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Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kutznetsov lateral dispersion

Juan Cristobal Latorre, A. A. Minzoni, C. A. Vargas, Noel F. Smyth

2006
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Chaos
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The effect of weak lateral dispersion of Zakharov-Kutznetsov-type on a Benjamin-Ono solitary wave is studied both asymptotically and numerically. The asymptotic solution is based on an approximate variational solution for the solitary wave, which is then modulated in time through the use of conservation equations. The effect of the dispersive radiation shed as the solitary wave evolves is also included in the modulation equations. It is found that the weak lateral dispersion produces a strongly
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... anisotropic, stable solitary wave which decays algebraically in the direction of propagation, as for the Benjamin-Ono solitary wave, and exponentially in the transverse direction. Moreover, it is found that initial conditions with amplitude above a threshold evolve into solitary waves, while those with amplitude below the threshold evolve as lumps for a short time, then merge into radiation. The modulation equations are found to give a quantitatively accurate description of the evolution of an initial condition into an anisotropic solitary wave. The existence of stable solitary waves is in contrast to previous studies of Benjamin-Ono-type equations subject to the stronger Kadomstev-Petviashvili or Benjamin-Ono-type lateral dispersion, for which the solitary waves either decay or collapse. The present study then completes the catalog of possible behaviors under lateral dispersion. There are a number of nonlinear wave equations which form natural extensions of the well-known Korteweg-de Vries equation to two space dimensions. Among these are the Kadomstev-Petviashvili (KP) and Zakharov-Kutznetsov equations. The Zakharov-Kutznetsov equation arises in, among other areas, the context of electromigration in nanoconductors. 1-3 The soliton solution of the Kadomstev-Petviashvili equation has algebraic decay at infinity, while the soliton solution of the Zakharov-Kutznetsov equation has exponential decay at infinity. In the present work a Zakharov-Kutznetsov Benjamin-Ono equation is considered as the soliton solution of this equation that has algebraic decay in the direction of propagation and exponential decay in the transverse direction. It is found that this anisotropic behavior has a profound effect on the stability of the soliton solution. Via a variational method it is shown that the soliton has a limited stability range and decays into radiation outside of this range. Approximate modulation equations governing the evolution of a soliton-like initial condition are derived, these equations incorporating the radiation shed as the soliton evolves, whose solutions are found to be in good agreement with numerical solutions. These modulation equations further show that the initial stages of the evo-lution are dominated by the shed radiation, in accordance with numerical solutions. As the Zakharov-Kutznetsov Benjamin-Ono equation has applications to thin nanoconductors on a dielectric substrate, this behavior has applications to electromigration. a͒

doi:10.1063/1.2355555
pmid:17199381
fatcat:bhnm3pydxrarfaho5mysqxzxem