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The Solution Operator of the Korteweg-de Vries Equation is Computable

2002
*
Electronical Notes in Theoretical Computer Science
*

*The*initial value problem

*of*

*the*

*Korteweg*-

*de*

*Vries*(KdV)

*equation*posted on

*the*real line R: defines a nonlinear map K R from

*the*space H s (R) to

*the*space C(R, H s (R)) for any given real numbers s ≥ ... In this paper we prove that

*the*map K R

*is*

*computable*for any integer s ≥ 3. 1 ...

*equation*was derived by

*Korteweg*-

*de*

*Vries*[9] as a model for long waves propagating in a channel. ...

##
###
The Variational Iteration Method to Solve the Korteweg-de Vries-Burgers Equation

2017
*
Figshare
*

Two case study problems

doi:10.6084/m9.figshare.5099773.v1
fatcat:yt5squhoz5c5dmt2gc4ctqohva
*of**Korteweg*-*de**Vries*-Burgers*equation*are solved by using*the*VIM and*the*exact*solutions*are obtained. ... To solve*the**Korteweg*-*de**Vries*-Burgers*Equation**equations**the*Variational Iteration Method (VIM)*is*used. ...*The**Korteweg*-*de**Vries*-Burgers*equation**is*written as: Where are real constants and*is*a positive integer. When β ,*the*Eq. (1.1)*is*turned into*the**Korteweg*-*de**Vries**equation*. ...##
###
Soliton, rational and special solutions of the Korteweg–de Vries hierarchy

2010
*
Applied Mathematics and Computation
*

*Solutions*

*of*

*the*

*Korteweg*-

*de*

*Vries*hierarchy are discussed. It

*is*shown that results by Wazwaz [Wazwaz A.M. ... Multiple -soliton

*solutions*

*of*

*the*perturbed KdV

*equation*, Commun Nonlinear Sci Numer Simulat, 2010; 15911: 3270 -3273] are

*the*well -known consequences

*of*

*the*full integrability for

*the*

*Korteweg*-

*de*

*Vries*... (9)

*is*

*the*partial case

*of*

*the*

*Korteweg*-

*de*

*Vries*hierarchy (1) . ...

##
###
Page 6001 of Mathematical Reviews Vol. , Issue 2003h
[page]

2003
*
Mathematical Reviews
*

, FL)
Complexiton

*solutions*to*the**Korteweg*-*de**Vries**equation*. ... Summary: “A novel class*of*explicit exact*solutions*to*the**Korteweg*-*de**Vries**equation**is*presented through its bilinear form. ...##
###
Page 2930 of Mathematical Reviews Vol. , Issue 85g
[page]

1985
*
Mathematical Reviews
*

(PL-PAN) 85g:35104 A note on singular

*solutions**of**the**Korteweg*-*de**Vries**equation*. Phys. Lett. A 100 (1984), no. 7, 321-324. ... Another so-called auto- Backlund transformation for*the*mapping*of*one*solution*to an- other*of**the*modified*Korteweg*-*de**Vries**equation*can be included. ...##
###
Matrix solitons solutions of the modified Korteweg-de Vries equation
[article]

2019
*
arXiv
*
pre-print

*The*focus

*is*on

*the*construction

*of*soliton

*solutions*admitted by matrix modified

*Korteweg*-

*de*

*Vries*

*equations*. ... Nonlinear non-Abelian

*Korteweg*-

*de*

*Vries*(KdV) and modified

*Korteweg*-

*de*

*Vries*(mKdV)

*equations*and their links via Baecklund transformations are considered. ... Introduction So called soliton

*equations*are widely investigated since

*the*late 1960's when

*the*first exact

*solution*

*of*

*the*

*Korteweg*-

*de*

*Vries*

*equation*was obtained, as reported, for instance, in

*the*well ...

##
###
Local existence of solutions for the modified Korteweg-de Vries (mKdV) equation in weighted Sobolev spaces

2021
*
South Florida Journal of Development
*

Our goal in this paper

doi:10.46932/sfjdv2n1-034
fatcat:gb6koh434ng5nmzqcnkwq46poi
*is*to give a proof*of*local existence*of**solutions*for*the*modified*Korteweg*-*de*-*Vries*(mKdV) in a weighted Sobolev space with initial data. ... A parabolic regularization*is*suggested, discussed and implemented for*the*approximation*of**solutions*for*the*mKdV*equation*with an appropriate class*of*nonlineariy. ... INTRODUCTION*The*purpose*of*this paper*is*to give a proof to*the*local existence*of**solutions*for*the*modified*Korteweg*-*de**Vries*(mKdV)*equation*in a weighted Sobolev space. ...##
###
Reciprocal link among the Qiao-Liu peakon equation and the modified Korteweg-de Vries equation
[article]

