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

Zbl 1281.35077
Full Text: