American Institute of Mathematical Sciences

Advanced
September  2004, 3(3): 417-431. doi: 10.3934/cpaa.2004.3.417

On the Cauchy problem for a coupled system of KdV equations

Felipe Linares 1, and  M. Panthee 2,

1. 

IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil
2. 

Instituto de Matemática Pura e Aplicada, 22460-320, Rio de Janeiro, Brazil

Received  August 2003 Revised  October 2003 Published  June 2004

We study some questions related to the well-posedness for the initial value problem associated to the system

$u_{t}+u_{x x x}+a_3 v_{x x x}+u u_{x}+a_1 v v_{x}+a_2(uv)_x =0,$

$b_1 v_{t}+v_{x x x}+b_2 a_3 u_{x x x}+v v_{x}+b_2 a_2 u u_{x}+b_2 a_1(uv)_x=0.$

Using recent methods, we prove a sharp local result in Sobolev spaces. We also prove global result under some conditions on the coefficients.

Keywords: Well-posedness, dispersive equations, Korteweg-de Vries equation.
Mathematics Subject Classification: 35Q35, 35Q53.
Citation: Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417
[1]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[2]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[3]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
[4]

M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22
[5]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[6]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[7]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[8]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[9]

Ahmat Mahamat Taboye, Mohamed Laabissi. Exponential stabilization of a linear Korteweg-de Vries equation with input saturation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021052
[10]

Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations & Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028
[11]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[12]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[13]

Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[14]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[15]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[16]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[17]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[18]

Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri. Non-standard boundary conditions for the linearized Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2625-2654. doi: 10.3934/dcdss.2021066
[19]

Julie Valein. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021039
[20]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (106)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy