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Register a new userTraveling Waves for the Nonlinear Variational Wave Equation
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We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and constant segments, glued together at points where at least one one-sided derivative is unbounded. Applying the method of proof to the Camassa--Holm equation, we recover some well-known results on its traveling wave solutions.
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We derive a new generalization of the nonlinear variational wave equation. We prove existence of local, smooth solutions for this system. As a limiting case, we recover the nonlinear variational wave equation.
We consider the quartic focusing Half Wave equation (HW) in one space dimension. We show first that that there exist traveling wave solutions with arbitrary small $H^{frac 12}(R)$ norm. This fact shows that small data scattering is not possible for (HW) equation and that below the ground state energy there are solutions whose energy travels as a localised packet and which preserve this localisation in time. This behaviour for (HW) is in sharp contrast with classical NLS in any dimension and with fractional NLS with radial data. The second result addressed is the non existence of traveling waves moving at the speed of light. The main ingredients of the proof are commutator estimates and a careful study of spatial decay of traveling waves profile using the harmonic extension to the upper half space.
We consider the 3D Gross-Pitaevskii equation begin{equation} onumber ipartial_t psi +Delta psi+(1-|psi|^2)psi=0 text{ for } psi:mathbb{R}times mathbb{R}^3 rightarrow mathbb{C} end{equation} and construct traveling waves solutions to this equation. These are solutions of the form $psi(t,x)=u(x_1,x_2,x_3-Ct)$ with a velocity $C$ of order $varepsilon|logvarepsilon|$ for a small parameter $varepsilon>0$. We build two different types of solutions. For the first type, the functions $u$ have a zero-set (vortex set) close to an union of $n$ helices for $ngeq 2$ and near these helices $u$ has degree 1. For the second type, the functions $u$ have a vortex filament of degree $-1$ near the vertical axis $e_3$ and $ngeq 4$ vortex filaments of degree $+1$ near helices whose axis is $e_3$. In both cases the helices are at a distance of order $1/(varepsilonsqrt{|log varepsilon|)}$ from the axis and are solutions to the Klein-Majda-Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross-Pitaevskii equation, namely the Ginzburg-Landau equation. To prove the existence of these solutions we use the Lyapunov-Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation.
We study the stability of traveling waves of nonlinear Schrodinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and 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 sub-critical or critical nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For traveling waves without vortices (i.e. nonvanishing) of general nonlinearity in any dimension, we find the sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable and find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of above cases, we construct unstable and stable invariant manifolds.
We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i pt_t u = sqrt{-Delta} , u - |u|^{p-1} u, quad mbox{with $(t,x) in R times R^d$} $$ with exponents $1 < p < infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{iomega t} Q_v(x-vt) $$ with frequency $omega in R$, velocity $v in R^d$, and some finite-energy profile $Q_v in H^{1/2}(R^d)$, $Q_v ot equiv 0$. We prove that traveling solitary waves for speeds $|v| geq 1$ do not exist. Furthermore, we generalize the non-existence result to the square root Klein--Gordon operator $sqrt{-DD+m^2}$ and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds $|v| < 1$. Finally, we discuss the energy-critical case when $p=(d+1)/(d-1)$ in dimensions $d geq 2$.
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