scholarly journals On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data

2015 ◽  
Vol 29 (3) ◽  
pp. 825-856 ◽  
Author(s):  
David Damanik ◽  
Michael Goldstein
Keyword(s):  
Initial Data ◽  
Existence And Uniqueness ◽  
Kdv Equation ◽  
Global Solutions ◽  
The Kdv Equation ◽  
Periodic Initial Data

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.

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 An extension of Bakhvalov’s theorem for systems of conservation laws with damping

2017 ◽  
Vol 14 (04) ◽  
pp. 703-719
Author(s):  
Hermano Frid
Keyword(s):  
Periodic Solution ◽  
Initial Data ◽  
Conservation Laws ◽  
Gas Dynamics ◽  
Global Solutions ◽  
Eulerian Formulation ◽  
Systems Of Conservation Laws ◽  
Periodic Initial Data ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas

For [Formula: see text] systems of conservation laws satisfying Bakhvalov conditions, we present a class of damping terms that still yield the existence of global solutions with periodic initial data of possibly large bounded total variation per period. We also address the question of the decay of the periodic solution. As applications, we consider the systems of isentropic gas dynamics, with pressure obeying a [Formula: see text]-law, for the physical range [Formula: see text], and also for the “non-physical” range [Formula: see text], both in the classical Lagrangian and Eulerian formulation, and in the relativistic setting. We give complete details for the case [Formula: see text], and also analyze the general case when [Formula: see text] is small. Further, our main result also establishes the decay of the periodic solution.

scholarly journals Global Solutions for the KdV Equation with Unbounded Data

1997 ◽  
Vol 139 (2) ◽  
pp. 339-364 ◽  
Author(s):  
Carlos E. Kenig ◽  
Gustavo Ponce ◽  
Luis Vega
Keyword(s):  
Kdv Equation ◽  
Global Solutions ◽  
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.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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