Scattering for 1D cubic NLS and singular vortex dynamics. (English) Zbl 1290.35235
Summary: We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions \(\chi_a(t, x)\) form a family of evolving regular curves in \(\mathbb R^3\) that develop a singularity in finite time, indexed by a parameter \(a > 0\). We consider curves that are small regular perturbations of \(\chi_a(t_0, x)\) for a fixed time \(t_0\). In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence for the binormal flow. Nevertheless, we construct solutions of the binormal flow with these initial data. Moreover, these solutions become also singular in finite time. Our approach uses the Hasimoto transform, which leads us to study the long-time behavior of a 1D cubic NLS equation with time-depending coefficients and small regular perturbations of the constant solution as initial data. We prove asymptotic completeness for this equation in appropriate function spaces.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B35 | Stability in context of PDEs |
| 35C06 | Self-similar solutions to PDEs |
| 35P25 | Scattering theory for PDEs |
| 47A40 | Scattering theory of linear operators |
| 76B47 | Vortex flows for incompressible inviscid fluids |
References:
| [1] | Alekseenko, S. V., Kuibin, P. A., Okulov, V. L.: Theory of Concentrated Vortices. An Intro- duction. Springer, Berlin (2007) · Zbl 1132.76001 |
| [2] | Banica, V., Vega, L.: On the Dirac delta as initial condition for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 97-711 (2008) · Zbl 1147.35092 · doi:10.1016/j.anihpc.2007.03.007 |
| [3] | Banica, V., Vega, L.: On the stability of a singular vortex dynamics. Comm. Math. Phys. 286, 593-627 (2009) · Zbl 1183.35029 · doi:10.1007/s00220-008-0682-3 |
| [4] | Batchelor, G. K.: An Introduction to Fluid Dynamics. Cambridge Univ. Press, Cambridge (1967) · Zbl 0152.44402 |
| [5] | Carles, R.: Geometric optics and long range scattering for one-dimensional nonlin- ear Schrödinger equations. Comm. Math. Phys. 220, 41-67 (2001) · Zbl 1029.35211 · doi:10.1007/s002200100438 |
| [6] | Christ, M.: Power series solution of a nonlinear Schrödinger equation. In: Mathematical As- pects of Nonlinear Dispersive Equations, Ann. of Math. Stud. 163, Princeton Univ. Press, Princeton, NJ, 131-155 (2007) · Zbl 1142.35084 |
| [7] | Da Rios, L. S.: On the motion of an unbounded fluid with a vortex filament of any shape. Rend. Circ. Mat. Palermo 22, 117-135 (1906) (in Italian) |
| [8] | de la Hoz, F.: Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane. Math. Z. 257, 61-80 (2007) · Zbl 1128.35099 · doi:10.1007/s00209-007-0115-6 |
| [9] | de la Hoz, F., García-Cervera, C., Vega, L.: A numerical study of the self-similar solutions of the Schrödinger map. SIAM J. Appl. Math. 70, 1047-1077 (2009) · Zbl 1219.65139 · doi:10.1137/080741720 |
| [10] | Grünrock, A.: Bi- and trilinear Schrödinger estimates in one space dimension with ap- plications to cubic NLS and DNLS. Int. Math. Res. Notices 2005, no. 41, 2525-2558 · Zbl 1088.35063 · doi:10.1155/IMRN.2005.2525 |
| [11] | Gustafson, S., Nakanishi, K., Tsai, T.-P.: Global dispersive solutions for the Gross- Pitaevskii equation in two and three dimensions. Ann. Henri Poincaré 8, 1303-1331 (2007) · Zbl 1375.35485 |
| [12] | Gustafson, S., Nakanishi, K., Tsai, T.-P.: Scattering theory for the Gross-Pitaevskii equa- tion in three dimensions. Comm. Contemp. Math. 11, 657-707 (2009) · Zbl 1180.35481 · doi:10.1142/S0219199709003491 |
| [13] | Gutiérrez, S., Rivas, J., Vega, L.: Formation of singularities and self-similar vortex motion under the localized induction approximation. Comm. Partial Differential Equations 28, 927- 968 (2003) · Zbl 1044.35089 · doi:10.1081/PDE-120021181 |
| [14] | Hasimoto, H.: A soliton on a vortex filament. J. Fluid Mech. 51, 477-485 (1972) · Zbl 0237.76010 · doi:10.1017/S0022112072002307 |
| [15] | Hayashi, N., Naumkin, P.: Domain and range of the modified wave operator for Schrödinger equations with critical nonlinearity. Comm. Math. Phys. 267, 477-492 (2006) · Zbl 1113.81121 · doi:10.1007/s00220-006-0057-6 |
| [16] | Nahmod, A., Shatah, J., Vega, L., Zeng, C.: Schrödinger maps and their associated frame systems. Int. Math. Res. Notices 2007, no. 21, art. ID rnm088, 29 pp. · Zbl 1142.35087 · doi:10.1093/imrn/rnm088 |
| [17] | Ozawa, T.: Long range scattering for nonlinear Schrödinger equations in one space dimension. Comm. Math. Phys. 139, 479-493 (1991) · Zbl 0742.35043 · doi:10.1007/BF02101876 |
| [18] | Ricca, R. L.: The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics. Fluid Dynam. Res. 18, 245-268 (1996) · Zbl 1006.01505 · doi:10.1016/0169-5983(96)82495-6 |
| [19] | Saffman, P. G.: Vortex Dynamics. Cambridge Monogr. Mech. Appl. Math., Cambridge Univ. Press, New York (1992) · Zbl 0777.76004 · doi:10.1017/CBO9780511624063 |
| [20] | Vargas, A., Vega, L.: Global wellposedness of 1D non-linear Schrödinger equation for data with an infinite L2 norm. J. Math. Pures Appl. 80, 1029-1044 (2001) · Zbl 1027.35134 · doi:10.1016/S0021-7824(01)01224-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
