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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Asymptotic stability of a Korteweg–de Vries equation with a two-dimensional center manifold

Shuxia Tang
  • Department of Mechanical & Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, USA; and UMR 7598 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris 6), 75005 Paris, France
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jixun Chu
  • Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China
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/ Peipei Shang / Jean-Michel Coron
Published Online: 2016-10-14 | DOI: https://doi.org/10.1515/anona-2016-0097

Abstract

Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg–de Vries equation posed on a finite interval [0,2π7/3]. The equation comes with a Dirichlet boundary condition at the left end-point and both the Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the associated linearized equation around the origin is not asymptotically stable. In this paper, the nonlinear Korteweg–de Vries equation is proved to be locally asymptotically stable around the origin through the center manifold method. In particular, the existence of a two-dimensional local center manifold is presented, which is locally exponentially attractive. Analyzing the Korteweg–de Vries equation restricted on the local center manifold, we obtain a polynomial decay rate of the solution.

Keywords: Korteweg–de Vries equation; nonlinearity; center manifold; asymptotic stability; polynomial decay rate

MSC 2010: 35Q53; 37L10; 93D05; 93D20

1 Introduction

The Korteweg–de Vries (KdV) equation

yt+yx+yyx+yxxx=0

was first derived by Boussinesq in [2, (283 bis)] and by Korteweg and de Vries in [14], for describing the propagation of small amplitude long water waves in a uniform channel. This equation is now commonly used to model unidirectional propagation of small amplitude long waves in nonlinear dispersive systems. An excellent reference to help understand both physical motivation and deduction of the KdV equation is the book by Whitham [22].

Rosier studied in [20] the following nonlinear Neumann boundary control problem for the KdV equation with homogeneous Dirichlet boundary conditions, posed on a finite spatial interval:

{yt+yx+yyx+yxxx=0,t(0,),x(0,L),y(t,0)=y(t,L)=0,yx(t,L)=u(t),t(0,),y(0,x)=y0(x),x(0,L),(1.1)

where L>0, the state is y(t,):[0,L], and u(t) denotes the controller. The equation comes with one boundary condition at the left end-point and two boundary conditions at the right end-point. Rosier first considered the first-order power series expansion of (y,u) around the origin, which gives the following corresponding linearized control system:

{yt+yx+yxxx=0,t(0,),x(0,L),y(t,0)=y(t,L)=0,yx(t,L)=u(t),t(0,),y(0,x)=y0(x),x(0,L).(1.2)

By means of multiplier technique and the Hilbert uniqueness method (HUM) [15], Rosier proved that (1.2) is exactly controllable if and only if the length of the spatial domain is not critical, i.e., L𝒩, where 𝒩 denotes the following set of critical lengths:

𝒩:={2πj2+l2+jl3;j,l}.(1.3)

Then, by employing the Banach fixed point theorem, he derived that the nonlinear KdV control system (1.1) is locally exactly controllable around 0 provided that L𝒩. In the cases with critical lengths L𝒩, Rosier demonstrated in [20] that there exists a finite dimensional subspace M of L2(0,L) which is unreachable for the linear system (1.2) when starting from the origin. In [8], Coron and Crépeau treated a critical case of L=2kπ (i.e., taking j=l=k in 𝒩), where k is a positive integer such that (see [7, Theorem 8.1 and Remark 8.2])

(j2+l2+jl=3k2 and j,l*)j=l=k.(1.4)

Here, the uncontrollable subspace M for the linear system (1.2) is one-dimensional. However, through a third-order power series expansion of the solution, they showed that the nonlinear term yyx always allows to “go” in small time into the two directions missed by the linearized control system (1.2), and then, using a fixed point theorem, they deduced the small-time local exact controllability around the origin of the nonlinear control system (1.1). In [4], Cerpa studied the critical case of L𝒩, where

𝒩:={2πj2+l2+jl3;j,l satisfying j>l and j2+jl+l2m2+mn+n2 for all m,n*{j}}.(1.5)

In this case, the uncontrollable subspace M for the linear system (1.2) is of dimension 2, and Cerpa used a second-order expansion of the solution to the nonlinear control system (1.1) to prove the local exact controllability in large time around the origin of the nonlinear control system (1.1) (the local controllability in small time for this length L is still an open problem). Furthermore, Cerpa and Crépeau considered in [5] the cases when the dimension of M for the linear system (1.2) is higher than 2. They implemented a second-order expansion of the solution to (1.1) for the critical lengths L2kπ for any k*, and implemented an expansion to the third order if L=2kπ for some k*. They showed that the nonlinear term yyx always allows to “go” into all the directions missed by the linearized control system (1.2) and then proved the local exact controllability in large time around the origin of the nonlinear control system (1.1).

