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

Klaus Weihrauch, Ning Zhong
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.  ... 
doi:10.1016/s1571-0661(04)80389-x fatcat:a7yoo6ldf5dqpfoirpqecods7m

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

Mohsen Soori
2017 Figshare  
Two case study problems 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.  ... 
doi:10.6084/m9.figshare.5099773.v1 fatcat:yt5squhoz5c5dmt2gc4ctqohva

Soliton, rational and special solutions of the Korteweg–de Vries hierarchy

Nikolai A. Kudryashov
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) .  ... 
doi:10.1016/j.amc.2009.11.024 fatcat:y6codvzvwbamlal5jhkpccyhqa

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]

Sandra Carillo, Mauro Lo Schiavo, Cornelia Schiebold
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  ... 
arXiv:1902.05823v3 fatcat:cdb3aqyjcfhdznpmz44cualx6i

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

Milton M. Cortez Gutiérrez, Hernan O. Cortez Gutiérrez, Girady I. Cortez Fuentes Rivera, Liv J. Cortez Fuentes Rivera, Deolinda E. Fuentes Rivera Vallejo
2021 South Florida Journal of Development  
Our goal in this paper 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.  ... 
doi:10.46932/sfjdv2n1-034 fatcat:gb6koh434ng5nmzqcnkwq46poi

Reciprocal link among the Qiao-Liu peakon equation and the modified Korteweg-de Vries equation [article]

Petr Vojcak
2010 arXiv   pre-print
This paper has been withdrawn by 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  ... 
arXiv:1010.1915v2 fatcat:awjmzq6ztjcwzm7bmvqaq6cqt4

A new integrable generalization of the Korteweg–de Vries equation

Ayşe Karasu-Kalkanlı, Atalay Karasu, Anton Sakovich, Sergei Sakovich, Refik Turhan
2008 Journal of Mathematical Physics  
A new integrable sixth-order nonlinear wave 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?  ... 
doi:10.1063/1.2953474 fatcat:vfugemqyanhhboshdcijhh7jqe

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]

Anastasia Doikou, Simon J.A. Malham, Ioannis Stylianidis, Anke Wiese
2022 arXiv   pre-print
Herein we extend 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.  ... 
arXiv:1905.05035v2 fatcat:tcawzbm46jarvkxts2e27g7bpm

Computing the Solutions of the Combined Korteweg-de Vries Equation by Turing Machines

Dianchen Lu, Qingyan Wang, Rui Zheng
2010 Electronic Proceedings in Theoretical Computer Science  
It 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.  ... 
doi:10.4204/eptcs.24.14 fatcat:ao36cdd5ovfizfaluiu3kuqdqe

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

Samaher marez Yassein, Asmaa Abd Aswhad
2019 Iraqi Journal of Science  
Since these 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] .  ... 
doi:10.24996/ijs.2019.60.7.17 fatcat:h75vibceq5fivom52vywli2ufq
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