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Stability of peakons for the Degasperis-Procesi equation Zhiwu Lin ∗and Yue Liu †Abstract The Degasperis-Procesi equation can be derived as a member of a one-parameter family of asymptotic shallow water approximations to the Eu-ler equations with the same asymptotic accuracy as that of the Camassa-Holm equation.In this paper,we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line.By con-structing a Liapunov function,we prove that the shapes of these peakon solitons are stable under small perturbations.Keywords:Stabiltiy;Degasperis-Procesi equation;Peakons Mathematics Subject Classification (2000):35G35,35Q51,35G25,35L051Introduction The Degasperis-Procesi (DP)equation y t +y x u +3yu x =0,x ∈R ,t >0,(1.1)with y =u −u xx ,was originally derived by Degasperis-Procesi [14]using the method of asymptotic integrability up to third order as one of three equations in the family of third order dispersive PDE conservation laws of the form u t −α2u xxt +γu xxx +c 0u x =(c 1u 2+c 2u 2x +c 3uu xx )x .(1.2)
The other two integrable equations in the family,after rescaling and applying a Galilean transformation,are the Korteweg-de Vries (KdV)equation
u t +u xxx +uu x =0
and the Camassa-Holm (CH)shallow water equation [2,15,18,22],
y t +y x u +2yu x =0,y =u −u xx .(1.3)
These three cases exhaust in the completely integrable candidates for(1.2)by Painlev´e analysis.Degasperis,Holm and Hone[13]showed the formal integra-bility of the DP equation as Hamiltonian systems by constructing a Lax pair and a bi-Hamiltonian structure.
The Camassa-Holm equation wasfirst derived by Fokas and Fuchassteiner [18]as a bi-Hamiltonian system,and then as a model for shallow water waves by Camassa and Holm[2].The DP equation is also in dimensionless space-time variables(x,t)an approximation to the incompressible Euler equations for shallow water under the Kodama transformation[20,21]and its asymptotic accuracy is the same as that of the Camassa-Holm(CH)shallow water equation, where u(t,x)is considered as thefluid velocity at time t in the spatial x-direction with momentum density y.
Recently,Liu and Yin[24]proved that thefirst blow-up infinite time to equation(1.1)must occur as wave breaking and shock waves possibly appear afterwards.It is shown in[24]that the lifespan of solutions
of the DP equation (1.1)is not affected by the smoothness and size of the initial profiles,but affected by the shape of the initial profiles(for the CH equation,see[1,8]).This can be viewed as a significant difference between the DP equation(or the CH equation )and the KdV.It is also noted that the KdV equation,unlike the CH equation or DP equation,does not have wave breaking phenomena[30].Under wave breaking we understand that development of singularities infinite time by which the wave remains bounded but its slope becomes unbounded[31].
It is well known that the KdV equation is an integrable Hamiltonian equation that possesses smooth solitons as traveling waves.In the KdV equation,the leading order asymptotic balance that confines the traveling wave solitons occurs between nonlinear steepening and linear dispersion.However,the nonlinear dispersion and nonlocal balance in the CH equation and the DP equation,even in the absence of linear dispersion,can still produce a confined solitary traveling waves
u(t,x)=cϕ(x−ct)(1.4) traveling at constant speed c>0,whereϕ(x)=e−|x|.Because of their shape (they are smooth except for a peak at their crest),these solutions are called the peakons[2,13].Peakons of both equations are true solitons that interact via elastic collisions under the CH dynamics,or the DP dynamics,respectively.The peakons of the CH equation are orbitally stable[12].For waves that approximate the peakons in a special way,a stability result was proved by a variation method [11].
Note that we can rewrite the DP equation as
北洋海军兴亡史u t−u txx+4uu x=3u x u xx+uu xxx,t>0,x∈R.(1.5) The peaked solitons are not classical solutions of(1.5).They satisfy the Degasperis-Procesi equation in the conservation law form
u t+∂x 12ϕ∗ 3
where∗stands for convolution with respect to the spatial variable x∈R.This is the exact meaning in which the peakons are solutions.
Recently,Lundmark and Szmigielski[26]presented an inverse scattering ap-proach for computing n-peakon solutions to equation(1.5).Holm and Staley[20] studied stability of solitons and peakons numerically to equation(1.5).Anal-ogous to the case of Camassa-Holm equation[6],Henry[19]and Mustafa[29] showed that smooth solutions to equation(1.5)have infinite speed of propaga-tion.
The following are three useful conservation laws of the Degasperis-Procesi equation.
E1(u)= R y dx,E2(u)= R yv dx,E3(u)= R u3dx,
where y=(1−∂2x)u and v=(4−∂2x)−1u,while the corresponding three useful conservation laws of the Camassa-Holm equation are the following:
F1(u)= R y dx,F2(u)= R(u2+u2x)dx,F3(u)= R(u3+uu2x)dx.(1.7)
The stability of solitary waves is one of the fundamental qualitative proper-ties of the solutions of nonlinear wave equations.Numerical simulations[13,25] suggest that the sizes and velocities of the peakons do not change as a result of collision so these patterns are expected to be stable.Furthermore,it is observed that the shape of the peakons remains approximately the same as time evolves. As far as we know,the case of stability of the peakons for the Camassa-Holm equation is well understood by now[11,12],while the Degasperis-Procesi equa-tion case is the subject of this paper.The goal of this paper is to establish a stability result of peaked solitons for equation(1.5).
It is found that the corresponding conservation laws of the Degasperis-Procesi equation are much weaker than those of the Camassa-Holm equation. In particular,one can see that the conservation law E2(u)for the DP equation
.In fact,by the Fourier transform,we have
is equivalent to u 2
L2
E2(u)= R yvdx= R1+ξ2
Let us denote
E2(u)= u 2X.
The following stability theorem is the principal result of the present paper. Theorem1(Stability)Let cϕbe the peaked soliton defined in(1.4)traveling with speed c>0.Then cϕis orbitally stable in the following sense.If u0∈H s
for some s>3/2,y0=u0−∂2x u0is a nonnegative Radon measure offinite total mass,and
u0−cϕ X<cε,|E3(u0)−E3(cϕ)|<c3ε,0<ε<1
6 ≤c√
ε.(1.10)
Remark1The state of affairs about these maxima/minima implied by the pre-vious theorem is a consequence of the assumption on y0,as shown in Lemma 3.1.For an initial profile u0∈H s,s>3/2,the
re exists a local solution u∈C([0,T),H s)of(1.5)with initial data u(0)=u0[32].Under the assump-tion y0=u0−∂2x u0≥0in Theorem1,the existence is global in time[24],that is T=+∞.For peakons cϕwith c>0,we have 1−∂2x (cϕ)=2cδ(hereδis the Dirac distribution).Hence the assumption on y0that it is a nonnegative measure is quite natural for a small perturbation of the peakons.Existence of global weak solution in H1of the DP equation is also proved in[16].Note that peakons cϕare not strong solutions,sinceϕ∈H s,only for s<3/2.
The above theorem of orbital stability states that any solution starting close to peakons cϕremains close to some translate of cϕin the norm X,at any later time.More information about this stability is contained in(1.9)and (1.10).Notice that for peakons cϕ,the function v cϕis single-humped with the height1
There are two standard methods to study stability issues of dispersive wave
equations.One is the variational approach which constructs the solitary waves as energy minimizers under appropriate constraints,and the stability automat-
ically follows.However,without uniqueness of the minimizer,one can only
规划成果obtain the stability of the set of minima.The variational approach is used in [11]for the CH equation.It is shown in[11]that the each peakon cϕis the unique minimum(ground state)of constrained energy,from which its orbital stability is proved for initial data u0∈H3with y0=(1−∂2x)u0≥0.Their proof strongly relies on the fact that the conserved energy F2in(1.7)of the CH
equation is the H1−norm of the solution.However,for the DP equation the energy E2in(1.8)is only the L2norm of the solution.Consequently,it is more
difficult to use such a variational approach for the DP equation.
Another approach to study stability is to linearize the equation around the solitary waves,and it is commonly believed that nonlinear stability is governed by the linearized equation.However,for the CH and DP equations,the non-linearity plays the dominant role rather than being a higher-order correction to linear terms.Thus it is unclear how one can get nonlinear stability of peakons by studying the linearized problem.Morover,the peaked solitons cϕare not differentiable,which makes it difficult to analyze the spectrum of the linearized operator around cϕ.
To establish the stability result for the DP equation,we extend the approach in[12]for the CH equation.The idea in[12]is to directly use the energy F2as the Liapunov functional.By expanding F2in(1.7)around the peakon cϕ,the error term is in the form of the difference of the maxima of cϕand the perturbed solution u.To estimate this difference,they establish two integral relations g2=F2(u)−2(max u)2and ug2=F3(u)−4
M3,M=max u(x)
3
and the error estimate|M−maxϕ|then follows from the structure of the above polynomial inequality.
To extend the above approach to nonlinear stability of the DP peakons,we have to overcome several difficulties.By expanding the energy E2(u)around the peakon cϕ,the error term turns out to be max v cϕ−max v u,with v u= (4−∂2x)−1u.We can derive the following two integral relations for M1=max v u, E2(u)and E3(u)by
g2=E2(u)−12M21and hg2=E3(u)−144M31
with some functions g and h related to v u.To get the required polynomial inequality from the above t
乌兹别克
wo identities,we need to show h≤18max v u.How-ever,since h is of the form−∂2x v u±6∂x v u+16v u,generally it can not be
5
bounded by v u.This new difficulty is due to the more complicated nonlinear
structure and weaker conservation laws of the DP equation.To overcome it,we introduce a new idea.By constructing g and h piecewise according to mono-
tonicity of the function v u,we then establish two new integral identities(3.7)
and(3.9)for E2,E3and all local maxima and minima of v u.The crucial es-timate h≤18max v u can now be shown by using this monotonicity structure
and properties of the DP solutions.This results in inequality(3.13)related to E2,E3and all local maxima and minima of v u.By analyzing the structure of
equality(3.13),we can obtain not only the error estimate|M1−max v cϕ|but more precise stability information from(1.10).We note that the same approach can also be used for the CH equation to gain more stability information(see
Remark2).
Although the DP equation is similar to the CH equation in several aspects, we would like to point out that these two equations are truly different.One of the novel features of the DP equation is it has not only peaked solitons[13], u(t,x)=ce−|x−ct|,c>0but also shock peakons[4,25]of the form
1
u(t,x)=−
for t≥T.
生死千里k+(t−T)
On the other hand,the isospectral problem in the Lax pair for equation(1.5)
is the third-order equation
ψx−ψxxx−λyψ=0
cf.[13],while the isospectral problem for the Camassa-Holm equation is the
手机乐讯网second order equation
ψxx−1
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