scholarly journals Local Well-Posedness of the KdV Equation with Quasi-Periodic Initial Data

2012 ◽  
Vol 44 (5) ◽  
pp. 3412-3428 ◽  
Author(s):  
Kotaro Tsugawa
Keyword(s):  
Initial Data ◽  
Kdv Equation ◽  
Well Posedness ◽  
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 WELL-POSEDNESS FOR A FAMILY OF PERTURBATIONS OF THE KdV EQUATION IN PERIODIC SOBOLEV SPACES OF NEGATIVE ORDER

2013 ◽  
Vol 15 (06) ◽  
pp. 1350005
Author(s):  
XAVIER CARVAJAL PAREDES ◽  
RICARDO A. PASTRAN
Keyword(s):  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Well Posedness ◽  
Initial Value ◽  
Negative Order ◽  
The Kdv Equation ◽  
De Vries ◽  
Sivashinsky Equation

We establish local well-posedness in Sobolev spaces Hs(𝕋), with s ≥ -1/2, for the initial value problem issues of the equation [Formula: see text] where η > 0, (Lu)∧(k) = -Φ(k)û(k), k ∈ ℤ and Φ ∈ ℝ is bounded above. Particular cases of this problem are the Korteweg–de Vries–Burgers equation for Φ(k) = -k2, the derivative Korteweg–de Vries–Kuramoto–Sivashinsky equation for Φ(k) = k2 - k4, and the Ostrovsky–Stepanyams–Tsimring equation for Φ(k) = |k| - |k|3.

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