Consider the case when there is no control, i.e., u=0, in (1.1), which gives the following initial-boundary-value KdV problem posed on a finite interval [0,L]:

{yt+yx+yxxx+yyx=0,t(0,),x(0,L),y(t,0)=y(t,L)=0,yx(t,L)=0,t(0,),y(0,x)=y0(x),x(0,L),(1.6)

where the boundary conditions are homogeneous. For the Lyapunov function

E(t)=12y(t,)L2(0,L)2=120Ly2(t,x)𝑑x,(1.7)

we have

E˙(t)=-0Ly(yx+yyx+yxxx)𝑑x=0Lyxyxx𝑑x=-12yx2(t,0)0.(1.8)

Thus, 0L2(0,L) is stable (see (P1) below for the definition of stable) for the KdV equation (1.6). Moreover, it has been proved in [18] that, if L𝒩, then 0 is exponentially stable for the corresponding linearized equation around the origin:

{yt+yx+yxxx=0,t(0,),x(0,L),y(t,0)=y(t,L)=0,yx(t,L)=0,t(0,),y(0,x)=y0(x),x(0,L),(1.9)

which gives the local asymptotic stability around the origin for the nonlinear equation (1.6). However, when L𝒩, Rosier pointed out in [20] that equation (1.9) is not asymptotically stable. Inspired by the fact that the nonlinear term yyx introduces the local exact controllability around the origin into the KdV control system (1.1) with L𝒩, we would like to discuss whether the nonlinear term yyx could introduce local asymptotic stability around the origin for (1.6).

This paper is devoted to investigating the local asymptotic stability of 0L2(0,L) for (1.6) with the critical length

L=2π7/3,

corresponding to j=2 and l=1 in (1.3). Let us recall that this local asymptotic stability means that the following two properties are satisfied.

  • (P1)

    Stability: for every ε>0, there exists η=η(ε)>0 such that y0L2(0,L)<η implies

    y(t,)L2(0,L)<εfor all t0.

  • (P2)

    (Local) attractivity: there exists ε0>0 such that y0L2(0,L)<ε0 implies

    limt+y(t,)L2(0,L)=0.

As mentioned above, the stability property (P1) is implied by (1.8). Our main concern is thus the local attractivity property (P2). We prove the following theorem, where the precise definition of a solution to (1.6) is given in Definition 2.7, and the precise definition of the finite dimensional vector space ML2(0,L) when L=2π7/3 is given in (2.8).

Theorem 1.1.

Consider the KdV equation (1.6) with L=2π7/3. There exist δ>0, K>0, ω>0 and a map g:MM, where ML2(0,L) is the orthogonal of M for the L2-scalar product, satisfying

gC3(M;M),(1.10)g(0)=0,g(0)=0,(1.11)

such that, with

G:={m+g(m);mM}L2(0,L),(1.12)

the following three properties hold for every solution y to (1.6) with y0L2(0,L)<δ:

  • (i)

    Local exponential attractivity of G :

    d(y(t,),G)Ke-ωtd(y0,G)for all t>0,(1.13)

    where d(χ,G) denotes the distance between χL2(0,L) and G :

    d(χ,G):=inf{χ-ψL2(0,L);ψG}.

  • (ii)

    Local invariance of G : If y0G , then y(t,)G for all t0.

  • (iii)

    If y0G , then there exists C>0 such that

    y(t,)L2(0,L)Cy0L2(0,L)1+ty0L2(0,L)2for all t0.(1.14)

In particular, 0L2(0,L) is locally asymptotically stable in the sense of the L2(0,L)-norm for (1.6).

Remark 1.2.

It can be derived from [9, Theorem 1 and comments] that, for every L>0, there are nonzero stationary solutions with the period of L to the following ordinary differential equation (ODE):

{f+ff+f′′′=0in [0,L],f(0)=f(L)=0,f(L)=0.

That is, besides the origin, there also exist other steady states of the nonlinear KdV equation (1.6). Therefore, 0L2(0,L) is not globally asymptotically stable for (1.6): Property (P2) does not hold for arbitrary ε0>0.

Our proof of Theorem 1.1 relies on the center manifold approach. This center manifold is G in Theorem 1.1. Center manifold theory plays an important role in studying dynamic properties of nonlinear systems near “critical situations”. The center manifold theorem was first proved for finite dimensional systems by Pliss [19] and Kelley [12], and the readers could refer to [13, 17] for more details of this theory. Analogous results are also established for infinite dimensional systems, such as partial differential equations (PDEs) [3, 1] and functional differential equations [10]. The center manifold method usually leads to a dimension reduction of the original problems. Then, in order to derive stability properties (asymptotic stability or unstability) of the full nonlinear equations, one only needs to analyze the reduced equation (restricted on the center manifold). When dealing with the infinite dimensional problems, this method can be extremely efficient if the center manifold is finite dimensional. Following the results on existence, smoothness and attractivity of a center manifold for evolution equations in [21], Chu, Coron and Shang studied in [6] the local asymptotic stability property of (1.6) with the critical length L=2kπ for any positive integer k such that (1.4) holds. They proved the existence of a one-dimensional local center manifold. By analyzing the resulting one-dimensional reduced equation, they obtained the local asymptotic stability of 0 for (1.6). For L=2π7/3, we get, following [6], the existence of a two-dimensional local center manifold. It is predictable that the two-dimensional local center manifold introduces more complexity than the one-dimensional local center manifold case.

The organization of this paper is as follows. In Section 2, some basic properties of the linearized KdV equation (1.9) and the KdV equation (1.6) are given. Then, in Section 3, we recall a theorem on the existence of a local center manifold for the KdV equation (1.6) and analyze the dynamics on the local center manifold. Theorem 1.1 follows from this analysis. In Section 4, we present the conclusion and some possible future works. We end this article with Appendix A containing computations which are important for the study of the dynamics on the center manifold.

2 Preliminaries

2.1 Some properties for the linearized equation of (1.6) around the origin

The origin y=0 is an equilibrium of the initial-boundary-value nonlinear KdV problem (1.6). In this subsection, we derive some properties for the linearized KdV equation (1.9) around the origin of (1.6) posed on the finite interval [0,L], where L=2π7/3𝒩, for which there exists a unique pair {j=2,l=1} satisfying (1.5).

Let 𝒜:D(𝒜)L2(0,L)L2(0,L) be the linear operator defined by

𝒜φ:=-φ-φ′′′,

with

D(𝒜):={φH3(0,L);φ(0)=φ(L)=φ(L)=0}L2(0,L).

Then the linearized equation (1.9) can be written as an evolution equation in L2(0,L):

dy(t,)dt=𝒜y(t,).

The following lemma can be immediately obtained.

Lemma 2.1.

𝒜-1 exists and is compact on L2(0,L). Hence, σ(A), the spectrum of A, consists of isolated eigenvalues only: σ(A)=σp(A), where σp(A) denotes the set of eigenvalues of A.

Proof.

By calculation, we get

𝒜-1φ=ψfor all φL2(0,L),

with

ψ:=-1-cos(x-L)1-cosL0L(1-cosy)φ(y)𝑑y+xL(1-cos(x-y))φ(y)𝑑y.

Hence we get the existence of 𝒜-1 and that, by the Sobolev embedding theorem, this operator is compact on L2(0,L). Therefore, σ(𝒜), the spectrum of 𝒜, consists of isolated eigenvalues only. ∎

The following proposition is proved.

Proposition 2.2 ([20, Proposition 3.1]).

