scholarly journals NUMERICAL SIMULATIONS OF THE ENERGY-SUPERCRITICAL NONLINEAR SCHRÖDINGER EQUATION

2010 ◽  
Vol 07 (02) ◽  
pp. 279-296 ◽  
Author(s):  
J. COLLIANDER ◽  
G. SIMPSON ◽  
C. SULEM
Keyword(s):  
Numerical Simulations ◽  
Wave Equations ◽  
Initial Conditions ◽  
A Priori ◽  
Sobolev Norm ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Long Time ◽  
Supercritical Nonlinearity ◽  
Homogeneous Norm

We present numerical simulations of the defocusing nonlinear Schrödinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig–Merle and Kilip–Visan who considered some energy supercritical wave equations and proved that if the solution is a priori bounded in the critical Sobolev space (i.e. the space whose homogeneous norm is invariant under the scaling leaving the equation invariant), then it exists for all time and scatters. In this paper, we numerically investigate the boundedness of the H2-critical Sobolev norm for solutions of the NLS equation in dimension five with quintic nonlinearity. We find that for a class of initial conditions, this norm remains bounded, the solution exists for long time, and scatters.

scholarly journals On Jensen-type Inequalities for Nonsmooth Radial Scattering Solutions of a Loglog Energy-Supercritical Schrödinger Equation

2018 ◽  
Vol 2020 (8) ◽  
pp. 2501-2541
Author(s):  
Tristan Roy
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
A Priori ◽  
Time Interval ◽  
Type Estimate ◽  
A Priori Bound ◽  
Long Time ◽  
Supercritical Nonlinearity ◽  
Gamma S ◽  
Radial Scattering

Abstract We prove scattering of solutions of the loglog energy-supercritical Schrödinger equation $i \partial _{t} u + \triangle u = |u|^{\frac{4}{n-2}} u g(|u|)$ with $g(|u|) := \log ^{\gamma } {( \log{(10+|u|^{2})} )}$, $0 < \gamma < \gamma _{n}$, n ∈ {3, 4, 5}, and with radial data $u(0) := u_{0} \in \tilde{H}^{k}:= \dot{H}^{k} (\mathbb{R}^{n})\,\cap\,\dot{H}^{1} (\mathbb{R}^{n})$, where $\frac{n}{2} \geq k> 1 \left(\text{resp.}\,\frac{4}{3}> k > 1\right)$ if n ∈ {3, 4} (resp. n = 5). The proof uses concentration techniques (see e.g., [ 2, 12]) to prove a long-time Strichartz-type estimate on an arbitrarily long time interval J depending on an a priori bound of some norms of the solution, combined with an induction on time of the Strichartz estimates in order to bound these norms a posteriori (see e.g., [ 8, 10]). We also revisit the scattering theory of solutions with radial data in $\tilde{H}^{k}$, $k> \frac{n}{2}$, and n ∈ {3, 4}; more precisely, we prove scattering for a larger range of $\gamma$ s than in [ 10]. In order to control the barely supercritical nonlinearity for nonsmooth solutions, that is, solutions with data in $\tilde{H}^{k}$, $k \leq \frac{n}{2}$, we prove some Jensen-type inequalities.

scholarly journals Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

2016 ◽  
Vol 13 (01) ◽  
pp. 1-105 ◽  
Author(s):  
Gustav Holzegel ◽  
Sergiu Klainerman ◽  
Jared Speck ◽  
Willie Wai-Yeung Wong
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Large Body ◽  
Small Data ◽  
Shock Formation ◽  
Time Behavior ◽  
Long Time ◽  
Remarkable Result ◽  
Quasilinear Wave Equations ◽  
Three Space

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small [Formula: see text]-initial conditions (with [Formula: see text] sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition.

scholarly journals The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere

2015 ◽  
Vol 786 ◽  
pp. 1-4 ◽  
Author(s):  
Paul K. Newton
Keyword(s):  
Numerical Simulations ◽  
Euler Equations ◽  
Large Scale ◽  
Initial Conditions ◽  
Infinite Family ◽  
Small Scale ◽  
Two Dimensional ◽  
Random Initial Conditions ◽  
Long Time ◽  
Initial Vorticity

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

scholarly journals A symplectic method for structure-preserving modelling of damped acoustic waves

2015 ◽  
Vol 471 (2183) ◽  
pp. 20150105 ◽  
Author(s):  
Xiaofan Li ◽  
Mingwen Lu ◽  
Shaolin Liu ◽  
Shizhong Chen ◽  
Huan Zhang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Numerical Simulations ◽  
Seismic Wave ◽  
Wave Equations ◽  
Superior Performance ◽  
Symplectic Method ◽  
Structure Preserving ◽  
Long Time

In this paper, a symplectic method for structure-preserving modelling of the damped acoustic wave equation is introduced. The equation is traditionally solved using non-symplectic schemes. However, these schemes corrupt some intrinsic properties of the equation such as the conservation of both precision and the damping property in long-term calculations. In the method presented, an explicit second-order symplectic scheme is used for the time discretization, whereas physical space is discretized by the discrete singular convolution differentiator. The performance of the proposed scheme has been tested and verified using numerical simulations of the attenuating scalar seismic-wave equation. Scalar seismic wave-field modelling experiments on a heterogeneous medium with both damping and high-parameter contrasts demonstrate the superior performance of the approach presented for suppression of numerical dispersion. Long-term computational experiments display the remarkable capability of the approach presented for long-time simulations of damped acoustic wave equations. Promising numerical results suggest that the approach is suitable for high-precision and long-time numerical simulations of wave equations with damping terms, as it has a structure-preserving property for the damping term.

