Nonlinear Schrödinger equations (NLSs) with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these waves are intimately connected to stability properties and long-time dynamics solutions NLSs. We study in detail, both analytically numerically.

Abstract We consider a linear Schrödinger equation with nonlinear perturbation in ℝ 3 . Assume that the Hamiltonian has exactly two bound states and its eigen‐values satisfy some resonance condition. prove if initial data is sufficiently small near ground state, then solution approaches to certain state as time tends infinity. Furthermore, difference between wave function solving asymptotic profile can have different types of decay: The resonance‐dominated solutions decay t −1/2 or...

Journal Article Lower Bound on the Blow-up Rate of Axisymmetric Navier–Stokes Equations Get access Chiun-Chuan Chen, Chen 1Department Mathematics and Taida Institute Mathematical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, 106 Center for Theoretical Taiwan, Taipei Office Correspondence to be sent to: ttsai@math.ubc.ca Search other works by this author on: Oxford Academic Google Scholar Robert M. Strain, Strain 2Department Mathematics, Harvard One Street,...

We study the global behavior of small solutions Gross–Pitaevskii equation in three dimensions. prove that disturbances from constant equilibrium with small, localized energy, disperse for large time, according to linearized equation. Translated defocusing nonlinear Schrödinger equation, this implies asymptotic stability all plane wave such disturbances. also every solution finite energy has a which is it. The key ingredients are: (1) some quadratic transforms solutions, effectively linearize...

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote axis symmetry and $r$ measure distance to z-axis. Suppose solution satisfies either $|v (x,t)| \le C_*{|t|^{-1/2}} $ or, for some $\e > 0$, C_* r^{-1+\epsilon} |t|^{-\epsilon /2}$ $-T_0\le t < 0$ $0<C_*<\infty$ allowed be large. We prove that $v$ is regular at time zero.

We study a class of nonlinear Schrödinger equations which admit families small solitary wave solutions. consider solutions are in the energy space H1, and decompose them into dispersive components. The goal is to establish asymptotic stability completeness wave. That is, we show that as t → ∞, component converges fixed wave, strongly H1 solution free equation.

In this article we consider nonlinear Schrödinger (NLS) equations in Rd for d=1, 2, and 3. We nonlinearities satisfying a flatness condition at zero such that solitary waves are stable. Let Rk(t,x) be K wave solutions of the equation with different speeds v1,v2,…,vK. Provided relative vk−vk−1 large enough no interaction two takes place positive time, prove sum Rk(t) is stable t≥0 some suitable sense H1. To result, use an energy method new monotonicity property on quantities related to...

We investigate the asymptotic behavior at time infinity of solutions close to a non-zero constant equilibrium for Gross-Pitaevskii (or Ginzburg-Landau-Schrödinger) equation.We prove that, in dimensions larger than 3, small perturbations can be approximated by linearized evolution, and wave operators are homeomorphic around 0 certain Sobolev spaces.Equation (1.2) with boundary condition (1.3) possesses, dimension d = 2, well-known static, spatially localized, topologically non-trivial vortex...

Relaxation of excited states in nonlinear Schrödinger equations Tai-Peng Tsai, Tsai Search for other works by this author on: Oxford Academic Google Scholar Horng-Tzer Yau International Mathematics Research Notices, Volume 2002, Issue 31, Pages 1629–1673, https://doi.org/10.1155/S1073792802201063 Published: 01 January 2002 Article history Received: 15 Revision received: April Accepted: 10 June

ABSTRACT We consider nonlinear Schrödinger equations in . Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset space of initial data, we construct solutions converging to excited states both non-resonant and resonant cases. In case, linearized operators around are non-self adjoint perturbations some with embedded eigenvalues. Although self-adjoint perturbation turns eigenvalues into resonances, this class turn an eigenvalue distance continuous...

We prove the existence of unique solutions for 3D incompressible Navier-Stokes equations in an exterior domain with small boundary data which do not necessarily decay time. As a corollary, time-periodic is shown. next show that spatial asymptotics periodic solution given by same Landau at all times. Lastly we if datum and initial asymptotically self-similar, then converges to sum vector field forward self-similar as time goes infinity.

For the incompressible Navier-Stokes equations in 3D half space, we show existence of forward self-similar solutions for arbitrarily large initial data.

We consider a nonlinear Schrodinger equation with bounded local potential in M 3 .The linear Hamiltonian is assumed to have two bound states the eigenvalues satisfying some resonance condition.Suppose that initial data are localized and small H 1 .We prove exactly three local-in-space behaviors can occur as time tends infinity: 1.The solutions vanish; 2. The converge ground states; 3. excited states.We also obtain upper bounds for relaxation all cases.In addition, matching lower was given...