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Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schr?dinger equations  [PDF]
Zaher Hani
Mathematics , 2012,
Abstract: We study the energy cascade problematic for some nonlinear Schr\"odinger equations on the torus $\T^2$ in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schr\"odinger nonlinearities that are equal or closely related to the standard polynomial one $|u|^2u$. Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the first Birkhoff normal form).
Orbital stability of standing waves for a class of Schr?dinger equations with unbounded potential  [PDF]
Guanggan Chen,Jian Zhang,Yunyun Wei
International Journal of Stochastic Analysis , 2006, DOI: 10.1155/jamsa/2006/57676
Abstract: This paper is concerned with the nonlinear Schrödinger equation with an unbounded potential iϕt=−Δϕ
Existence of trajectories with unbounded consumption for a model with innovations
Zaslavski Alexander J
Journal of Inequalities and Applications , 2006,
Abstract: We consider a model of economic dynamics with discrete innovations and establish the existence of trajectories with unbounded consumption.
Existence of trajectories with unbounded consumption for a model with innovations
Alexander J. Zaslavski
Journal of Inequalities and Applications , 2006,
Abstract: We consider a model of economic dynamics with discrete innovations and establish the existence of trajectories with unbounded consumption.
Logarithmic bounds on Sobolev norms for time-dependent linear Schr?dinger equations  [PDF]
W. -M. Wang
Mathematics , 2008,
Abstract: We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schr\"odinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the Sobolev norms are unbounded, which indicates that the log is almost surely necessary. In [W], the author showed that the Sobolev norms remain bounded for an explicit time periodic potential. This is in the exceptional set in the sense of [N]. The present paper together with [N, W] give a rather complete picture of time dependent linear Schr\"odinger equations on the circle.
Fractional Sobolev-Poincaré and fractional Hardy inequalities in unbounded John domains  [PDF]
Ritva Hurri-Syrj?nen,Antti V. V?h?kangas
Mathematics , 2013,
Abstract: We prove fractional Sobolev-Poincar\'e inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.
Orbital stability of standing waves for a class of Schr dinger equations with unbounded potential
Jian Zhang,Yunyun Wei
International Journal of Stochastic Analysis , 2006,
Abstract: This paper is concerned with the nonlinear Schr dinger equation with an unbounded potential i t = Δ + V ( x ) μ | | p 1 λ | | q 1 , x ∈ N , t ≥ 0 , where μ > 0 , λ > 0 , and 1 < p < q < 1 + 4 / N . The potential V ( x ) is bounded from below and satisfies V ( x ) → ∞ as | x | → ∞ . From variational calculus and a compactness lemma, the existence of standing waves and their orbital stability are obtained.
Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains  [PDF]
Serena Boccia,Sara Monsurrò,Maria Transirico
International Journal of Mathematics and Mathematical Sciences , 2008, DOI: 10.1155/2008/582435
Abstract: We study in this paper a class of second-order linear elliptic equations in weighted Sobolev spaces on unbounded domains of ?, ≥3. We obtain an a priori bound, and a regularity result from which we deduce a uniqueness theorem.
Global Schr?dinger maps  [PDF]
Ioan Bejenaru,Alexandru D. Ionescu,Carlos E. Kenig,Daniel Tataru
Mathematics , 2008,
Abstract: We consider the Schr\"{o}dinger map initial-value problem in dimension two or greater. We prove that the Schr\"{o}dinger map initial-value problem admits a unique global smooth solution, provided that the initial data is smooth and small in the critical Sobolev space. We prove also that the solution operator extends continuously to the critical Sobolev space.
Bohm Trajectories as Approximations to Properly Fluctuating Quantum Trajectories  [PDF]
Pisin Chen,Hagen Kleinert
Physics , 2013,
Abstract: We explain the approximate nature of particle trajectories in Bohm's quantum mechanics. They are streamlines of a superfluid in Madelung's reformulation of the Schr\"{o}dinger wave function, around which the proper particle trajectories perform their quantum mechanical fluctuations to ensure Heisenberg's uncertainty relation between position and momentum.
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