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Register a new userMass-transfer instability of ground-states for Hamiltonian Schrodinger systems
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We study generic semilinear Schrodinger 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.
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We consider systems of weakly coupled Schrodinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.
In this work we consider the weakly coupled Schrodinger cubic system [ begin{cases} displaystyle -Delta u_i+lambda_i u_i= mu_i u_i^{3}+ u_isum_{j eq i}b_{ij} u_j^2 u_iin H^1(mathbb{R}^N;mathbb{R}), quad i=1,ldots, d, end{cases} ] where $1leq Nleq 3$, $lambda_i,mu_i >0$ and $b_{ij}=b_{ji}>0$ for $i eq j$. This system admits semitrivial solutions, that is solutions $mathbf{u}=(u_1,ldots, u_d)$ with null components. We provide optimal qualitative conditions on the parameters $lambda_i,mu_i$ and $b_{ij}$ under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the $d=2$ equations case. For $dgeq 3$ equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case $lambda_iequiv lambda$ and $b_{ij}equiv b$. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the $d=2$ case.
We study the existence of ground states for the coupled Schrodinger system begin{equation} left{begin{array}{lll} displaystyle -Delta u_i+lambda_i u_i= mu_i |u_i|^{2q-2}u_i+sum_{j eq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i u_iin H^1(mathbb{R}^n), quad i=1,ldots, d, end{array}right. end{equation} $ngeq 1$, for $lambda_i,mu_i >0$, $b_{ij}=b_{ji}>0$ (the so-called symmetric attractive case) and $1<q<n/(n-2)^+$. We prove the existence of a nonnegative ground state $(u_1^*,ldots,u_d^*)$ with $u_i^*$ radially decreasing. Moreover we show that, for $1<q<2$, such ground states are positive in all dimensions and for all values of the parameters.
We consider linear stability of steady states of 1(1/2) and 3D Vlasov-Maxwell systems for collisionless plasmas. The linearized systems can be written as separable Hamiltonian systems with constraints. By using a general theory for separable Hamiltonian systems, we recover the sharp linear stability criteria obtained previously by different approaches. Moreover, we obtain the exponential trichotomy estimates for the linearized Vlasov-Maxwell systems in both relativistic and nonrelativistic cases.
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