STABILITY OF SMOOTH PERIODIC TRAVELING WAVES IN THE DEGASPERISPROCESI EQUATION ANNA GEYER AND DMITRY E. PELINOVSKY

2025-05-03 0 0 1.19MB 33 页 10玖币

侵权投诉

STABILITY OF SMOOTH PERIODIC TRAVELING WAVES

IN THE DEGASPERIS–PROCESI EQUATION

ANNA GEYER AND DMITRY E. PELINOVSKY

Abstract. We derive a precise energy stability criterion for smooth periodic waves in

the Degasperis–Procesi (DP) equation. Compared to the Camassa-Holm (CH) equation,

the number of negative eigenvalues of an associated Hessian operator changes in the

existence region of smooth perodic waves. We utilize properties of the period function

with respect to two parameters in order to obtain a smooth existence curve for the family

of smooth periodic waves of a ﬁxed period. The energy stability condition is derived on

parts of this existence curve which correspond to either one or two negative eigenvalues of

the Hessian operator. We show numerically that the energy stability condition is satisﬁed

on either part of the curve and prove analytically that it holds in a neighborhood of the

boundary of the existence region of smooth periodic waves.

1. Introduction

The Degasperis-Procesi (DP) equation

ut−utxx + 4uux= 3uxuxx +uuxxx (1.1)

has a special role in the modeling of ﬂuid motion. It was derived in [8] as a transformation

of the integrable hierarchy of KdV equations, with the same asymptotic accuracy as the

Camassa–Holm (CH) equation [1]. Although a more general family of model equations can

also be derived by using this method [6, 9], only the DP and CH equations are integrable

with the use of the inverse scattering transform. It was shown in [3, 21, 23] that the

DP and CH equations describe the horizontal velocity u=u(t, x) for the unidirectional

propagation of waves of a shallow water ﬂowing over a ﬂat bed at a certain depth. A

review of applicability of these model equations as approximations of peaked waves in

ﬂuids was recently given in [33].

In the present paper, we are concerned with smooth traveling wave solutions, for which

the DP and CH equations have been justiﬁed as model equations in hydrodynamics [3].

Existence of smooth periodic traveling waves has been well understood by using ODE

methods [27, 28]. However, stability of smooth periodic traveling waves was considered

to be a diﬃcult problem in the functional-analytic framework, even though integrabil-

ity implies their stability due to the structural stability of the Floquet spectrum of the

Date: October 7, 2022.

arXiv:2210.03063v1 [math.AP] 6 Oct 2022

2 ANNA GEYER AND DMITRY E. PELINOVSKY

associated linear system [29]. Only very recently in [14], we derived an energy stabil-

ity criterion for the smooth periodic traveling waves of the CH equation by using its

Hamiltonian formulation.

For smooth solitary waves, orbital stability was obtained for the CH equation in [4]

and spectral and orbital stability for the DP equation was obtained in [30, 31]. The

energy stability criterion for the smooth solitary waves was derived for the entire family

of the generalized CH equations [26] and was shown to be satisﬁed asymptotically and

numerically. A recent work [32] used the period function to show that the energy stability

criterion is satisﬁed analytically for the entire family of smooth solitary waves.

The purpose of this work is to derive an energy stability criterion for the smooth peri-

odic traveling waves in the DP equation.

Let us brieﬂy comment on the various Hamiltonian formulations which exist both for

the CH and DP equations. These two equations belong to a larger class of generalized CH

equations, the so-called b-family, which reduces to CH for b= 2 and to DP for b= 3. As

far as we know, only one Hamiltonian formulation exists for general b, which was obtained

in [7] and used in the stability analysis of smooth solitary waves in [26], while one more

(alternative) formulation exists for b= 3 and two more alternative formulations exist for

b= 2. In [14], we used the two alternative formulations to study spectral stability of the

smooth periodic waves. Here we will only use the alternative formulation which exists for

b= 3. Whether the Hamiltonian formulation from [7] can also be adopted to the study

of spectral stability of smooth periodic waves for the b-family is left for further studies.

We consider the DP equation (1.1) in the periodic domain TL:= [0, L] of length L > 0.

For notational simplicity, we write Hs

per instead of Hs(TL) for the Sobolev space of L-

periodic functions with index s≥0. The DP equation (1.1) on TLformally conserves the

mass, momentum, and energy given respectively by

M(u) = Iudx, (1.2)

E(u) = 1

2Iu(1 −∂2

x)(4 −∂2

x)−1udx, (1.3)

and

F(u) = 1

6Iu3dx. (1.4)

The standard Hamiltonian structure for the DP equation (1.1) is given by

dt =JδF

δu , J =−(1 −∂2

x)−1(4 −∂2

x)∂x,(1.5)

where Jis a well-deﬁned operator from Hs+1

per to Hs

per for every s≥0 and δF

δu =1

2u2.