𝒜 generates a C0-semigroup of contractions {S(t)}t0 on L2(0,L), that is, for any given initial data y0L2(0,L), S(t)y0 is the mild solution of the linearized equation (1.9), and

S(t)y0L2(0,L)y0L2(0,L)for all t0.

Moreover, Re(λ)0 for every λσ(A).

If Re(λ)<0 for all λσ(𝒜), then it follows directly from the ABLP theorem (Arendt–Batty–Lyubich–Phong) [16] that the semigroup S(t) is asymptotically stable on L2(0,L). Since we only have Re(λ)0 for all λσ(𝒜), the main concern needs to be put on the eigenvalues on the imaginary axis and their corresponding eigenfunctions. Following the proofs of [6, Lemma 2.6] and [20, Lemma 3.5] yields the next lemma.

Lemma 2.3.

There exists a unique pair of conjugate eigenvalues of A on the imaginary axis, that is,

σp(𝒜)i={λ=±iq;q=202121}.

Moreover, the corresponding eigenfunctions of A with respect to λ=±iq are

φ:=C(φ1iφ2),

respectively, where C is an arbitrary constant, and φ1,φ2 are two nonzero real-valued functions:

φ1(x)=Θ(cos(521x)-3cos(121x)+2cos(421x)),(2.1)φ2(x)=Θ(-sin(521x)-3sin(121x)+2sin(421x)),(2.2)

with

Θ:=114π3/74.(2.3)

Remark 2.4.

The equations satisfied by φ1 and φ2 are

{φ1+φ1′′′=-qφ2,φ1(0)=φ1(L)=0,φ1(0)=φ1(L)=0,(2.4)

and

{φ2+φ2′′′=qφ1,φ2(0)=φ2(L)=0,φ2(0)=φ2(L)=0.(2.5)

Remark 2.5.

We have

0Lφ1(x)φ2(x)𝑑x=0,(2.6)

and, with the definition of Θ given in (2.3),

φ1L2(0,L)=φ2L2(0,L)=1.(2.7)

From the results in Lemma 2.1, Proposition 2.2 and Lemma 2.3, we obtain the following corollary.

Corollary 2.6.

λ=±i202121 is the unique eigenvalue pair of A on the imaginary axis, and all the other eigenvalues of A have negative real parts which are uniformly bounded away from the imaginary axis, i.e., there exists r>0 such that any of the nonzero eigenvalues of A has a real part which is less than -r.

Let us define

M:=span{φ1,φ2}={m1φ1+m2φ2;𝐦=(m1,m2)2}L2(0,L),(2.8)

where φ1,φ2 are defined in (2.1), (2.2) and (2.3). Then the following decomposition holds:

L2(0,L)=MM,

with

M:={φL2(0,L);0Lφ(x)φ1(x)dx=0,0Lφ(x)φ2(x)dx=0}.(2.9)

2.2 Some properties of the KdV equation (1.6)

By considering equation (1.6) as a special case (with f=0 and u=0) of [7, (4.6)–(4.8)], we give the following definition for a solution to equation (1.6), which follows from [7, Definition 4.1].

Definition 2.7.

Let T>0 and y0L2(0,L). A solution to the Cauchy problem (1.6) on [0,T] is a function

y:=C0([0,T];L2(0,L))L2(0,T;H1(0,L))

such that, for every τ[0,T] and for every ϕC3([0,τ]×[0,L]) satisfying

ϕ(t,0)=ϕ(t,L)=ϕx(t,0)=0for all t[0,τ],(2.10)

one has

-0τ0L(ϕt+ϕx+ϕxxx)y𝑑x𝑑t+0τ0Lϕyyx𝑑x𝑑t+0Ly(τ,x)ϕ(τ,x)𝑑x-0Ly0(x)ϕ(0,x)𝑑x=0.(2.11)

A solution to the Cauchy problem (1.6) on [0,+) is a function

yC0([0,+);L2(0,L))Lloc2([0,+);H1(0,L))

such that, for every T>0, y restricted to [0,T]×(0,L) is a solution to (1.6) on [0,T].

Then by considering equation (1.6) as a special case of [8, (A.1)] (with f=0 and u=0), the following two propositions about the existence and uniqueness of the solutions to (1.6) follow directly from [8, Propositions 14 and 15].

