Evolution of geometric properties on
solution curve of KdV equation

Ling XU

doi:https://doi.org/10.14445/22315373/IJMTT-V43P505

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 43 | Number 1 | Year 2017 | Article Id. IJMTT-V43P505 | DOI : https://doi.org/10.14445/22315373/IJMTT-V43P505

Evolution of geometric properties on solution curve of KdV equation

Ling XU

Abstract

This paper discusses some properties of solution curve of the Cauchy problem on the KdV equation by Lagrange coordinate, obtained the evolution consequence of monotonicity, concavity and the isolated extreme points. Namely under the enough smooth condition, the isolated extreme points , monotonicity and convexity of initial solution can be inherited to the solution curve at any t  0. .

Keywords

KdV equation ; solution curve ; monotonicity ; convexity ; isolated extreme point.

References

[1] T Kato,On the Cauchy problem for the (generalized) Korteweg-de Vries equation,Advances in Mathematics supplementary studies,studies in Applied Math.8, pp.93-128,1989.
[2] T Kato,K.Masuda,Nonlinear Evolution Equations and Analyticity.I,Analyse nonlinéaire,pp.455-467,1986.
[3] ANDERS SJÖBERG，On the Korteweg-de Vries Equation:Existence and Uniqueness,J.Math.Anal.Appl.29, pp.569-579 ,1970.
[4] KENIG C E, PONCE G , VEGA L,Oscillatory integrals and regularity of dispersive ,equations[J].Indiana Univ Math,40(1),pp.33-69,1991.
[5] Jennifer Gorsky，A.Alexandrou Himonas，Well-posedness of KdV with higher ,dispersion.Mathematics and Computers in Simulation 80,pp.173–183,2009.
[6] Junfeng Li,Shaoguang Shi,Local well-posedness for the dispersion generalized periodic KdV equation,J.Math.Anal.Appl.379 ,pp.706-718,2011.
[7] Chulkwang Kwak，Local well-posedness for the fifth-order KdV equations on T,Journal of Differential Equations 260,pp.7683-7737,2016.
[8] Sunghyun Hong, Chulkwang Kwak，Global well-posedness and nonsqueezing propertyfor the higher-order KdV-type flow，J.Math.Anal.Appl.441 pp.140-166,2016.
[9] Heather Hannah, A Alexandrou Himonas , Gerson Petronilho，Gevrey regularity of the periodic gKdV equation，Journal of Differential Equations 250,pp. 2581-2600,2011.
[10] M.E.Gage,Curve shortening makes convex curves circular,Invent. Math. 76, pp.357-364,1984.
[11] Marcos Craizer , Ralph Teixeira,Evolution of an extremum by curvature motion,J.Math.Anal.Appl.293,pp.721-737,2004.
[12] Luc Molinet,F rancis Ribaud,On the Cauchy problem for the generalized Korteweg-de Vries equation,Communications in Partial Differential Equations.28(11-12),pp.2065-2091,2003.

Citation :

Ling XU, "Evolution of geometric properties on solution curve of KdV equation," International Journal of Mathematics Trends and Technology (IJMTT), vol. 43, no. 1, pp. 20-27, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V43P505