The evolution problem (1.5) is well-deﬁned for local solutions u∈C((−t0, t0), Hs

per)∩

C1((−t0, t0), Hs−1

per ) with s > 3

2, see [40], where t0>0 is the local existence time.

SMOOTH PERIODIC WAVES IN THE DEGASPERIS–PROCESI EQUATION 3

Smooth traveling waves of the form u(t, x) = φ(x−ct) with c−φ > 0 are obtained

from the critical points of the augmented energy functional

Λc,b(u) := cE(u)−F(u)−b

4M(u),(1.6)

where bis a parameter obtained after integration of the third-order diﬀerential equation

(2.1) satisﬁed by the traveling wave proﬁle φ, see Section 2. After two integrations of the

third-order equation (2.1) with integration constants aand b, all smooth periodic wave

solutions with the proﬁle φcan be found from the ﬁrst-order invariant

(c−φ)2(φ02−φ2−b) + a= 0.(1.7)

The second variation of the augmented energy functional (1.6) is determined by an

associated Hessian operator L:L2

per →L2

per given by

L:= c−φ−3c(4 −∂2

x)−1.(1.8)

The operator Lis self-adjoint and bounded as the sum of the bounded multiplication

operator (c−φ) and the compact operator −3c(4 −∂2

x)−1in L2

per. Since c−φ > 0, the

continuous spectrum of Lis strictly positive, hence Lhas ﬁnitely many negative eigenval-

ues of ﬁnite algebraic multiplicities and a zero eigenvalue of ﬁnite algebraic multiplicity.

The ﬁrst result of this paper is about the existence of smooth periodic traveling waves

with proﬁle φsatisfying the ﬁrst-order invariant (1.7), and the number of negative eigen-

values of Lgiven by (1.8).

Theorem 1.1. For a ﬁxed c > 0, smooth periodic solutions of the ﬁrst-order invariant

(1.7) with proﬁle φ∈H∞

per satisfying c−φ > 0exist in an open, simply connected region

on the (a, b)plane enclosed by three boundaries:

•a= 0 and b∈(−c2,0), where the periodic solutions are peaked,

•a=a+(b)and b∈(0,1

8c2), where the solutions have inﬁnite period,

•a=a−(b)and b∈(−c2,1

8c2), where the solutions are constant,

where a+(b)and a−(b)are smooth functions of b. For every point inside the region, the

periodic solutions are smooth functions of (a, b)and their period is strictly increasing in

bfor every ﬁxed a∈(0,27

256 c4). There exists a smooth curve a=a0(b)for b∈(−2

9c2,0)

in the interior of the existence region such that the Hessian operator Lin L2

per has only

one simple negative eigenvalue above the curve and two simple negative eigenvalues (or a

double negative eigenvalue) below the curve. The rest of its spectrum for a6=a0(b)includes

a simple zero eigenvalue and a strictly positive spectrum bounded away from zero. Along

the curve a=a0(b)the Hessian operator Lhas only one simple negative eigenvalue, a

double zero eigenvalue, and the rest of its spectrum is strictly positive.

The three curves bounding the existence region of smooth periodic waves in Theorem

1.1 are shown in Figure 1.1 for c= 1. The curve in the interior of the existence region

is the curve a=a0(b), which was found numerically by plotting the period function of

4 ANNA GEYER AND DMITRY E. PELINOVSKY

0 0.05 0.1 a

-1

-0.8

-0.6

-0.4

-0.2

0.2

solitary waves

a=a+(b)

a=a0(b)

constant waves

a=a−(b)

peaked waves

Figure 1.1. The existence region of smooth periodic solutions of the ﬁrst-

order invariant (1.7) on the parameter plane (a, b) for c= 1 enclosed by

three boundaries (red lines). The blue line shows the curve a=a0(b) which

separates the cases of one and two negative eigenvalue of L.

the periodic solutions of Theorem 1.1 versus afor ﬁxed band detecting its maximum if

it exists, see Lemmas 3.2 and 4.3 below.

The transformation

φ(x) = cϕ(x), b =c2β, a =c4α(1.9)

normalizes the parameter cto unity with ϕ,β, and αsatisfying the same equation (1.7)

but with c= 1. Hence, the smooth periodic waves are uniquely determined by the free

parameters (a, b) and c= 1 can be used everywhere. Similarly, although we only consider

the case of right-propagating waves with c > 0, all results can be extended to the left-

propagating waves with c < 0 by using the scaling transformation (1.9).

Spectral stability of smooth periodic travelling waves with respect to co-periodic per-

turbations is determined by the spectrum of the linearized operator JLin L2

per, with J

given in (1.5). Since Jis a skew-adjoint operator and Lis self-adjoint, the spectrum of

the linearized operator JLis symmetric with respect to iR[20]. Therefore, the periodic

wave is spectrally stable if the spectrum of JLin L2

per is located on iR. The second result

of this paper gives the energy criterion for the spectral stability of the smooth periodic

waves in the DP equation (1.1).

SMOOTH PERIODIC WAVES IN THE DEGASPERIS–PROCESI EQUATION 5

Theorem 1.2. For a ﬁxed c > 0and a ﬁxed period L > 0, there exists a C1mapping

a7→ b=BL(a)for a∈(0, aL)with some L-dependent aL∈(0,27

256 c4)and a C1mapping

a7→ φ= ΦL(·, a)∈H∞

per of smooth L-periodic solutions along the curve b=BL(a). Let

ML(a) := M(ΦL(·, a)) and FL(a) := F(ΦL(·, a)),

where M(u)and F(u)are given by (1.2) and (1.4). The L-periodic wave with proﬁle

φ= ΦL(·, a)such that B0

L(a)6= 0 is spectrally stable if the mapping

a7→ FL(a)

ML(a)3(1.10)

is strictly decreasing and, for B0

L(a)<0, if additionally the mapping a7→ ML(a)is

strictly increasing. The stability criterion holds true for every point in a neighborhood of

the boundary a=a−(b).

Figure 1.2 shows the numerically computed mappings a7→ FL(a)/M3

L(a) and a7→

ML(a) for four values of ﬁxed L. The parameter ais chosen in (0, aL), where aLdepends

on L. It follows that the stability criterion of Theorem 1.2 is satisﬁed for all cases. This

property has been analytically proven only near the boundary a=a−(b) by means of the

Stokes expansion, see Lemma 5.6.

It is harder to check the stability criterion of Theorem 1.2 near the other two boundaries

of the existence region of Theorem 1.1 where the waves are either peaked or solitary.

The perturbation theory becomes singular in these two asymptotic limits because c−φ

vanishes for the peaked periodic waves and the period function diverges for the solitary

waves. Nevertheless, some relevant results are available in these two limits:

•For the boundary a= 0 and b∈(−c2,0), where the periodic solutions are peaked,

the spectral stability problem for the DP equation (1.1) needs to be set up by

using a weak formulation of the evolution problem. This setup was elaborated

for a generalized CH equation in [25], building on previous work in [36], to show

spectral instability of peaked solitary waves. Linear and nonlinear instability of

peaked periodic waves with respect to peaked periodic perturbations was shown

for the CH equation in [35]. Spectral and linear instability of peaked periodic

waves for the reduced Ostrovsky equation was proven in [15, 16]. Instability of

peaked periodic waves in the DP equation or in the generalized CH equation is

still open for further studies.

•For the boundary a=a+(b) and b∈(0,1

8c2), where the periodic solutions have

inﬁnite period, spectral stability of solitary waves over a nonzero background was

shown for the general b-family in [26] and for the DP equation in [30]. The methods

in [26, 30] are not related to the energy stability criterion (1.10), and it remains

open to show the equivalence of the three diﬀerent stability criteria for smooth

solitary waves over a nonzero background.

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：

STABILITYOFSMOOTHPERIODICTRAVELINGWAVESINTHEDEGASPERIS{PROCESIEQUATIONANNAGEYERANDDMITRYE.PELINOVSKYAbstract.WederiveapreciseenergystabilitycriterionforsmoothperiodicwavesintheDegasperis{Procesi(DP)equation.ComparedtotheCamassa-Holm(CH)equation,thenumberofnegativeeigenvaluesofanassociatedHessianoper...

展开>> 收起<<

STABILITY OF SMOOTH PERIODIC TRAVELING WAVES IN THE DEGASPERISPROCESI EQUATION ANNA GEYER AND DMITRY E. PELINOVSKY.pdf

共33页,预览5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！

STABILITY OF SMOOTH PERIODIC TRAVELING WAVES IN THE DEGASPERISPROCESI EQUATION ANNA GEYER AND DMITRY E. PELINOVSKY

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: