Soliton interactions with an external forcing the modiﬁed Korteweg-de Vries framework Marcelo V. Flamarion1and Eﬁm Pelinovsky23

2025-05-03 0 0 1.66MB 10 页 10玖币

侵权投诉

Soliton interactions with an external forcing: the modiﬁed

Korteweg-de Vries framework

Marcelo V. Flamarion1and Eﬁm Pelinovsky2,3

1Unidade Acadêmica do Cabo de Santo Agostinho,

UFRPE/Rural Federal University of Pernambuco, BR 101 Sul, Cabo de Santo Agostinho-PE, Brazil, 54503-900

marcelo.ﬂamarion@ufrpe.br

2Institute of Applied Physics, 46 Uljanov Str., Nizhny Novgorod 603155, Russia.

3National Research University–Higher School of Economics, Moscow, Russia.

Abstract

The aim of this work is to study asymptotically and numerically the interaction of solitons with an

external forcing with variable speed using the forced modiﬁed Korteweg-de Vries equation (mKdV). We

show that the asymptotic predictions agree well with numerical solutions for forcing with constant speed

and linear variable speed. Regarding forcing with linear variable speed, we ﬁnd regimes in which the solitons

are trapped at the external forcing and its amplitude increases or decreases in time depending on whether

the forcing accelerates or decelerates.

1 Introduction

The forced Korteweg-de Vries equation (fKdV) is broadly used to describe the propagation of waves of small

amplitudes produced by moving sources with constant speed and small amplitudes [1, 30, 29, 31, 24, 12, 13,

14, 3, 6]. For instance this equation has been used to investigate ﬂows past obstacles in hydrodynamics, ship

wakes, trapped waves [23, 17, 15, 22, 20, 21, 11], internal waves in stratiﬁed ﬂuid ﬂows and so on [2]. In many

practical applications, for example, ship wakes or atmospheric ﬂows, the speed of the external force may vary

along the time or depend on the medium. Consequently, it can produce surface or internal solitary waves whose

speed depend on the medium in which these waves propagate. For an external force with a time-dependent

speed, a complete asymptotic and numerical study of the fKdV equation was ﬁrst given by Grimshaw et al.

[16]. The authors showed a good agreement of results predicted by both theories and conditions for trapping

were determined within the asymptotic framework. Similar studies in the in the context of gravity-capillary

waves were done later by Flamarion [7] and in the non-integrable Whitham equation in [8].

When waves have larger amplitude or the nonlinear coeﬃcient is small, nonlinearity is dominant in the

dynamic and the fKdV equation fails to predict many phenomena. In that case, nonlinear terms of higher-order

have to be taken into account and the forced modiﬁed KdV equation (mKdV), which incorporates a cubic

nonlinearity, arises. Among the problems that can be investigated in this framework, we mention waves on the

surface of conductive ﬂuid in an electric ﬁeld [27], wave in quantum-dimensional ﬁlms, elastic waves in solids

[25]. Although the mKdV equation is integrable, the nonlinear dynamics is more complicated than the fKdV

equation and the sign of the cubic nonlinearity plays a fundamental role on the qualitative behaviour of the

solutions. For instance, when cubic nonlinearity is positive, the mKdV equation admits breathers (traveling

oscillating moving wave packets) and solitons of both polarities as solutions. This equation has been widely

used to study breathers and solitons with diﬀerent polarities [26, 4]. In particular, an asymptotic study on

trapped waves in this framework was done by Pelinovsky [26] and the results were compared qualitatively with

the fKdV equation. Asymptotic results predict that trapped waves can occur only when the forcing and soliton

have the same polarity, however fully numerical results were not reported.

The goal of this article is to investigate numerically and asymptotically the interaction of solitons with an

external forcing with variable speed using the forced modiﬁed Korteweg-de Vries equation with positive cubic

nonlinearity. Asymptotic and numerical results are compared qualitatively and quantitatively and conditions of

trapping are discussed. In particular, we show that for forcing with linear variable speed, it is possible to trap

a soliton at the external forcing and that its amplitude increases or decreases in time depending on whether the

external force accelerates or decelerates.

This article is organized as follows. In section 2 we present the mathematical formulation of the problem.

Asymptotic results are presented in section 3 and numerical results in section 4. The conclusion is presented in

section 5.

arXiv:2210.02180v1 [physics.flu-dyn] 5 Oct 2022

2 The forced modiﬁed Korteweg-de Vries equation

We consider the modiﬁed Korteweg-de Vries equation with a forcing term of variable speed (v(t)) in canonical

form

ut+cux+ 6u2ux+uxxx =fxx−Zv(t)dt.(1)

to investigate the interaction of solitons with a external forcing ﬁeld. Here, we denote by u(x, t)the wave ﬁeld,

f(x)the external forcing that travels with variable speed v(t),cthe long-wave propagation speed and a small

positive parameter. It is convenient to rewrite equation (1) in the forcing moving frame. To this end, we write

the traveling variables

x0=x−Zv(t)dt, t0=t.

In the new coordinate system, dropping the primes equation (1) reads

ut+ ∆(t)ux+ 6u2ux+uxxx =fx(x),(2)

where

∆(t) = c−v(t)(3)

is the deviation speed. This important parameter measures the diﬀerence between the linear long-wave speed

and the speed of the external forcing [16]. Assuming that the forcing vanishes at inﬁnity, equation (2) conserves

the total mass (M(t)), with

dt = 0,where M(t) = Z∞

−∞

u(x, t)dx. (4)

Additionally, the rate of change of momentum (P(t)) is balanced by the external forcing through the equation

dt =Z∞

−∞

u(x, t)df(x)

dx dx, where P(t) = 1

2Z∞

−∞

u2(x, t)dx. (5)

In the absence of an external forcing, the mKdV admits solitons as solutions [26], which are given by the

expressions

u(x, t) = asech(aΦ),where Φ=x−qt −x0, q =c−v+a2,(6)

where ais amplitude of the soliton, x0is its the initial position of the crest. Here, ais allowed to be negative,

which represents depression solitary waves.

3 Asymptotic results

Asymptotic results on the interaction of a solitons with an external forcing were ﬁrst reported Pelinovsky [26]

for elevation waves. Here, we present his main results and additionally we expand formally his analysis to

depression solitons and solutions of dynamical systems are obtained numerically. With this in mind, we assume

that the wave ﬁeld is close to the soliton, but the parameters vary slowly in time. The soliton is described by

the formulas

u(Φ, T ) = a(T) sech(a(T)Φ),where Φ=x−Ψ(T),and Ψ(T) = x0+1

Zq(T)dT, (7)

where functions aand qare determined from the interaction between the wave ﬁeld and the external ﬁeld. We

formally introduce the “slow time" by considering the new variable T=t. Assuming a weak force (1), we

seek for a solution in the form of the asymptotic expansion

u(Φ, T ) = u0+u1+2u2+··· ,

q(t) = q0+q1+2q2+··· .(8)

At the lowest order of the perturbation theory, the solutions u0and q0are deﬁned as in equation (7) with the

position of the initial crest given by x0=c−v−a2.

The momentum balance equation (5) and kinematic condition yields the dynamical system

dΨ

dt =aZ∞

−∞

sech(aΦ)df

dΦ(Φ+Ψ)dΦ,

dt = ∆(T) + a2,

(9)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Solitoninteractionswithanexternalforcing:themodiedKorteweg-deVriesframeworkMarceloV.Flamarion1andEmPelinovsky2;31UnidadeAcadêmicadoCabodeSantoAgostinho,UFRPE/RuralFederalUniversityofPernambuco,BR101Sul,CabodeSantoAgostinho-PE,Brazil,54503-900marcelo.amarion@ufrpe.br2InstituteofAppliedPhysics,46Ul...

展开>> 收起<<

Soliton interactions with an external forcing the modiﬁed Korteweg-de Vries framework Marcelo V. Flamarion1and Eﬁm Pelinovsky23.pdf

共10页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Soliton interactions with an external forcing the modiﬁed Korteweg-de Vries framework Marcelo V. Flamarion1and Eﬁm Pelinovsky23

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: