Related papers: The derivation of the conservation law for defocus…
For nonlinear Schroedinger equations with a power nonlinearity, a new approach to derive the conservation law of the momentum and the pseudo conformal conservation law is obtained. Since this approach does not contain approximating…
We analize the Nonlinear Schr\"odinger Equation.
We revisit the derivation of the virial identity for nonlinear Schr\"odinger equations. In \cite{O06, FM17}, several conservation laws, such as for the charge and the energy, were derived without constructing a sequence of approximate…
The conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived. For discrete models the conserved charges are constructed explicitly. The applications of the general…
We prove an "almost conservation law" to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schr\"odinger equation in H^s(R^n) when n = 2, 3 and s > 4/7, 5/6, respectively.
Compared with macroscopic conservation law for the solution of the derivative nonlinear Schr\"odingger equation (DNLS) with small mass in \cite{KlausS:DNLS}, we show the corresponding microscopic conservation laws for the Schwartz solutions…
Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux.
Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux and with initial data being a sum of periodic function and a function…
We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schr\"odinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation…
The dynamics of nonlinear conservation laws have long posed fascinating problems. With the introduction of some nonlinearity, e.g. Burgers' equation, discontinuous behavior in the solutions is exhibited, even for smooth initial data. The…
All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted…
In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schr\"odinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same…
We consider a nonlinear Schrodinger equation with power nonlinearity, either on a compact manifold without boundary, or on the whole space in the presence of harmonic confinement, in space dimension one and two. Up to introducing an extra…
Conservation laws are computed for various nonlinear partial differential equations that arise in elasticity and acoustics. Using a scaling homogeneity approach, conservation laws are established for two models describing shear wave…
In this paper we consider the Schr{\"o}dinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on…
We present a unified framework for the rigorous derivation of conservation laws and related identities for nonlinear Schr\"odinger equations with power-type nonlinearities. This approach treats the equation in its Duhamel form and uses the…
Consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial…
We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equation in one space dimension. Under a suitable weakly dissipative condition on the nonlinearity, we show that the small data solution has a logarithmic…
Consider the initial value problem for systems of cubic derivative nonlinear Schr\"odinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under…
In this paper, non-variational systems of differential equations containing small terms are considered, and a consistent approach for deriving approximate conservation laws through the introduction of approximate Lagrange multipliers is…