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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.

MSC:

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

Citations:

Zbl 1281.35077
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