Perturbative expansion and the initial-value problem of the kdv equation

1980 ◽  
Vol 27 (17) ◽  
pp. 571-574
Author(s):  
G. Tuechetti
Keyword(s):  
Initial Value Problem ◽  
Kdv Equation ◽  
Perturbative Expansion ◽  
Initial Value ◽  
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 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.

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.

scholarly journals Complexiton solutions for complex KdV equation by optimal Homotopy Asymptotic Method

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6195-6211 ◽  
Author(s):  
Samina Zuhra ◽  
Noor Khan ◽  
Saeed Islam ◽  
Rashid Nawaz
Keyword(s):  
Asymptotic Method ◽  
Initial Value Problem ◽  
Kdv Equation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Initial Value ◽  
Kdv Equations ◽  
Hyperbolic Form ◽  
Optimal Homotopy Asymptotic Method ◽  
Position Solution

In this article an innovative technique named as Optimal Homotopy Asymptotic Method has been explored to treat system of KdV equations computed from complex KdV equation. By developing special form of initial value problems to complex KdV equation, three different types of semi analytic complextion solutions fromcomplexKdVequation have been achieved. First semi analytic position solution received fromtrigonometric form of initial value problem, second is semi analytic negation solution received by hyperbolic form of initial value problem and third one is special type of semi analytic solution expressed by the combination of trigonometric and hyperbolic functions. It was proved that only first order OHAM solution is accurate to the closed-form solution.

Averaging principle for the KdV equation with a small initial value

Nonlinearity ◽  
2016 ◽  
Vol 29 (2) ◽  
pp. 603-656 ◽  
Author(s):  
Xiaoping Yuan ◽  
Jing Zhang
Keyword(s):  
Kdv Equation ◽  
Averaging Principle ◽  
Initial Value ◽  
The Kdv Equation

The equation A(t, u(t))′ + B(t, u(t)) = 0

1976 ◽  
Vol 79 (3) ◽  
pp. 545-561 ◽  
Author(s):  
Bruce Calvert
Keyword(s):  
Differential Operator ◽  
Banach Spaces ◽  
Lower Order ◽  
Kdv Equation ◽  
Initial Value ◽  
Order Elliptic Operator ◽  
The Kdv Equation ◽  
Unbounded Subset ◽  
Image Position ◽  
Bbm Equation

The initial value problem for the equationhas been studied recently as a model for long waves in nonlinear dispersive systems. Benjamin, Bona and Mahony (2) introduced this equation as an alternative to the KdV equation of Korteweg-de Vries. Hence, it is referred to as the BBM equation. They studied solutions u(x, t) of the BBM equation for t ≥ 0 and x∈(− ∞, ∞), satisfying u(x, 0) = g(x). Bona and Bryant(1) carried through the study of the BBM equation for t ≥ 0 and x ∈ [0, ∞), satisfying u(x, 0) = g(x) and u(0, t) = h(t). The aim of this paper is to study the equationwhere At and Bt are mappings defined on subsets of Banach spaces, especially when At is a second order elliptic operator and B is a differential operator of lower order, defined on an unbounded subset Ω of .

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