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We study the focusing NLS equation in $$R\mathbb{R}^N$$ in the mass-supercritical and energy-subcritical (or intercritical ) regime, with $H^1$ data at the mass-energy threshold $$\mathcal{ME}(u_0)=\mathcal{ME}(Q)$$, where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the $H^1$-critical case, in dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, $$Q^\pm$$, besides the standing wave $$e^{it}Q$$, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blow-up occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the $H^1$-critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around the standing wave and some tailored approximate solutions to the NLS equation. We establish important decay properties of functions associated to the spectrum of the linearized Schrödinger operator, which, in combination with modulational stability and coercivity for the linearized operator on special subspaces, allows us to use a fixed-point argument to show the existence of special solutions. Finally, we prove the uniqueness by studying exponentially decaying solutions to a sequence of linearized equations.
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Mixed dispersion nonlinear Schrödinger equation in higher dimensions: theoretical analysis and numerical computations
Abstract In the present work we provide a characterization of the ground states of a higher-dimensional quadratic-quartic model of the nonlinear Schrödinger class with a combination of a focusing biharmonic operator with either an isotropic or an anisotropic defocusing Laplacian operator (at the linear level) and power-law nonlinearity. Examining principally the prototypical example of dimension d = 2, we find that instability arises beyond a certain threshold coefficient of the Laplacian between the cubic and quintic cases, while all solutions are stable for powers below the cubic. Above the quintic, and up to a critical nonlinearity exponent p , there exists a progressively narrowing range of stable frequencies. Finally, above the critical p all solutions are unstable. The picture is rather similar in the anisotropic case, with the difference that even before the cubic case, the numerical computations suggest an interval of unstable frequencies. Our analysis generalizes the relevant observations for arbitrary combinations of Laplacian prefactor b and nonlinearity power p .
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- PAR ID:
- 10345209
- Date Published:
- Journal Name:
- Journal of Physics A: Mathematical and Theoretical
- Volume:
- 55
- Issue:
- 26
- ISSN:
- 1751-8113
- Page Range / eLocation ID:
- 265701
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1751-8121/ac7019
