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We study the stability of traveling waves of the nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models for this are the Gross–Pitaevskii (GP) equation and the cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For dimension two, the non-degeneracy condition is also proved for these slow traveling waves. For general traveling waves without vortices (that is nonvanishing) and with general nonlinearity in any dimension, we find a sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable, and we find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of all of the above cases, we construct unstable and stable invariant manifolds.

Light bullets are spatially localized ultra-short optical pulses in more than one space dimensions. They contain only a few electromagnetic oscillations under their envelopes and propagate long distances without essentially changing shapes. Light bullets of femtosecond durations have been observed in recent numerical simulation of the full Maxwell systems. The sineGordon (SG) equation comes as an asymptotic reduction of the two level dissipationless MaxwellBloch system. We derive a new and complete nonlinear Schrdinger (NLS) equation in two space dimensions for the SG pulse envelopes so that it is globally well-posed and has all the relevant higher order terms to regularize the collapse of the standard critical NLS (CNLS). We perform a modulation analysis and found that SG pulse envelopes undergo focusingdefocusing cycles. Numerical results are in qualitative agreement with asymptotics and

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field, which could be either vacuum or nonvacuum. The initial density is allowed to vanish, and the spatial measure of the set of vacuum can be arbitrarily large; in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states. © 2012 Wiley Periodicals, Inc