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Center Stable Manifolds Around Line Solitary Waves of the Zakharov–Kuznetsov Equation

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Abstract

In this paper, we construct center stable manifolds of unstable line solitary waves for the Zakharov–Kuznetsov equation on \({\mathbb {R}} \times {\mathbb {T}}_L\) and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag (SIAM J Math Anal 44:1175–1210, 2012). Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod (Ann Inst H Poincaré Anal Non Lineaire 32:347–371, 2015) and modifying the mobile distance in Nakanishi and Schlag (2012), we construct a contraction map on the graph space.

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References

  1. Alexander, J.C., Pego, R.L., Sachs, R.L.: On the transverse instability of solitary waves in the Kadomtsev–Petviashvili equation. Phys. Lett. A 226, 187–192 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bates, P.W., Jones, C.K.R.T.: Invariant manifolds for semilinear partial differential equations. In: Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported. A Series in Dynamical Systems and Their Application, vol. 2, pp. 1–38. Wiley, Chichester (1989)

    Google Scholar 

  3. Beceanu, M.: A critical center-stable manifold for Schrödinger’s equation in three dimensions. Commun. Pure Appl. Math. 65, 431–507 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beceanu, M.: A center-stable manifold for the energy-critical wave equation in \(\mathbb{R} ^3\) in the symmetric setting. J. Hyperbolic Differ. Equ. 11, 437–476 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benjamin, T.B.: The stability of solitary waves. Proc. R. Soc. (Lond.) Ser. A 328, 153–183 (1972)

    MathSciNet  Google Scholar 

  6. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I. Geom. Funct. Anal. 3, 107–156 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bridges, T.J.: Universal geometric conditions for the transverse instability of solitary waves. Phys. Rev. Lett. 84, 2614–2617 (2000)

    Article  Google Scholar 

  8. Faminskii, A.V.: The Cauchy problem for the Zakharov–Kuznetsov equation. Transl. Differ. Equ. A 31, 1002–1012 (1995)

    MathSciNet  Google Scholar 

  9. Ginibre, J., Tsutsumi, Y., Velo, G.: On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151, 384–436 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grünrock, A.: A remark on the modified Zakharov–Kuznetsov equation in three space dimensions. Math. Res. Lett. 21, 127–131 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grünrock, A., Herr, S.: The Fourier restriction norm method for the Zakharov–Kuznetsov equation. Discrete Contin. Dyn. Syst. 34, 2061–2068 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jin, J., Lin, Z., Zeng, C.: Invariant manifolds of traveling waves of the 3D Gross–Pitaevskii equation in the energy space. Commun. Math. Phys. (2018). https://doi.org/10.1007/s00220-018-3189-6

    Article  MathSciNet  MATH  Google Scholar 

  13. Johnson, K.: The transverse instability of periodic waves in Zakharov–Kuznetsov type equations. Stud. Appl. Math. 124, 323–345 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970)

    MATH  Google Scholar 

  15. Krieger, J., Schlag, W.: Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Am. Math. Soc. 19, 815–920 (2006)

    Article  MATH  Google Scholar 

  16. Krieger, J., Nakanishi, K., Schlag, W.: Threshold phenomenon for the quintic wave equation in three dimensions. Commun. Math. Phys. 327, 309–332 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Krieger, J., Nakanishi, K., Schlag, W.: Center-stable manifold of the ground state in the energy space for the critical wave equation. Math. Ann. 361, 1–50 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lannes, D., Linares, F., Saut, J.-C.: The Cauchy problem for the Euler–Poisson system and derivation of the Zakharov–Kuznetsov equation. In: Cicognani, M., Colombini, F., Del Santo, D. (eds.) Studies in Phase Space Analysis with Applications to PDEs, Progress in Nonlinear Differential Equations and Applications, vol. 84, pp. 181–213. Springer, New York (2013)

    MATH  Google Scholar 

  19. Linares, F., Pastor, A.: Well-posedness for the two-dimensional modified Zakharov–Kuznetsov equation. SIAM J. Math. Anal. 41, 1323–1339 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Linares, F., Pastor, A.: Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation. J. Funct. Anal. 206, 1060–1085 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Linares, F., Pastor, A., Saut, J.-C.: Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton. Commun. Partial Differ. Equ. 35, 1674–1689 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Linares, F., Saut, J.-C.: The Cauchy problem for the 3D Zakharov–Kuznetsov equation. Discrete Contin. Dyn. Syst. 24, 547–565 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157, 219–254 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Martel, Y., Merle, F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18, 55–80 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Martel, Y., Merle, F.: Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Math. Ann. 341, 391–427 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Martel, Y., Merle, F., Nakanishi, K., Raphaël, P.: Codimension one threshold manifold for the critical gKdV equation. Commun. Math. Phys. 342, 1075–1106 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mizumachi, T.: Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations. SIAM J. Math. Anal. 32, 1050–1080 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mizumachi, T.: Stability of line solitons for the KP-II equation in \(\mathbb{R} ^2\). Mem. Am. Math. Soc. 238, vii+95 (2015)

