Limit behaviors of standing waves for NLS equation with rotation: mass supercritical case

We consider focusing nonlinear Schrödinger equations (NLS), in the $L^2$-critical and supercritical cases. We present a systematic numerical investigation of the dependence of the blow-up time on properties of the data or on the (parameters of the) equation in three cases: dependence on the strength of the nonlinearity in the equation when the initial data is fixed; dependence on the strength of a damping term in the equation when the initial data is fixed; and dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed. For some cases, analytic results are available and presented. In most situations our numerical counterexamples show that monotonicity in the evolution of the blow-up time does not occur. In addition they show that in certain regimes the blow-up time is very sensitive to the different parameters that we modulate. <br>&nbsp; &nbsp Our numerical solutions are very reliable since not only we test independence on the precise setting of the numerical problem (size of the periodic domain, discretization etc.) but we compare the same simulations with two different methods in two independent codes: a spectral time splitting code and a relaxation method, with results identical at the order of precision.

Long distant behavior of the mode of supercritical collapse in the Nonlinear Schroedinger Equation is analyzed both analytically and numerically. The boundary of region of self-similarity of a weak collapse is determined. Exponential decay of the solution at large distances and convergence of wave energy integral are demonstrated.

Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as t \to +\infty , were constructed for the L^2 critical and subcritical (NLS) and (gKdV) equations in previous works (see [Merle, F.: Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990), no. 2, 223-240], [Martel, Y.: Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math. 127 (2005), no. 5, 1103-1140] and [Martel, Y. and Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849-864]). In this paper, we extend the construction of multi-soliton solutions to the L^2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.

We consider the solutionuɛ(t) of the saturated nonlinear Schrodinger equation $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} u + \varepsilon \left| u \right|^{q - 1} uandu(0,.) = \varphi (.)$$ (1) where\(N \geqslant 2,\varepsilon > 0,1 + 4/N 0 such that\(\mathop {\lim }\limits_{t \to T} \left\| {u(t)} \right\|_{H^1 } = + \infty \), whereu(t) is solution of $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} uandu(0,.) = \varphi (.)$$ (1) For ɛ>0 fixed,uɛ(t) is defined for all time. We are interested in the limit behavior as ɛ→0 ofuɛ(t) fort≥T. In the case where there is no loss of mass inuɛ at infinity in a sense to be made precise, we describe the behavior ofuɛ as ɛ goes to zero and we derive an existence result for a solution of (1) after the blow-up timeT in a certain sense. Nonlinear Schrodinger equation with supercritical exponents are also considered.

We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Meleard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound is established via exponential tightness, the difficulty being that the marginals of our measure-valued processes are not of bounded masses. The local minoration is proved by linking the trajectories of the action functional's domain to the solutions of perturbations of the PDE obtained in the large population limit. The use of Girsanov theorem then leads us to regularize these perturbations. As an application, we study the logistic age-structured population. In the super-critical case, the deterministic approximation admits a non trivial stationary stable solution, whereas the stochastic microscopic process gets extinct almost surely. We establish estimates of the time during which the microscopic process stays in the neighborhood of the large population equilibrium by generalizing the works of Freidlin and Ventzell to our measure-valued setting.

The authors study, by applying and extending the methods developed by Cazenave (2003), Dias and Figueira (2014), Dias et al. (2014), Glassey (1994–1997), Kato (1987), Ohta and Todorova (2009) and Tsutsumi (1984), the Cauchy problem for a damped coupled system of nonlinear Schrodinger equations and they obtain new results on the local and global existence of H 1-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases.

In this paper, we introduce some references on the limit behaviors of Galton-Watson branching process with immigration in subcritical, critical and supercritical cases. Some research work and new progress of the author and collaborators are presented and discussed, including the large deviation and upper deviation for the branching processes with immigration in critical case; the harmonic moment, large deviation and lower deviation for the branching processes with immigration in supercritical case.

