# American Institute of Mathematical Sciences

May  2015, 14(3): 843-859. doi: 10.3934/cpaa.2015.14.843

## Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves

 1 Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan

Received  April 2014 Revised  December 2014 Published  March 2015

We consider the fourth order nonlinear Schrödinger type equation (4NLS) which arises in context of the motion of vortex filament. The purposes of this paper are twofold. Firstly, we consider the initial value problem for (4NLS) under the periodic boundary condition. By refining the modified energy method used in our previous paper [23], we prove the unique existence of the global solution for (4NLS) in the energy space $H_{p e r}^2(0,2L)$ with $L>0$. Secondly, we study the stability property of periodic standing waves for (4NLS). Using the spectrum properties of the Schrödinger operators associated to the periodic standing wave developed by Angulo [1], we prove that standing wave of dnoidal type is orbitally stable under the time evolution by (4NLS).
Citation: Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure and Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
##### References:
 [1] P. J. Angulo, Nonlinear stability of periodic travelling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations, J. Differential Equations, 235 (2007), 1-30. doi: 10.1016/j.jde.2007.01.003. [2] T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153-183. [3] J. L. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. London Ser. A, 344 (1975), 363-374. [4] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. [5] T. J. Bridges and G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal., 33 (2002), 1356-1378. doi: 10.1137/S0036141099361494. [6] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, 1971. [7] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. [8] L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend, Circ. Mat. Palermo., 22 (1906), 117-135 [in Italian]. [9] Y. Fukumoto, Motion of a curved vortex filament: higher-order asymptotics, In Proc. of IUTAM Symposium on Geometry and Statistics of Turbulence (eds. T. Kambe, T. Nakano and T. Miyauchi). (2001), 211-216, Kluwer. doi: 10.1007/978-94-015-9638-1_25. [10] Y. Fukumoto and H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part I. A higher-order asymptotic formula for the velocity, J. Fluid. Mech., 417 (2000), 1-45. doi: 10.1017/S0022112000008995. [11] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. [12] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477-485. [13] S. M. Hoseini and T. R. Marchant, Solitary wave interaction for a higher-order nonlinear Schrödinger equation, IMA J. Appl. Math., 72 (2007), 206-222. doi: 10.1093/imamat/hxl034. [14] Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Comm. Partial Differential Equations, 32 (2007), 1493-1510. doi: 10.1080/03605300701629385. [15] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. math J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003. [16] C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3. [17] S. Kida, A vortex filament moving without change of form, J. Fluid Mech., 112 (1981), 397-409. doi: 10.1017/S0022112081000475. [18] S. Kwon, On the fifth-order KdV equation: local well-posedness and lack of uniform continuity of the solution map, J. Differential Equations, 245 (2008), 2627-2659. doi: 10.1016/j.jde.2008.03.020. [19] J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci., 1 (1991), 71-93. doi: 10.1007/BF01209148. [20] S. Levandosky, Stability of solitary waves of a fifth-order water wave model, Phys. D, 227 (2007), 162-172. doi: 10.1016/j.physd.2007.01.006. [21] M. Maeda and J. Segata, Existence and stability of standing waves of fourth order nonlinear Schrödinger type equation related to vortex filament, Funkcial. Ekvac., 54 (2011), 1-14. doi: 10.1619/fesi.54.1. [22] A. Moyua and L. Vega, Bounds for the maximal function associated to periodic solutions of one-dimensional dispersive equations, Bull. Lond. Math. Soc., 40 (2008), 117-128. doi: 10.1112/blms/bdm096. [23] J. Segata, Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrödinger type equation on torus, J. Differential Eq., 252 (2012), 5994-6011. doi: 10.1016/j.jde.2012.02.016. [24] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034. [25] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.

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##### References:
 [1] P. J. Angulo, Nonlinear stability of periodic travelling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations, J. Differential Equations, 235 (2007), 1-30. doi: 10.1016/j.jde.2007.01.003. [2] T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153-183. [3] J. L. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. London Ser. A, 344 (1975), 363-374. [4] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. [5] T. J. Bridges and G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal., 33 (2002), 1356-1378. doi: 10.1137/S0036141099361494. [6] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, 1971. [7] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. [8] L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend, Circ. Mat. Palermo., 22 (1906), 117-135 [in Italian]. [9] Y. Fukumoto, Motion of a curved vortex filament: higher-order asymptotics, In Proc. of IUTAM Symposium on Geometry and Statistics of Turbulence (eds. T. Kambe, T. Nakano and T. Miyauchi). (2001), 211-216, Kluwer. doi: 10.1007/978-94-015-9638-1_25. [10] Y. Fukumoto and H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part I. A higher-order asymptotic formula for the velocity, J. Fluid. Mech., 417 (2000), 1-45. doi: 10.1017/S0022112000008995. [11] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. [12] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477-485. [13] S. M. Hoseini and T. R. Marchant, Solitary wave interaction for a higher-order nonlinear Schrödinger equation, IMA J. Appl. Math., 72 (2007), 206-222. doi: 10.1093/imamat/hxl034. [14] Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Comm. Partial Differential Equations, 32 (2007), 1493-1510. doi: 10.1080/03605300701629385. [15] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. math J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003. [16] C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3. [17] S. Kida, A vortex filament moving without change of form, J. Fluid Mech., 112 (1981), 397-409. doi: 10.1017/S0022112081000475. [18] S. Kwon, On the fifth-order KdV equation: local well-posedness and lack of uniform continuity of the solution map, J. Differential Equations, 245 (2008), 2627-2659. doi: 10.1016/j.jde.2008.03.020. [19] J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci., 1 (1991), 71-93. doi: 10.1007/BF01209148. [20] S. Levandosky, Stability of solitary waves of a fifth-order water wave model, Phys. D, 227 (2007), 162-172. doi: 10.1016/j.physd.2007.01.006. [21] M. Maeda and J. Segata, Existence and stability of standing waves of fourth order nonlinear Schrödinger type equation related to vortex filament, Funkcial. Ekvac., 54 (2011), 1-14. doi: 10.1619/fesi.54.1. [22] A. Moyua and L. Vega, Bounds for the maximal function associated to periodic solutions of one-dimensional dispersive equations, Bull. Lond. Math. Soc., 40 (2008), 117-128. doi: 10.1112/blms/bdm096. [23] J. Segata, Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrödinger type equation on torus, J. Differential Eq., 252 (2012), 5994-6011. doi: 10.1016/j.jde.2012.02.016. [24] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034. [25] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.
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