Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation

2019 ◽  
Vol 21 (08) ◽  
pp. 1850061 ◽  
Author(s):  
Achenef Tesfahun
Keyword(s):  
Lower Bound ◽  
Initial Data ◽  
Recent Result ◽  
Analytic Functions ◽  
Conservation Law ◽  
Kdv Equation ◽  
Space Of Analytic Functions ◽  
Positive Formula ◽  
The Kdv Equation ◽  
Fixed Radius

It is shown that the uniform radius of spatial analyticity [Formula: see text] of solutions at time [Formula: see text] to the KdV equation cannot decay faster than [Formula: see text] as [Formula: see text] given initial data that is analytic with fixed radius [Formula: see text]. This improves a recent result of Selberg and da Silva, where they proved a decay rate of [Formula: see text] for arbitrarily small positive [Formula: see text]. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear [Formula: see text] estimates associated with the KdV equation.

scholarly journals Nonanalytic solutions of the KdV equation

2004 ◽  
Vol 2004 (6) ◽  
pp. 453-460 ◽  
Author(s):  
Peter Byers ◽  
A. Alexandrou Himonas
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Kdv Equation ◽  
Initial Value ◽  
The Kdv Equation

We construct nonanalytic solutions to the initial value problem for the KdV equation with analytic initial data in both the periodic and the nonperiodic cases.

scholarly journals Evolution equations having conservation laws with flux characteristics

2007 ◽  
Vol 49 (1) ◽  
pp. 39-52
Author(s):  
B. Van Brunt ◽  
M. Vlieg Hulstman
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Divergence Form ◽  
Subject Classification ◽  
The Kdv Equation

A class of evolution equations in divergence form is studied in this paper. Specifically, we develop conditions under which the spatial divergence term, the flux, corresponds to the characteristic of a conservation law. The KdV equation is a prominent example of an equation having a flux term that is also a characteristic for a conservation law. We show that the flux term must be self-adjoint. General equations for the corresponding conservation laws and Hamiltonian densities are derived and supplemented with examples. 2000 Mathematics subject classification: primary 35K.

THE N = 3 SUPERSYMMETRIC KdV HIERARCHY

1993 ◽  
Vol 08 (12) ◽  
pp. 1161-1169 ◽  
Author(s):  
C. M. YUNG
Keyword(s):  
Conservation Law ◽  
Hamiltonian Structure ◽  
Kdv Equation ◽  
Superconformal Algebra ◽  
Supersymmetric Extension ◽  
Kdv Hierarchy ◽  
The Kdv Equation

An N = 3 supersymmetric extension of the KdV equation, whose Hamiltonian structure is the classical O(3)-extended superconformal algebra is presented. Integrability of the equation is argued based on the existence of a non-trivial conservation law which reduces to the first non-trivial conservation law of the KdV equation.

scholarly journals The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments

1974 ◽  
Vol 65 (2) ◽  
pp. 289-314 ◽  
Author(s):  
Joseph L. Hammack ◽  
Harvey Segur
Keyword(s):  
Initial Data ◽  
Water Waves ◽  
Entire Range ◽  
Vries Equation ◽  
Kdv Equation ◽  
Asymptotic Shape ◽  
Wide Range ◽  
The Kdv Equation ◽  
De Vries ◽  
Wave Forms

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.

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