Abstract
In this paper we will study the stability properties of self-similar solutions of \(1\)D cubic NLS equations with time-dependent coefficients of the form
The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation
As a by-product of our results we prove that Eq. (0.1) is well-posed in appropriate function spaces when the initial datum is given by \(u(0,x)= z_0 \mathrm p.v \frac{1}{x}\) for some values of \(z_0\in \mathbb{C }\setminus \{ 0\}\), and \(A\) is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution.
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Notes
Notice that if \(a=0\), then \(|f(0)|=|f^{\prime }(0)|=0\), so that \(f\equiv 0\).
If \(|f|_{+\infty }=|f|_{-\infty }\) (\(|f^{\prime }|_{+\infty }=|f^{\prime }|_{-\infty }\)), then we will denote \(|f|_{\pm \infty }\) by \(|f|_{\infty }\) (respectively, \(|f^{\prime }|_{\pm \infty }\) by \(|f^{\prime }|_{\infty }\)).
In order to simplify notation, in what follows we will write simply \(Y\) to denote the space \(Y_{t_{0}}^{\nu }\).
Conversely, using the parallel frame defined by the system (2.53), it can be also proved that if \(\mathbf{X }(t,x)\) is a regular solution of LIE, and define the function \(u=\alpha +i \beta \), then \(u\) solves the \(1\)d-cubic Schrödinger equation
$$\begin{aligned} \displaystyle { iu_t+ u_{xx}+\frac{u}{2} (|u|^2-A(t))=0 } \end{aligned}$$with \(A(t)= -|u|^2(0,t)/2- \langle \partial _t\mathbf e_{1} , \mathbf{e_2 }\rangle (0,t)\).
Recall that for odd solutions of LIA, the third component of the associated tangent vector, \(T_{3}\), is an even function. Thus, in particular \(T_3(+\infty )=T_3(-\infty )\).
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Acknowledgments
We would like to thank the anonymous referee for a number of helpful suggestions. L. Vega is funded in part by the grant MTM 2007-82186 of MEC (Spain) and FEDER. Part of this work was done while the first author was visiting the Universidad del País Vasco under the PIC program. S. Gutiérrez was partially supported by the grant MTM 2007-82186 of MEC (Spain). Financial support from the program “Euclidean Harmonic Analysis, Nilpotent Lie Groups and PDEs”, held in the Centro di Ricerca Matematica Ennio De Giorgi in Pisa, is also kindly acknowledged by S. Gutiérrez.
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Gutierrez, S., Vega, L. On the stability of self-similar solutions of 1D cubic Schrödinger equations. Math. Ann. 356, 259–300 (2013). https://doi.org/10.1007/s00208-012-0847-4
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DOI: https://doi.org/10.1007/s00208-012-0847-4