DEGENERACY OF CNOIDAL WAVES IN UNBOUNDED SOLUTIONS OF THE KORTEWEG - DE VRIES EQUATION

 

A. P. Chernyaev ¹, S. A. Chernyaeva ²

¹ Moscow Institute of Physics and Technology (State University), 9 Institutsky Per., Dolgoprudny, Moscow region 141700, .Russia

² Bauman Moscow State Technical University, 2/18, Rubtsovskaya Naberezhnaya Str., Moscow, 105082, Russia

 

 The paper is received on May 26, 2018

 

Abstract. The Korteweg – de Vries equation (KdV equation) is a third-order nonlinear partial differential equation that plays a very important role in the theory of nonlinear waves. It was obtained by Boussinesq in 1877, but a detailed analysis was done by Korteweg and de Vries in 1895. Firstly, the KdV equation was initiated by the needs of hydrodynamics, however, over time, it has penetrated not only into various branches of mathematical physics, but also in numerous fields of scientific knowledge. For the KdV equation, a large number of exact solutions, representing stationary nonlinear waves, are found. However, there are unbounded solutions of this equation, the physical content of that is not yet clear. In the present paper, we consider the process of transformation of cnoidal waves in unbounded periodic solution of the KdV equation. Cnoidal waves are mathematically described by elliptic integrals with parameters defining the amplitudes and periods. They are obtained by the procedure for search KdV equation solutions of traveling wave type. Through this, the KdV equation is reduced firstly to the third order ordinary equation, and then the order of the ordinary differential equation is reduced up to the first order. This ordinary equation is integrated in elliptic functions and, therefore, the cnoidal waves through these elliptic functions are expressed. It is worth to note that in the case of transformation of cnoidal waves into unbounded periodic solutions, the last are expressed through elementary functions.

Key words: Korteweg – de Vries, cnoidal wave, unbounded periodic solutions, elliptic functions.

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For citation:

A. P. Chernyaev, S. A. Chernyaeva. Degeneracy of cnoidal waves in unbounded solutions of the Korteweg - de Vries equation. Zhurnal Radioelektroniki - Journal of Radio Electronics. 2018. No. 6. Available at http://jre.cplire.ru/jre/jun18/5/text.pdf

DOI  10.30898/1684-1719.2018.6.5