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Register a new userLong-time dynamics for a simple aggregation equation on the sphere
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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.
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We study the McKean-Vlasov equation [ partial_t varrho= beta^{-1} Delta varrho + kappa abla cdot (varrho abla (W star varrho)) , , ] with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller--Segel model for bacterial chemotaxis, and the noisy Hegselmann--Krausse model for opinion dynamics.
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.
In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data. Employing the $bar{partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the long time asymptotic behavior of the solution $q(x,t)$ in space-time soliton region. Based on the resulting asymptotic behavior, the asymptotic approximation of the WKI equation is characterized with the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ leading order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$. Our results also confirm the soliton resolution conjecture for the WKI equation with finite density initial data.
The long-time asymptotic behavior of solutions to the focusing nonlinear Schrodinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the presence of discrete spectrum. The results of the analysis provide the first rigorous characterization of the nonlinear interactions between solitons and the coherent oscillating structures produced by localized perturbations in a modulationally unstable medium. The study makes crucial use of the inverse scattering transform for the focusing NLS equation with nonzero boundary conditions, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. Previously, it was shown that in the absence of discrete spectrum the $xt$-plane decomposes asymptotically in time into two types of regions: a left far-field region and a right far-field region, where to leading order the solution equals the condition at infinity up to a phase shift, and a central region where the asymptotic behavior is described by slowly modulated periodic oscillations. Here, it is shown that in the presence of a conjugate pair of discrete eigenvalues in the spectrum a similar coherent oscillatory structure emerges but, in addition, three different interaction outcomes can arise depending on the precise location of the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a mixed regime in which the soliton transmission or trapping is accompanied by the formation of an additional, nondispersive localized structure akin to a soliton-generated wake. The soliton-induced position and phase shifts of the oscillatory structure are computed, and the analytical results are validated by a set of accurate numerical simulations.
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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