Three-Level Difference Schemes on Non-Uniform in Time Grids

2001 ◽  
Vol 1 (3) ◽  
pp. 265-284 ◽  
Author(s):  
Piotr Matus ◽  
Elena Zyuzina
Keyword(s):  
Hilbert Spaces ◽  
Wave Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Approximation ◽  
Difference Schemes ◽  
Nonuniform Grids ◽  
Priori Estimates ◽  
Long Time ◽  
The Right

Abstract In this work, a stability of three-level operator-difference schemes on nonuniform in time grids in Hilbert spaces is studied. A priori estimates of a long time stability (for t → ∞) in the sense of the initial data and the right-hand side are obtained in different energy norms without demanding the quasiuniformity of the grid. New difference schemes of the second order of local approximation on nonuniform grids both in time and space on standard stencils for parabolic and wave equations are adduced.

scholarly journals Transition of Liesegang Precipitation Systems: Simulations with an Adaptive Grid PDE Method

2011 ◽  
Vol 10 (4) ◽  
pp. 867-881 ◽  
Author(s):  
Paul A. Zegeling ◽  
István Lagzi ◽  
Ferenc Izsák
Keyword(s):  
Pattern Formation ◽  
Numerical Simulations ◽  
Initial Conditions ◽  
Qualitative Change ◽  
Adaptive Grid ◽  
Time Behavior ◽  
Centrally Symmetric ◽  
Type Pattern ◽  
Long Time ◽  
Pde Method

AbstractThe dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of an adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments.

scholarly journals Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

2021 ◽  
Author(s):  
Mamoru Okamoto ◽  
Kota Uriya
Keyword(s):  
Fourth Order ◽  
Decay Estimate ◽  
The Self ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Self Similar Solution ◽  
Long Time ◽  
Self Similar

AbstractWe consider the long-time behavior of solutions to a fourth-order nonlinear Schrödinger (NLS) equation with a derivative nonlinearity. By using the method of testing by wave packets, we construct an approximate solution and show that the solution for the fourth-order NLS has the same decay estimate for linear solutions. We prove that the self-similar solution is the leading part of the asymptotic behavior.

scholarly journals The local wave phenomenon in the quintic nonlinear Schrödinger equation by numerical methods

2021 ◽  
Author(s):  
Yaning Tang ◽  
Zaijun Liang ◽  
Wenxian Xie
Keyword(s):  
Optical Fibers ◽  
Analytical Solutions ◽  
Initial Conditions ◽  
Fourier Method ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Light Pulses ◽  
Propagation Of Light ◽  
Wide Range ◽  
Local Wave

Abstract The nonlinear Schrodinger hierarchy has a wide range of applications in modeling the propagation of light pulses in optical fibers. In this paper, we focus on the integrable nonlinear Schrodinger (NLS) equation with quintic terms, which play a prominent role when the pulse duration is very short. First, we investigate the spectral signatures of the spatial Lax pair with distinct analytical solutions and their periodized wavetrains by Fourier oscillatory method. Then, we numerically simulate the wave evolution of the quintic NLS equation from different initial conditions through the symmetrical split-step Fourier method. We find many localized high-peak structures whose profiles are very similar to the analytical solutions, and we analyze the formation of rouge waves (RWs) in different cases. These results may be helpful to understand the excitation of nonlinear waves in some nonlinear fields, such as optical fibers, oceanography and so on.

Low Energy Scattering for Nonlinear Schrödinger Equations in Fractional Order Sobolev Spaces

1997 ◽  
Vol 09 (03) ◽  
pp. 397-410 ◽  
Author(s):  
M. Nakamura ◽  
T. Ozawa
Keyword(s):  
Scattering Problem ◽  
Sobolev Norm ◽  
Asymptotic Completeness ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Wave Operators ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Homogeneous Norm

We consider the scattering problem for the nonlinear Schrödinger equations with interactions behaving as a power p at zero. In the critical and subcritical cases (s≥n/2-2/(p-1)≥0), we prove the existence and asymptotic completeness of wave operators in the sense of Sobolev norm of order s on a set of asymptotic states with small homogeneous norm of order n/2-2/(p-1) in space dimension n≥1.

scholarly journals Quasi-stability and continuity of attractors for nonlinear system of wave equations

2021 ◽  
Vol 8 (1) ◽  
pp. 27-45
Author(s):  
M. M. Freitas ◽  
M. J. Dos Santos ◽  
A. J. A. Ramos ◽  
M. S. Vinhote ◽  
M. L. Santos
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Global Attractors ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Small Perturbations ◽  
Recent Approach ◽  
Nonlinear Coupled System ◽  
Long Time ◽  
External Forces

Abstract In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

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