We study nonlinear dispersive wave systems described by hyperbolic PDE's in R^{d} and difference equations on the lattice Z^{d}. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi-Pasta-Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.
We consider the logarithmic Schrödinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson-a time-independent Gaussian function-is an orbitally stable solution. In the general case in dimension $d = 1$, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the…
In this paper, we consider the large time asymptotic nonlinear stability of a superposition of shock waves with contact discontinuities for the one dimensional Jin-Xin relaxation system with small initial perturbations, provided that the strengths of waves are small with the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and an estimate on the heat equation.
This paper is concerned with the large-time behavior of solutions for the one-dimensional compressible Navier-Stokes system. We show that the combination of viscous contact wave with rarefaction waves for the non-isentropic polytropic gas is stable under \emph{large} initial perturbation without the condition that the adiabatic exponent $γ$ is close to 1, provided the strength of the combination waves is suitably small.
Stability of Superposition of Viscous Contact Wave and Rarefaction Waves for Compressible Navier-Stokes System with Degenerate Heat-Conductivity and Large-Data
We prove a superposition principle in the spirit of Crandall-Zhang and Lindqvist-Manfredi for a class of second order quasilinear equations. Riesz potentials of nonnegative and compactly supported continuous functions are either subsolutions or supersolutions for the operators associated to the stationary form of the doubly nonlinear diffusion equation. This class of operators includes both the p-Laplacian as well as the stationary porous medium equation.
This paper is concerned with the study of nonlinear stability of superposition of boundary layer and rarefaction wave on the two-fluid Navier-Stokes-Poisson system in the half line $\mathbb{R}_{+}=:(0,+\infty)$. On account of the quasineutral assumption and the absence of the electric field for the large time behavior, we successfully construct the boundary layer and rarefaction wave, and then we give the rigorous proofs of the stability theorems on the superposition of boundary layer and rarefaction wave under small perturbations for the corresponding initial boundary value problem of the Nav…
A class of differentiable solutions is proved for the isentropic Euler equations in two and three space dimensions. The solutions are explicitly given in terms of solutions to inviscid Burgers equations, and several directions of propagation. The relative orientation of the directions is critical. Within the directional constraints, the Burgers solutions are arbitrary. The several velocities add, and the pressures combine nonlinearly. These solutions cannot exist beyond the time when shocks develop in any of the Burgers solutions.
Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters ar…
Nonlinear Stability for the Superposition of Viscous Contact Wave and Rarefaction Waves to Non-isentropic Compressible Navier-Stokes System with General Initial Perturbations
Spectral analysis, maximum principles and shape optimization for nonlinear superposition operators of mixed fractional order
Ghent University（根特大学）；National Institute of Technology Calicut（卡利卡特国立理工学院）；Indian Institute of Technology (BHU)（印度技术学院（贝拿勒斯 Hindu 大学））；Queen Marry University of London（伦敦玛丽女王大学）
We study the large-time asymptotic behavior of solutions toward the combination of a viscous contact wave with two rarefaction waves for the compressible non-isentropic Navier-Stokes equations coupling with the Maxwell equations through the Lorentz force (called the Navier-Stokes-Maxwell equations). It includes the electrodynamic effects into the dissipative structure of the hyperbolic-parabolic system and turns out to be more complicated than that in the simpler compressible Navier-Stokes equations. Based on a new observation of the specific structure of the Maxwell equations in the Lagrangia…
We study a class of nonlinear fractional differential equations with multiple delays, which is represented by the Voigt creep fractional model of viscoelasticity. We discuss two Voigt models, the first being linear and the second being nonlinear. The linear Voigt model give us the physical interpretation and is associated with important results since the creep function characterizes the viscoelastic behavior of stress and strain. For the nonlinear model of Voigt, our theoretical study and analysis provides existence and stability, where time delays are expressed in terms of Boltzmann's superpo…
Motivated by classical results for Gevrey spaces and their applications to nonlinear partial differential equations we define so-called Gevrey-modulation spaces. We establish analytic as well as non-analytic superposition results on Gevrey-modulation spaces. These results are extended to a special weighted modulation space where the weight increases stronger than any polynomial but less than as in the Gevrey case.
We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for one-dimensional bipolar Vlasov-Poisson-Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micro-macro type decomposition around the local Maxwellian related to the bipolar VPB system in our previous work [26], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to 1D bipolar VPB system under some smallness cond…