Proposition 2.8.

Let T(0,+). There exist ε=ε(T)>0 and C=C(T)>0 such that, for every y0L2(0,L) with y0L2(0,L)<ε(T), there exists at least one solution y to equation (1.6) on [0,T] which satisfies

y:=maxt[0,T]y(t,)L2(0,L)+(0Ty(t,)H1(0,L)2dt)1/2C(T)y0L2(0,L).

Proposition 2.9.

Let T(0,+). There exists C>0 such that, for each pair of solutions (y1,y2), corresponding to each pair of initial conditions (y10,y20)(L2(0,L))2, to equation (1.6) on [0,T], the following inequalities hold:

0T0L(y1x(t,x)-y2x(t,x))2𝑑x𝑑t0L(y10(x)-y20(x))2𝑑xexp(C(1+y1L2(0,T;H1(0,L))2+y2L2(0,T;H1(0,L))2)),0L(y1(t,x)-y2(t,x))2𝑑x0L(y10(x)-y20(x))2𝑑xexp(C(1+y1L2(0,T;H1(0,L))2+y2L2(0,T;H1(0,L))2)),

for all t[0,T].

Let us also mention that for every solution y to (1.6) on [0,T] or on [0,+),

ty(t,)L2(0,L)2 is a non-increasing function.(2.12)

This can be easily seen by multiplying the first equation of (1.6) with y, integrating on [0,L] and performing integration by parts. One then gets, if y is smooth enough,

ddt0Ly(t,x)2𝑑x=-yx(t,0)2,

which gives (2.12). The general case follows from a smoothing argument. As a consequence of Proposition 2.8, Proposition 2.9 and (2.12), one sees that (1.6) has one and only one solution defined on [0,+) if y0L2(0,L)<ε(1).

3 Existence of a center manifold and dynamics on this manifold

Let us start this section by recalling why, as it is classical, the property “0L2(0,L) is locally asymptotically stable in the sense of the L2(0,L)-norm for (1.6)” stated at the end of Theorem 1.1 is a consequence of the other statements in this theorem. For convenience, let us recall the argument. Let y0L2(0,L) be such that y0L2(0,L)<δ and let y be the solution to (1.6). It suffices to check that

y(t,)0in L2(0,L)as t+.(3.1)

By (1.13), (2.12) and the fact that M is of finite dimension, there exists an increasing sequence of positive real numbers (tn)n and z0L2(0,L) such that

tn+as n+,y(tn,)z0in L2(0,L)as n+,(3.2)z0Gandz0L2(0,L)<δ.(3.3)

Let z:[0,+)×(0,L) be the solution to (1.6) satisfying the initial condition z(0,)=z0. It follows from (1.14) and (3.3) that

z(t,)0in L2(0,L)as t+.(3.4)

Let η>0. By (3.4), there exists τ>0 such that

z(τ,)L2(0,L)η2.(3.5)

By Proposition 2.9 and (3.2),

y(tn+τ,)z(τ,)in L2(0,L)as n+.(3.6)

By (3.5) and (3.6), there exists n0 such that

y(tn0+τ,)L2(0,L)<η,

which, together with (2.12), implies that

y(t,)L2(0,L)<ηfor all ttn0+τ,

which concludes the proof of (3.1).

The remaining parts of this section are organized as follows. We first recall in Section 3.1 a theorem (Theorem 3.1) on the existence of a local center manifold for (1.6). Then in Section 3.2 we analyze the dynamics of (1.6) on this center manifold and deduce Theorem 1.1 from this analysis.

3.1 Existence of a local center manifold

In [6, Theorem 3.1], following [21], the existence of a center manifold for (1.6) was proved for the first critical length, i.e., L=2π. The same proof applies for our L (i.e., L=2π7/3) and allows us to get the following theorem.

Theorem 3.1.

There exist δ(0,ε(1)), K>0, ω>0 and a map g:MM satisfying (1.10) and (1.11) such that, with G defined by (1.12), the following two properties hold for every solution y(t,x) to (1.6) with y0L2(0,L)<δ:

  • (i)

    Local exponential attractivity of G :

    d(y(t,),G)Ke-ωtd(y0,G)for all t>0,

    where d(χ,G) denotes the distance between χL2(0,L) and G :

    d(χ,G):=inf{χ-ψL2(0,L);ψG}.

