Stable manifolds for an orbitally unstable nonlinear Schrödinger equation. (English) Zbl 1180.35490

The paper is concerned with the focusing cubic nonlinear Schrödinger equation (1) \(i\partial_t\psi+\Delta\psi=-|\psi|^2\psi\) in \(\mathbb{R}^3\) which is locally well-posed in \(H^1(\mathbb{R}^3)\). It is well known that for each \(\alpha>0\) there exists a unique positive radial solution \(\phi=\phi(\cdot,\alpha)\) of \(-\Delta\phi+\alpha^2\phi=\phi^3\), the ground state. This gives rise to the ground state soliton solution \(\psi=e^{it\alpha^2}\phi\) of (1). It is also well-known that this solution is orbitally unstable. The two main theorems of the paper describe a codimension one stable manifold of initial conditions.
The author imposes the condition that the matrix operator \[ \mathcal{H}(\alpha)= \begin{pmatrix} -\Delta+\alpha^2-2\phi^2(\cdot,\alpha)\quad & -\phi^2(\cdot, \alpha)\\ \phi^2(\cdot,\alpha)\quad & \Delta-\alpha^2+2\phi^2(\cdot,\alpha) \end{pmatrix} \] does not have embedded eigenvalues in the essential spectrum \((-\infty,-\alpha^2]\cup[\alpha^2,\infty)\). This is true in the one-dimensional case as shown by the author and J. Krieger [J. Am. Math. Soc. 19, No. 4, 815–920 (2006; Zbl 1281.35077), and it holds for “generic perturbations” of the cubic nonlinearity.
The first theorem states the existence of a codimension nine manifold \(\mathcal{M}\) consisting of initial conditions \(\psi(0)=\phi(\cdot,\alpha_0)+R_0+\Phi(R_0)\) where \(\alpha_0\) is fixed, and \(R_0\) lies in a neighborhood of \(0\) in a codimension nine linear subspace of \(W^{1,2}\cap W^{1,1}(\mathbb{R}^3)\). The parametrization \(\Phi\) of \(\mathcal{M}\) is Lipschitz continuous. The solutions \(\psi(t)\) with these initial conditions are global \(H^1\) solutions and have the form \(\psi(t)=W(t,\cdot)+R(t)\) with \[ W(t,x)=e^{i\theta(t,x)}\phi(x-y(t), \alpha(t)) \] which is a “solution with moving parameters”. \(R\) is a small perturbation and there is scattering \[ R(t)=e^{i t\Delta}f_0+o_{L^2}(t)\qquad\text{ as } t\to\infty \] for some \(f_0\in L^2(\mathbb R^3)\).
Applying all symmetries of the NLS equation to \(\mathcal{M}\) yields a codimension one manifold \(\mathcal{N}\). Six additional dimensions come from the Galilei transforms, one from modulation, and one from scaling. This is the content of Theorem 2.


35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems


Zbl 1281.35077
Full Text: DOI arXiv Link