    MATH  Google Scholar 

  29. Mizumachi, T.: Stability of line solitons for the KP-II equation in \(\mathbb{R}^2\), II. Proc. R. Soc. Edinb. Sect. A 148, 149–198 (2018)

    Article  MATH  Google Scholar 

  30. Mizumachi, T., Tzvetkov, N.: Stability of the line soliton of the KP-II equation under periodic transverse perturbations. Math. Ann. 352, 659–690 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Molinet, L., Pilod, D.: Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications. Ann. Inst. H. Poincaré Anal. Non Lineaire 32, 347–371 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Molinet, L., Saut, J.-C., Tzvetkov, N.: Global well-posedness for the KP-II equation on the background of a non-localized solution. Ann. Inst. H. Poincaré Anal. Non Lineaire 28, 653–676 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nakanishi, K., Schlag, W.: Global dynamics above the ground state energy for the cubic NLS equation in 3D. Calc. Var. Partial Differ. Equ. 44, 1–45 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nakanishi, K., Schlag, W.: Invariant manifolds around soliton manifolds for the nonlinear Klein–Gordon equation. SIAM J. Math. Anal. 44, 1175–1210 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pego, R., Weinstein, M.I.: Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pelinovsky, D.: Normal form for transverse instability of the line soliton with a nearly critical speed of propagation. Math. Model. Nat. Phenom. 13, 1–20 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ribaud, F., Vento, S.: Well-posedness results for the three-dimensional Zakharov–Kuznetsov equation. SIAM J. Math. Anal. 44, 2289–2304 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rousset, F., Tzvetkov, N.: Transverse nonlinear instability of solitary waves for some Hamiltonian PDE’s. J. Math. Pures. Appl. 90, 550–590 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rousset, F., Tzvetkov, N.: Transverse nonlinear instability for two-dimensional dispersive models. Ann. I. Poincaré-AN 26, 477–496 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rousset, F., Tzvetkov, N.: Stability and instability of the KdV solitary wave under the KP-I flow. Commun. Math. Phys. 313, 155–173 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Schlag, W.: Stable manifolds for an orbitally unstable nonlinear Schrödinger equation. Ann. Math. 2(169), 139–227 (2009)

    Article  MATH  Google Scholar 

  42. Villarroel, J., Ablowitz, M.J.: On the initial value problem for the KPII equation with data that do not decay along a line. Nonlinearity 17, 1843–1866 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yamazaki, Y.: Transverse instability for nonlinear Schrödinger equation with a linear potential. Adv. Differ. Equ. 21, 429–462 (2016)

    MATH  Google Scholar 

  44. Yamazaki, Y.: Stability for line solitary waves of Zakharov–Kuznetsov equation. J. Differ. Equ. 262, 4336–4389 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zakharov, V.E.: Instability and nonlinear oscillations of solitons. JETP Lett. 22, 172–173 (1975)

    Google Scholar 

  46. Zakharov, V.E., Kuznetsov, E.A.: On three dimensional solitons. Sov. Phys. JETP 39, 285–286 (1974)

    Google Scholar 

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Acknowledgements

The author would like to express his great appreciation to Professor Yoshio Tsutsumi for a lot to helpful advices and encouragements. The author would like to thank Professor Nikolay Tzvetkov for his helpful encouragements. The author is supported by JSPS Research Fellowships for Young Scientists under Grant 18J00947.

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo, Kyoto, 606-8502, Japan

    Yohei Yamazaki

  2. Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Yohei Yamazaki

  3. University of Cergy-Pontoise, UMR CNRS 8088, 95000, Cergy-Pontoise, France

    Yohei Yamazaki

  4. Hiroshima University, 1-3-2 Kagamiyama, Higashi-Hiroshima City, Hiroshima, 739-8511, Japan

    Yohei Yamazaki

  5. Faculty of Mathematics, Kyushu University, Fukuoka, Japan

    Yohei Yamazaki

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Correspondence to Yohei Yamazaki.

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The author is supported by JSPS Research Fellowships for Young Scientists under Grant 18J00947.

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Yamazaki, Y. Center Stable Manifolds Around Line Solitary Waves of the Zakharov–Kuznetsov Equation. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10329-4

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  • DOI: https://doi.org/10.1007/s10884-023-10329-4

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