Adiabatic law for self-focusing of optical beams,” Opt. Lett. 21, 1735–1737 (1996).

    Article  Google Scholar 

  2. A. Hasegawa and Y. Kodama, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, No. 5, 510–524 (1987).

    Article  Google Scholar 

  3. Y. Kodama, “Optical solitons in a monomode fiber,” J. Stat. Phys. 39, No. 5-6, 597–614 (1985).

    Article  MathSciNet  Google Scholar 

  4. H. Kumar and F. Chand, “Dark and bright solitary waves solutions of the higher order nonlinear Schrödinger equation with self-steeping and self-frequency shift effects,” J. Nonlinear Opt. Phys. Mater. 22, No. 1, Article ID 1350001 (2013).

  5. C. Laurey, “The Cauchy problem for a third order nonlinear Schr¨odinger equation,” Nonlinear Anal., Theory Methods Appl. 29, No. 2, 121–158 (1997).

  6. G. Staffilani, “On the generalized Korteweg–de Vries-type equations,” Differ. Integral Equ. 10, No. 4, 777–796 (1997).

    MathSciNet  MATH  Google Scholar 

  7. M. Alves, M. Sepúlveda, and O. Vera, “Smoothing properties for the higher-order nonlinear Schrödinger equation with constant coefficients,” Nonlinear Anal., Theory Methods Appl. 71, No. 3-4, 948–966 (2009).

  8. A. Batal, T. Özsari, and K. C. Yilmaz, “Stabilization of higher order linear and nonlinear Schrödinger equations on a finite domain. I,” Evol. Equ. Control Theory 10, No. 4, 861–919 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Bisognin, V. Bisognin, and O. P. V. Villagran, “Stabilization of solutions to higher-order nonlinear Schrödinger equation with localized damping,” Electron. J. Differ. Equ. 2007, Paper No 6 (2007).

  10. V. Bisognin and O. P. V. Villagran, “On the unique continuation property for the higher order nonlinear Schrödinger equation with constant coefficients,” Turk. J. Math. 30, No. 1–38 (2006).

  11. R. A. Capistrano-Filho and L. Soares de Souza, “Control results with overdetermination condition for higher order dispersive system,” J. Math. Anal. Appl. 506, No. 1, Article ID 125546 (2022).

  12. X. Carvajal, “Sharp global well-posedness for a higher order Schrödinger equation,” J. Fourier Anal. Appl. 12, No. 1, 53–70 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  13. X. Carvajal and F. Linares, “A higher order nonlinear Schrödinger equation with variable coefficients,” Differ. Integral Equ. 16, No. 9, 1111–1130 (2003).

    MATH  Google Scholar 

  14. X. Carvajal and W. Neves, “Persistence of solutions to higher order nonlinear Schrödinger equation,” J. Differ. Equations 249, No. 9, 2214–2236 (2010).

    Article  MATH  Google Scholar 

  15. X. Carvajal and M. Panthee, “Unique continuation for a higher order nonlinear Schrödinger equation,” J. Math. Anal. Appl. 303, No. 1, 188–207 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  16. M. M. Cavalcanti et al., “Well-posedness and asymptotic behavior of a generalized higher order nonlinear Schrödinger equation with localized dissipation,” Comput. Math. Appl. 96, 188–208 (2021).

    MathSciNet  MATH  Google Scholar 

  17. M. Chen, “Stabilization of the higher order nonlinear Schrödinger equation with constant coefficients,” Proc. Indian Acad. Sci., Math. Sci. 128, No. 3, Paper No. 39 (2018).

  18. V. Ceballos et al., “Exact boundary controllability for higher order nonlinear Schrödinger equations with constant coefficients,” Electron. J. Differ. Equ. 2005, Paper No. 122 (2005).

  19. A. V. Faminskii, “Controllability problems for the Korteweg–de Vries equation with integral overdetermination,” Differ. Equ. 55, No. 1, 126–137 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  20. A. V. Faminskii, “Control problems with an integral condition for Korteweg–de Vries equation on unbounded domains,” J. Optim. Theory Appl. 180, No. 1, 290–302 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. V. Faminskii, “On one control problem for Zakharov–Kuznetsov equation,” In: Analysis, Probability, Applications, and Computation, pp. 305–313, Birkh¨auser, Charm (2019).

  22. A. V. Faminskii, “The higher order nonlinear Schrödinger equation with quadratic nonlinearity on the real axis,” Adv. Differ. Equ. 28, No. 5–6, 413–466 (2023).

    MathSciNet  MATH  Google Scholar 

  23. A. V. Faminskii and N. A. Larkin, “Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval,” Electron. J. Differ. Equ. 2010, Paper No. 01 (2010).

  24. J. Fan and S. Jiang, “Well-posedness of an inverse problem of a time-dependent Ginzburg–Landau model for superconductivity,” Commun. Math. Sci. 3, No. 3, 393–401 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Fan and G. Nakamura, “Local solvability of an inverse problem to the density-dependent Navier–Stokes equations,” Appl. Anal. 87, No. 10–11, 1255-1265 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Lu, M. Chen and Q. Lui, “A nonlinear inverse problem of the Korteweg–de Vries equation,” Bull. Math. Sci. 9, No. 3, Article ID 1950014 (2019).

  27. E. V. Martynov, “Inverse problems for the generalized Kawahara equation,” Lobachevskii J. Math. 43, No. 10, 2714–2742 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York (2000).

    MATH  Google Scholar 

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Authors and Affiliations

  1. RUDN University, 6, Miklukho-Maklaya St, Moscow, 117198, Russia

    A. V. Faminskii & E. V. Martynov

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  1. A. V. Faminskii
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  2. E. V. Martynov
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Correspondence to A. V. Faminskii.

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Faminskii, A.V., Martynov, E.V. Inverse Problems for the Higher Order Nonlinear Schrödinger Equation. J Math Sci 274, 475–492 (2023). https://doi.org/10.1007/s10958-023-06614-8

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