We study the periodic non-linear Schrodinger equations with odd integer power nonlinearities, for initial data which are assumed to be small in some negative order Sobolev space, but which may have large L^2 mass. These equations are known to be illposed in H^s for all negative s, in the sense that the solution map fails to be uniformly continuous from H^s to itself, even for short times and small norms. Here we show that these equations are even more unstable. For the cubic equation, the solution map is discontinuous from H^s to even the space of distributions. For the quintic and higher order nonlinearities, there exist pairs of solutions which are uniformly bounded in H^s, are arbitrarily close in any C^N norm at time zero, and fail to be close in the distribution topology at an arbitrarily small positive time.
In the present paper, we establish the existence and orbital instability results of cnoidal periodic waves for the quintic Klein-Gordon and nonlinear Schrödinger equations. The spectral analysis for the corresponding linearized operator is established by using the Floquet theory. The orbital instability is determined by applying an abstract result due to Shatah and Strauss.
We study generic semilinear Schrödinger systems which may be written in Hamiltonian form. In the presence of a single gauge invariance, the components of a solution may exchange mass between them while preserving the total mass. We exploit this feature to unravel new orbital instability results for ground-states. More precisely, we first derive a general instability criterion and then apply it to some well-known models arising in several physical contexts. In particular, this mass-transfer instability allows us to exhibit $L^2$-subcritical unstable ground-states.
We show the strong instability of radial ground state standing waves for the focusing $L^2$-supercritical nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu + c|x|^{-2} u = - |u|^α u, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $u: \mathbb{R} \times \mathbb{R}^d \rightarrow \mathbb{C}$, $c\ne 0$ satisfies $c<λ(d):=\left(\frac{d-2}{2}\right)^2$ and $\frac{4}{d} <α<\frac{4}{d-2}$. This result extends a recent result of Bensouilah-Dinh-Zhu [{\it On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse…
We study the strong instability of ground-state standing waves $e^{iωt}ϕ_ω(x)$ for $N$-dimensional nonlinear Schrödinger equations with double power nonlinearity. One is $L^2$-subcritical, and the other is $L^2$-supercritical. The strong instability of standing waves with positive energy was proven by Ohta and Yamaguchi (2015). In this paper, we improve the previous result, that is, we prove that if $\partial_λ^2S_ω(ϕ_ω^λ)|_{λ=1}\le0$, the standing wave is strongly unstable, where $S_ω$ is the action, and $ϕ_ω^λ(x)\mathrel{\mathop:}=λ^{N/2}ϕ_ω(λx)$ is the $L^2$-invariant scaling.
We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger operator with a point interaction describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The perturbed Laplace operator always has a unique simple negative eigenvalue. We prove that if the frequency of the standing wave is close to the negative eigenvalue, it is stable. Moreover, if the frequency is sufficiently large, we have the stability i…
We study the strong instability of standing waves $e^{iωt}ϕ_ω(x)$ for nonlinear Schrödinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $ω\in\mathbb{R}$ is a frequency, and $ϕ_ω\in H^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrödinger equations with a harmonic potential, Ohta (2018) proved that if $\partial_λ^2S_ω(ϕ_ω^λ)|_{λ=1}\le0$, then the standing wave is strongly unstable, where $S_ω$ is the action, and $ϕ_ω^λ(x)\mathrel{\mathop:}=λ^{N/2}ϕ_ω(λx)$ is the scaling, which does …
We consider the focusing $L^2$-supercritical fractional nonlinear Schrödinger equation \[ i\partial_t u - (-Δ)^s u = -|u|^αu, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d\geq 2, \frac{d}{2d-1} \leq s <1$ and $\frac{4s}{d}<α<\frac{4s}{d-2s}$. By means of the localized virial estimate, we prove that the ground state standing wave is strongly unstable by blow-up. This result is a complement to a recent result of Peng-Shi [J. Math. Phys. 59 (2018), 011508] where the stability and instability of standing waves were studied in the $L^2$-subcritical and $L^2$-critical cases.
We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2σ}\partial_x u=0, $$ where $1<σ<2$. The equation has a two-parameter family of solitary wave solutions of the form $$ u_{ω,c}(t,x)=e^{iωt+i\frac c2(x-ct)-\frac{i}{2σ+2}\int_{-\infty}^{x-ct}φ^{2σ}_{ω,c}(y)dy}φ_{ω,c}(x-ct). $$ The stability theory in the frequency region of $|c|<2\sqrtω$ was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case $c=2\sqrtω$.
We study the stability/instability of standing waves for the one dimensional nonlinear Schrödinger equation with double power nonlinearities: \begin{align*} &i\partial_t u +\partial_x^2 u -|u|^{p-1}u +|u|^{q-1}u=0, \quad (t,x)\in \mathbb{R}\times\mathbb{R} ,~1<p<q. \end{align*} When $q<5$, the stability properties of standing waves $e^{iωt}ϕ_ω$ may change for the frequency $ω$. A sufficient condition for yielding instability for small frequencies are obtained in previous results, but it has not been known what the sharp condition is. In this paper we completely calculate the explicit formula o…
A nonlinear Schrödinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schrödinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in $L^2$ to those of the NLS as the grid size $h>0$ approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.
We consider the following nonlinear Schrödinger equation with derivative: \begin{equation} iu_t =-u_{xx} -i |u|^{2}u_x -b|u|^4u , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this equation is a gauge equivalent form of the well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of DNLS satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in our equation only affects the coefficient in front of the quintic term in the elliptic equation, so in this sense the additional no…
We consider the focusing nonlinear Schrödinger equation with inverse square potential \[ i\partial_t u + Δu + c|x|^{-2} u = - |u|^αu, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d \geq 3$, $c\ne 0$, $c<λ(d)=\left(\frac{d-2}{2}\right)^2$ and $0<α\leq \frac{4}{d}$. Using the profile decomposition obtained recently by the first author \cite{Bensouilah}, we show that in the $L^2$-subcritical case, i.e. $0<α<\frac{4}{d}$, the sets of ground state standing waves are orbitally stable. In the $L^2$-critical case, i.e. $α=\frac{4}{d}$, we show that ground state…
We consider the equation $Δu=Vu$ in exterior domains in $\mathbb{R}^2$ and $\mathbb{R}^3$, where $V$ has certain periodicity properties. In particular we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation $Δu=Vu$ is studied as part of a broader class of elliptic evolution equations.
We consider the periodic non-linear Schrödinger equation with non-linearity given by $|u|^{p-1}u$ for odd $p > 1$ in dimension $1$. We first establish that the difference between the non-linear evolution and a phase rotation of the the linear evolution is in a smoother space. We then study forced and damped defocusing non-linear Schrödinger equations of the above type and establish an analogous smoothing statement that extends globally in time. As a corollary we establish both existence and smootheness for global attractors in the energy space.
A polynomial-in-time growth bound is established for global Sobolev $H^s(\mathbb T)$ solutions to the derivative nonlinear Schrödinger equation on the circle with $s>1$. These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.
