Solitary Waves and Compactons in a class of Generalized Korteweg-DeVries Equations

Details Solitary Waves and Compactons in a class of Generalized Korteweg-DeVries Equations Solitary Waves and Compactons in a class of Generalized Korteweg-DeVries Equations

1993-05-18

arxiv.org/abs/patt-sol/9305006v1

Nonlinear Sciences

Pattern Formation and Solitons

16 pages. LaTeX. Figures available upon request (Postscript or hard copy)

Scientific paper

10.1103/PhysRevE.48.4027

We study the class of generalized Korteweg-DeVries equations derivable from the Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - { {(\vp_{x})^{l}} \over {l(l-1)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right) dx, $ where the usual fields $u(x,t)$ of the generalized KdV equation are defined by $u(x,t) = \vp_{x}(x,t)$. This class contains compactons, which are solitary waves with compact support, and when $l=p+2$, these solutions have the feature that their width is independent of the amplitude. We consider the Hamiltonian structure and integrability properties of this class of KdV equations. We show that many of the properties of the solitary waves and compactons are easily obtained using a variational method based on the principle of least action. Using a class of trial variational functions of the form $u(x,t) = A(t) \exp \left[-\beta (t) \left|x-q(t) \right|^{2n} \right]$ we find soliton-like solutions for all $n$, moving with fixed shape and constant velocity, $c$. We show that the velocity, mass, and energy of the variational travelling wave solutions are related by $ c = 2 r E M^{-1}$, where $ r = (p+l+2)/(p+6-l)$, independent of $n$.\newline \newline PACS numbers: 03.40.Kf, 47.20.Ky, Nb, 52.35.Sb

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