2010
*
arXiv
*
pre-print

This paper has been withdrawn by

arXiv:1010.1915v2
fatcat:awjmzq6ztjcwzm7bmvqaq6cqt4
*the*author.*The*result*is*already known. ... Acknowledgments*The*author*is*very grateful to Dr. A. Sergyeyev for useful discussions. ... This research was supported by Silesian University in Opava under*the*student grant project SGS/18/2010 and by*the*Ministry*of*Education, Youth and Sports*of*Czech Republic under*the*grant MSM4781305904 ...##
###
A new integrable generalization of the Korteweg–de Vries equation

2008
*
Journal of Mathematical Physics
*

A new integrable sixth-order nonlinear wave

doi:10.1063/1.2953474
fatcat:vfugemqyanhhboshdcijhh7jqe
*equation**is*discovered by means*of**the*Painleve analysis, which*is*equivalent to*the**Korteweg*-*de**Vries**equation*with a source. ... A Lax representation and a Backlund self-transformation are found*of**the*new*equation*, and its travelling wave*solutions*and generalized symmetries are studied. ... •*Is*there any other recursion*operator**of**the*KdV6, different from*the*recursion*operator*(52)*of**the*potential*Korteweg*-*de**Vries**equation*? ...##
###
Page 4413 of Mathematical Reviews Vol. , Issue 89H
[page]

1989
*
Mathematical Reviews
*

*The*corresponding optimisation problem

*is*solved.

*The*

*equations*

*of*

*the*problem are reduced to

*the*perturbed

*Korteweg*-

*de*

*Vries*

*equation*. B. G. ... Summary: “We investigate

*the*stability

*of*a solitary wave

*solution*

*of*

*the*

*Korteweg*-

*de*

*Vries*

*equation*when a fifth-order spatial deriv- ative term

*is*added. ...

##
###
Page 3799 of Mathematical Reviews Vol. , Issue 2004e
[page]

2004
*
Mathematical Reviews
*

Summary: “In

*the*long-wave, weakly nonlinear limit a generic model for*the*interaction*of*two waves with nearly coincident lin- ear phase speeds*is*a pair*of*coupled*Korteweg*-*de**Vries**equations*. ... Summary: “*The*paper focuses on*the*details*of**the*emergence*of**Korteweg*-*de**Vries*solitons from an initial harmonic excitation. ...##
###
Applications of Grassmannian flows to integrable systems
[article]

2022
*
arXiv
*
pre-print

Herein we extend

arXiv:1905.05035v2
fatcat:tcawzbm46jarvkxts2e27g7bpm
*the*work*of*Poppe and demonstrate how*solution*flows*of**the*non-commutative potential*Korteweg**de**Vries*and nonlinear Schrodinger systems are examples*of*such Grassmannian flows. ... For example, it*is*well-known that*solutions*to classical integrable partial differential systems can be generated by solving a corresponding linear partial differential system for*the*scattering data ... We would also like to thank*the*anonymous referees for their insightful comments and suggestions that helped to significantly improve*the*original manuscript. ...##
###
Computing the Solutions of the Combined Korteweg-de Vries Equation by Turing Machines

2010
*
Electronic Proceedings in Theoretical Computer Science
*

It

doi:10.4204/eptcs.24.14
fatcat:ao36cdd5ovfizfaluiu3kuqdqe
*is*shown that, for any integer s>2,*the*nonlinear*solution**operator*which maps an initial condition data to*the**solution**of**the*Combined KdV*equation*can be*computed*by a Turing machine. ... In this paper, we study*the**computability**of**the*initial value problem*of**the*Combined KdV*equation*. ... In this paper, we investigate a variation*of**Korteweg*-*de**Vries**equation*: u t + uu x + u 2 u x + u xxx = 0. This*is*often called Combined*Korteweg*-*de**Vries*(CKdV)*equation*. ...##
###
Page 7059 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

Summary: “For

*the*fifth-order*Korteweg*-*de**Vries**equation*it*is*demonstrated that*the*Hamiltonian*is*bounded from below for fixed momentum. ... [Kuznetsov, Evgenii Aleksandrovich] (RS-AOS-L; Moscow) On*the*nonlinear stability*of*solitary wave*solutions**of**the*fifth-order*Korteweg*-*de**Vries**equation*. (English summary) Phys. Lett. ...##
###
Efficient Iterative Method for Solving Korteweg-de Vries Equations

2019
*
Iraqi Journal of Science
*

Since these

doi:10.24996/ijs.2019.60.7.17
fatcat:h75vibceq5fivom52vywli2ufq
*equations*arise in many applications and physical phenomena, it*is*officially showed that this*equation*has solitary waves as*solutions*,*The**Korteweg*-*de**Vries**equation**is*utilized to characterize ...*The**Korteweg*-*de**Vries**equation*plays an important role in fluid physics and applied mathematics. This*equation**is*a fundamental within study*of*shallow water waves. ...*The**Korteweg*-*de**Vries**equation*(KdV)*equation*has been arisen in*the*study*of*shallow water waves, [1] . ...
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