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

Journal Article · Mon Nov 01 04:00:00 EST 1993 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States)

DOI:https://doi.org/10.1103/PhysRevE.48.4027· OSTI ID:5836016

Cooper, F ^[1]; Shepard, H ^[2]; Sodano, P ^[3]

---

1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States) Physics Department, University of New Hampshire, Durham, New Hampshire, 03824 (United States)
2. Physics Department, University of New Hampshire, Durham, New Hampshire 03824 (United States)
3. Dipartimento di Fisica and Sezione INFN Universita di Perugia, 06100 Perugia (Italy)

We study the class of generalized Korteweg--de Vries (KdV) equations derivable from the Lagrangian: [ital L]([ital l],[ital p]) =[integral][1/2[ital cphi][sub [ital x]cphi[ital t]] [minus]([ital cphi][sub [ital x]])[sup [ital l]]/[ital l]([ital l][minus]1) +[alpha]([ital cphi][sub [ital x]])[sup [ital p]]([ital cphi][sub [ital x][ital x]])[sup 2]][ital dx], where the usual fields [ital u]([ital x],[ital t]) of the generalized KIdV equation are defined by [ital u]([ital x],[ital t])=[ital cphi][sub [ital x]]([ital x],[ital t]). This class contains compactons, which are solitary waves with compact support, and when [ital l]=[ital 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 [ital u]([ital x],[ital t])=[ital A]([ital t])exp[[minus][beta]([ital t])[vert bar][ital x][minus][ital q]([ital t])[vert bar][sup 2[ital n]]] we find solitonlike solutions for all [ital n], moving with fixed shape and constant velocity, [ital c]. We show that the velocity, mass, and energy of the variational traveling-wave solutions are related by [ital c]=2[ital rEM][sup [minus]1], where [ital r]=([ital p]+[ital l]+2)/([ital p]+6[minus][ital l]), independent of [ital n].

• OSTI ID: 5836016
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States), Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States) Vol. 48:5; ISSN 1063-651X; ISSN PLEEE8
• Country of Publication: United States
• Language: English

Related Subjects

661100* -- Classical & Quantum Mechanics-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
AMPLITUDES
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
HAMILTONIANS
KORTEWEG-DE VRIES EQUATION
LAGRANGIAN FUNCTION
MATHEMATICAL OPERATORS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS
QUASI PARTICLES
SOLITONS
VARIATIONAL METHODS