  • (ii)

    Local invariance of G : If y0G , then y(t,)G for all t0.

3.2 Dynamics on the local center manifold

In this section we study the dynamics of (1.6) on Gδ with

Gδ:={ζ(x)G;ζL2(0,L)<δ}.

Let

Ω:={(m1,m2)2;m1φ1+m2φ2+g(m1φ1+m2φ2)Gδ},

then Ω is a bounded open subset of 2 which contains (0,0)2. Let 𝐦0=(m10,m20)Ω, and let y be the solution of (1.6) on [0,+) for the initial data y0:=m10φ1+m20φ2+g(m10φ1+m20φ2). It follows from (2.12) and Theorem 3.1 that y(t,)Gδ for every t[0,+). Hence we can define, for t[0,+), 𝐦(t)=(m1(t),m2(t))Ω by requiring that

y(t,)=m1(t)φ1+m2(t)φ2+g(m1(t)φ1+m2(t)φ2).(3.7)

Since yC0([0,+);L2(0,L)), we have 𝐦C0([0,+);2). Let T>0 and uC0(0,T). We apply (2.11) with τ=T and ϕ(t,x):=u(t)φ1(x) (note that, by (2.4), (2.10) holds). We get

-0T0L(u˙(t)φ1(x)+u(t)φ1(x)+u(t)φ1′′′(x))y(t,x)𝑑x𝑑t+0T0Lu(t)φ1(x)(yyx)(t,x)𝑑x𝑑t=0.(3.8)

From (2.4), (2.9), (3.7) and (3.8), we have

-0T(m1(t)u˙(t)-qm2(t)u(t))𝑑t-120T0Ly2(t,x)φ1(x)u(t)𝑑x𝑑t=0.

Hence, in the sense of distributions on (0,T),

m˙1=-qm2+120L(m1φ1+m2φ2+g(m1φ1+m2φ2))2φ1𝑑x.

Similarly, in the sense of distributions on (0,T),

m˙2=qm1+120L(m1φ1+m2φ2+g(m1φ1+m2φ2))2φ2𝑑x.

Hence, if we define F:Ω2, 𝐦=(m1,m2)F(𝐦) by

F(𝐦):=(-qm2+120L(m1φ1+m2φ2+g(m1φ1+m2φ2))2φ1𝑑xqm1+120L(m1φ1+m2φ2+g(m1φ1+m2φ2))2φ2𝑑x),(3.9)

then

𝐦˙=F(𝐦).(3.10)

Note that, by (1.10) and (3.9), FC3(Ω;2), which, together with (3.10), implies that

𝐦C4([0,+);2).(3.11)

We now estimate g close to 0M. Let ψC3([0,L]) be such that

ψ(0)=ψ(L)=ψ(0)=0.(3.12)

Using Definition 2.7 with ϕ(t,x):=ψ(x), (3.12) and integration by parts, we get

-1τ0τ0L(ψ+ψ′′′)y𝑑x𝑑t-12τ0τ0Lψy2𝑑x𝑑t+0L1τ(y(τ,x)-y0(x))ψ(x)𝑑x=0.(3.13)

Letting τ0+ in (3.13), and using (3.9), (3.10) and (3.11), we get

-0L(ψ+ψ′′′)y0𝑑x-120Lψy02𝑑x+0L(m˙1(0)φ1(x)+m˙2(0)φ2(x)+gm1(𝐦0)m˙1(0)+gm2(𝐦0)m˙2(0))ψ𝑑x=0.(3.14)

We expand g in a neighborhood of 0M. Using (1.10) and (1.11), there exist

aM,bM,cM(3.15)

such that

g(αφ1+βφ2)=α2a+αβb+β2c+o(α2+β2)in L2(0,L)as α2+β20,(3.16)gm1(αφ1+βφ2)=2αa+βb+o(|α|+|β|)in L2(0,L)as |α|+|β|0,(3.17)gm2(αφ1+βφ2)=αb+2βc+o(|α|+|β|)in L2(0,L)as |α|+|β|0.(3.18)

As usual, by (3.16), we mean that, for every ς1>0, there exists ς2>0 such that

α2+β2ς1g(αφ1+βφ2)-(α2a+αβb+β2c)L2(0,L)ς2(α2+β2).