<p style='text-indent:20px;'>We consider the focusing <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle <inline-formula><tex-math id="M3">\begin{document}$ \Theta \subset \mathbb{R}^3 $\end{document}</tex-math></inline-formula>. We construct a solution behaving asymptotically as a solitary wave on <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^3, $\end{document}</tex-math></inline-formula> for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R. Killip, M. Visan and X. Zhang, which is the same as the one on the whole Euclidean space given by S. Roundenko and J. Holmer in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.

We study a nonlinear Schrodinger equation which arises as an effective single particle model in X-ray free electron lasers (XFEL). This equation appears as a first principles model for the beam-matter interactions that would take place in an XFEL molecular imaging experiment in [A. Fratalocchi and G. Ruocco, Phys. Rev. Lett., 106 (2011), 105504]. Since XFEL are more powerful by several orders of magnitude than more conventional lasers, the systematic investigation of many of the standard assumptions and approximations has attracted increased attention. In this model the electrons move under a rapidly oscillating electromagnetic field, and the convergence of the problem to an effective time-averaged one is examined. We use an operator splitting pseudospectral method to investigate numerically the behavior of the model versus that of its time-averaged version in complex situations, namely the energy subcritical/mass supercritical case and in the presence of a periodic lattice. We find the time-averaged mode...

This paper discusses the weakly coupled nonlinear Schrodinger equations in the supercritical case. With the best constant of the Gagliardo-Nirenberg inequality, we derive a sufficient condition for the global existence of solutions; this condition is expressed in terms of stationary solutions (nonlinear ground state).

In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schrödinger(NLS) equation driven by multiplicative noise. First, we show the global existence of the unique solution for the damped stochastic NLS equation in critical case. Meanwhile, the exponential integrability of the solution is proved, which implies the continuous dependence on the initial data. Then, we analyze the effect of the damped term and noise on the blow-up phenomenon. By modifying the associated energy, momentum and variance identity, we deduce a sharp blow-up condition for damped stochastic NLS equation in supercritical case. Moreover, we show that when the damped effect is large enough, the damped effect can prevent the blow-up of the solution with high probability.

Let \(\mathbf {Z}_{n}=(Z_{n}^{(1)},Z_{n}^{(2)},\cdots ,Z_{n}^{(d)})\) be a d-type (d < ∞) Galton-Watson branching process. For a positive integer k ≥ 2. Pick k individuals at random from the nth generation by simple random sampling without replacement. Trace their lines of descent backward in time till they meet. Let Xn,k be the generation number of the coalescence time of these k individuals of the nth generation. We call the common ancestor of these chosen individuals in the Xn,kth generation their last common ancestor. In this paper, the limit behaviors of the distributions of Xn,k, for any integer k ≥ 2, is studied for the supercritical cases. Also, we investigate the limit distribution of joint distribution of the generation number and the type of the last common ancestor of these randomly chosen individuals and their types in the supercritical case.

We study the nonlinear Schrödinger equation with an arbitrary real potential $$V(x)\in (L^1+L^\infty )(\Gamma )$$ on a star graph $$\Gamma $$ . At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength $$-\gamma <0$$ . We show the existence of ground states $$\varphi _{\omega }(x)$$ as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on V(x). Moreover, for $$V(x)=-\dfrac{\beta }{x^{\alpha }}$$ , in the supercritical case, we prove that the standing waves $$e^{i\omega t}\varphi _{\omega }(x)$$ are orbitally unstable in $$H^{1}(\Gamma )$$ when $$\omega $$ is large enough. Analogous result holds for an arbitrary $$\gamma \in {\mathbb {R}}$$ when the standing waves have symmetric profile.

In this paper a bisexual Galton–Watson branching process allowing the immigration of mating units is introduced and its limit behaviour is investigated. For the supercritical case, i.e., asymptotic growth rate greater than one, necessary and sufficient conditions for the almost sure and L 1 convergence of the suitably normed underlying sequences are given. *Research supported by the Consejerı´a de Educación y Juventud de la Junta de Extremadura and the Fondo Social Europeo, grant IPR98A023.

Concentration of coupled cubic nonlinear Schrödinger equation