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Register a new userLong time dynamics for semirelativistic NLS and Half Wave in arbitrary dimension
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We consider the Cauchy problems associated with semirelativistc NLS (sNLS) and half wave (HW). In particular we focus on the following two main questions: local/global Cauchy theory; existence and stability/instability of ground states. In between other results, we prove the existence and stability of ground states for sNLS in the $L^2$ supercritical regime. This is in sharp contrast with the instability of ground states for the corresponding HW, which is also established along the paper, by showing an inflation of norms phenomenon. Concerning the Cauchy theory we show, under radial symmetry assumption the following results: a local existence result in $H^1$ for energy subcritical nonlinearity and a global existence result in the $L^2$ subcritical regime.
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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 give a complete study of the asymptotic behavior of a simple model of alignment of unit vectors, both at the level of particles , which corresponds to a system of coupled differential equations, and at the continuum level, under the form of an aggregation equation on the sphere. We prove unconditional convergence towards an aligned asymptotic state. In the cases of the differential system and of symmetric initial data for the partial differential equation, we provide precise rates of convergence.
We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schr{o}dinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below the ground states to determine the global behavior (blow-up/scattering) of the solution. Our proof of scattering is based on the varia-tional characterization of the ground states, localized virial estimates, linear profile decomposition and nonlinear profiles.
We consider the half-wave maps equation $$ partial_t vec{S} = vec{S} wedge | abla| vec{S}, $$ where $vec{S}= vec{S}(t,x)$ takes values on the two-dimensional unit sphere $mathbb{S}^2$ and $x in mathbb{R}$ (real line case) or $x in mathbb{T}$ (periodic case). This an energy-critical Hamiltonian evolution equation recently introduced in cite{LS,Zh}, which formally arises as an effective evolution equation in the classical and continuum limit of Haldane-Shastry quantum spin chains. We prove that the half-wave maps equation admits a Lax pair and we discuss some analytic consequences of this finding. As a variant of our arguments, we also obtain a Lax pair for the half-wave maps equation with target $mathbb{H}^2$ (hyperbolic plane).
The Sasa-Satsuma equation with $3 times 3 $ Lax representation is one of the integrable extensions of the nonlinear Schr{o}dinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data. Based on the Rieamnn-Hilbert problem characterization for the Cauchy problem and the $overline{partial}$-nonlinear steepest descent method, we find qualitatively different long time asymptotic forms for the Sasa-Satsuma equation in three solitonic space-time regions: (1) For the region $x<0, |x/t|=mathcal{O}(1)$, the long time asymptotic is given by $$q(x,t)=u_{sol}(x,t| sigma_{d}(mathcal{I})) + t^{-1/2} h + mathcal{O} (t^{-3/4}). $$ in which the leading term is $N(I)$ solitons, the second term the second $t^{-1/2}$ order term is soliton-radiation interactions and the third term is a residual error from a $overlinepartial$ equation. (2) For the region $ x>0, |x/t|=mathcal{O}(1)$, the long time asymptotic is given by $$ u(x,t)= u_{sol}(x,t| sigma_{d}(mathcal{I})) + mathcal{O}(t^{-1}).$$ in which the leading term is $N(I)$ solitons, the second term is a residual error from a $overlinepartial$ equation. (3) For the region $ |x/t^{1/3}|=mathcal{O}(1)$, the Painleve asymptotic is found by $$ u(x,t)= frac{1}{t^{1/3}} u_{P} left(frac{x}{t^{1/3}} right) + mathcal{O} left(t^{2/(3p)-1/2} right), qquad 4<p < infty.$$ in which the leading term is a solution to a modified Painleve $mathrm{II}$ equation, the second term is a residual error from a $overlinepartial$ equation.
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