Solitary waves in a class of generalized Korteweg–de Vries equations

Abstract

We study the class of generalized Korteweg--de Vries (KdV) equations derivable from the Lagrangian: L(l,p) =F[1/2${\mathit{cphi}}_{\mathit{x}}$${\mathit{cphi}}_{\mathit{t}}$ -(${\mathit{cphi}}_{\mathit{x}}$${)}^{\mathit{l}}$/l(l-1) +\ensuremath{\alpha}(${\mathit{cphi}}_{\mathit{x}}$${)}^{\mathit{p}}$(${\mathit{cphi}}_{\mathit{x}\mathit{x}}$${)}^{2}$]dx, where the usual fields u(x,t) of the generalized KIdV equation are defined by u(x,t)=${\mathit{cphi}}_{\mathit{x}}$(x,t). This class contains compactons, which are solitary waves with compact support, and when l=p+2, these solutions have the feature that their width is independent of the amplitude. We consider the Hamiltonian structure and integrability properties of this class of KdV equations. We show that many of the properties of the solitary waves and compactons are easily obtained using a variational method based on the principle of least action. Using a class of trial variational functions of the form u(x,t)=A(t)exp[-\ensuremath{\beta}(t)\ensuremath{\Vert}x-q(t)${\mathrm{\ensuremath{\Vert}}}^{2\mathit{n}}$] we find solitonlike solutions for all n, moving with fixed shape and constant velocity, c. We show that the velocity, mass, and energy of the variational traveling-wave solutions are related by c=2${\mathit{rEM}}^{\mathrm{\ensuremath{-}}1}$, where r=(p+l+2)/(p+6-l), independent of n.

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