2,913 research outputs found
Rigorous justification of the short-pulse equation
We prove that the short-pulse equation, which is derived from Maxwell
equations with formal asymptotic methods, can be rigorously justified. The
justification procedure applies to small-norm solutions of the short-pulse
equation. Although the small-norm solutions exist for infinite times and
include modulated pulses and their elastic interactions, the error bound for
arbitrary initial data can only be controlled over finite time intervals.Comment: 15 pages, no figure
Modulation equations near the Eckhaus boundary: the KdV equation
We are interested in the description of small modulations in time and space
of wave-train solutions to the complex Ginzburg-Landau equation \begin{align*}
\partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi
|\Psi|^2, \end{align*} near the Eckhaus boundary, that is, when the wave train
is near the threshold of its first instability. Depending on the parameters , a number of modulation equations can be derived, such as
the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau
based amplitude equations. Here we establish error estimates showing that the
KdV approximation makes correct predictions in a certain parameter regime. Our
proof is based on energy estimates and exploits the conservation law structure
of the critical mode. In order to improve linear damping we work in spaces of
analytic functions.Comment: 44 pages, 8 figure
Justification of Leading Order Quasicontinuum Approximations of Strongly Nonlinear Lattices
We consider the leading order quasicontinuum limits of a one-dimensional
granular medium governed by the Hertz contact law under precompression. The
approximate model which is derived in this limit is justified by establishing
asymptotic bounds for the error with the help of energy estimates. The
continuum model predicts the development of shock waves, which are also studied
in the full system with the aid of numerical simulations. We also show that
existing results concerning the Nonlinear Schrodinger (NLS) and Korteweg
de-Vries (KdV) approximation of FPU models apply directly to a precompressed
granular medium in the weakly nonlinear regime
Justification of the NLS Approximation for the KdV Equation Using the Miura Transformation
It is the purpose of this paper to give a simple proof of the fact
that solutions of the KdV equation can be approximated via solutions of the NLS equation. The proof is based on an elimination of the quadratic terms of the KdV equation via the Miura transformation
The KdV approximation for a system with unstable resonances
The KdV equation can be derived via multiple scaling analysis for the approximate description of long waves in dispersive systems with a conservation law. In this paper we justify this approximation for a system with unstable resonances by proving estimates between the KdV approximation and true solutions of the original system. We expect that the approach will allow to handle more complicated systems without a detailed discussion of the resonances and without finding a suitable energy
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