A nonlinear system of Euler–Lagrange equations. Reduction to the Korteweg–de Vries equation and periodic solutions

1975 ◽  
Vol 16 (8) ◽  
pp. 1573-1579 ◽  
Author(s):  
Walter Zielke
Keyword(s):  
Nonlinear System ◽  
Periodic Solutions ◽  
Vries Equation ◽  
Lagrange Equations ◽  
De Vries

scholarly journals Control and stabilization of the periodic fifth order Korteweg-de Vries equation

2019 ◽  
Vol 25 ◽  
pp. 38
Author(s):  
Cynthia Flores ◽  
Derek L. Smith
Keyword(s):  
Exponential Stability ◽  
Periodic Solutions ◽  
Vries Equation ◽  
Regularity Property ◽  
Contraction Principle ◽  
Smoothing Effect ◽  
Dissipative Term ◽  
De Vries ◽  
Fifth Order

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in Hs(𝕋), s > 2. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.

Periodic solutions of the (2 + 1)-dimensional complex modified Korteweg-de Vries equation

2020 ◽  
Vol 34 (18) ◽  
pp. 2050202 ◽  
Author(s):  
Feng Yuan ◽  
Ying Jiang
Keyword(s):  
Periodic Solutions ◽  
Darboux Transformation ◽  
Vries Equation ◽  
Extreme Values ◽  
Dynamical Characteristics ◽  
Wave Solutions ◽  
De Vries

The periodic solutions for the [Formula: see text]-dimensional complex modified Korteweg–de Vries (cmKdV) equation are obtained by using the Darboux transformation (DT). Starting with a periodic seed, we get two types of periodic solutions that are respectively, induced by different properties of the parameter [Formula: see text], namely, the periodic line wave solutions for [Formula: see text] and the breather solutions for [Formula: see text]. Especially, for each type, two different kinds of solutions can be obtained via choosing different constrains, namely, [Formula: see text] or [Formula: see text]. The detailed dynamical characteristics of these solutions are also analyzed, including period, velocity, extreme values, amputations and the rules of temporal evolutions.

scholarly journals Periodic Solutions of the Korteweg--de Vries Equation Driven by White Noise

2005 ◽  
Vol 36 (3) ◽  
pp. 815-855 ◽  
Author(s):  
A. De Bouard ◽  
A. Debussche ◽  
Y. Tsutsumi
Keyword(s):  
White Noise ◽  
Periodic Solutions ◽  
Vries Equation ◽  
De Vries

Periodic and decay mode solutions of the generalized variable-coefficient Korteweg–de Vries equation

2019 ◽  
Vol 33 (20) ◽  
pp. 1950234 ◽  
Author(s):  
Yufeng Zhang ◽  
Jiangen Liu
Keyword(s):  
Periodic Solutions ◽  
Decay Mode ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Kdv Equation ◽  
Variation Of Parameters ◽  
Triply Periodic ◽  
De Vries ◽  
Mode Solution ◽  
First Time

In this paper, we can obtain periodic and decay mode solutions of the generalized variable-coefficient Korteweg–de Vries (gvc-KdV) equation for the first time by using the multivariate transformation technique. Periodic solutions are doubly and triply periodic solutions, and the decay mode solutions include the 1-decay solution and the 2-decay mode solution. Furthermore, we will also study the effect of variation of parameters for solutions systematically, relating to different forms of expression, through graphic display.

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