Similar definitions are used in (3.17), (3.18) and later on. We now expand the left-hand side of (3.14) in terms of m10, m20, (m10)2, m10m20 and (m20)2 as |m10|+|m20|0.

For the functions φ1 and φ2 defined by (2.1), (2.2) and (2.3), the following equalities can be derived from (2.4), (2.5) and using integrations by parts:

0Lφ1(x)φ2(x)𝑑x=10721,0Lφ2(x)φ1(x)𝑑x=-10721,(3.19)0Lφ12(x)φ1(x)𝑑x=0,0Lφ22(x)φ2(x)𝑑x=0,(3.20)0Lφ12(x)φ2(x)𝑑x=-2c1,0Lφ22(x)φ1(x)𝑑x=23c1,(3.21)0Lφ1(x)φ2(x)φ1(x)𝑑x=c1,0Lφ1(x)φ2(x)φ2(x)𝑑x=-3c1,(3.22)

where the constant c1 is defined by

c1:=177147392392π1/(2π)3/74.

Looking successively at the terms in (m10)2, m10m20 and (m20)2 in (3.14) as |m10|+|m20|0, we get, using (3.9), (3.10), (3.16)–(3.18) as well as (3.19)–(3.22),

-0L(ψx+ψxxx)a𝑑x-120Lψxφ12𝑑x+0L(-c1φ2+qb)ψ𝑑x=0,(3.23)-0L(ψx+ψxxx)b𝑑x-0Lψxφ1φ2𝑑x+0L(c1φ1-3c1φ2-2qa+2qc)ψ𝑑x=0,(3.24)-0L(ψx+ψxxx)c𝑑x-120Lψxφ22𝑑x+0L(3c1φ1-qb)ψ𝑑x=0.(3.25)

Since (3.23)–(3.25) must hold for every ψC3([0,L]) satisfying (3.12), one gets that a, b and c are of class C on [0,L] and satisfy

{a+a′′′+φ1φ1-c1φ2+qb=0,a(0)=a(L)=0,a(L)=0,(3.26){b+b′′′+φ1φ2+φ1φ2+c1φ1-3c1φ2-2qa+2qc=0,b(0)=b(L)=0,b(L)=0,(3.27){c+c′′′+φ2φ2+3c1φ1-qb=0,c(0)=c(L)=0,c(L)=0.(3.28)

In Appendix A, we derive the unique functions a:[0,L], b:[0,L] and c:[0,L] which are solutions to (3.26), (3.27) and (3.28). From (3.9) and (3.16), and using (3.20)–(3.22), we get that, as 𝐦𝟎2,

F(𝐦)=(-qm2+3c1m22+c1m1m2+A1m13+B1m12m2+C1m1m22+D1m23qm1-c1m12-3c1m1m2+A2m13+B2m12m2+C2m1m22+D2m23)+o(|𝐦|3),(3.29)

with

A1:=0Laφ1φ1𝑑x,B1:=0Lbφ1φ1𝑑x+0Laφ2φ1𝑑x,(3.30)C1:=0Lcφ1φ1𝑑x+0Lbφ2φ1𝑑x,D1:=0Lcφ2φ1𝑑x,(3.31)A2:=0Laφ1φ2𝑑x,B2:=0Lbφ1φ2𝑑x+0Laφ2φ2𝑑x,(3.32)C2:=0Lcφ1φ2𝑑x+0Lbφ2φ2𝑑x,D2:=0Lcφ2φ2𝑑x.(3.33)

Let us now study the local asymptotic stability property of 𝟎2 for (3.10). We propose two methods for that. The first one is a more direct one, which relies on normal forms for dynamical systems on 2. The second one, which relies on a Lyapunov approach related to the physics of (1.6), is less direct. However, there is a reasonable hope that this second method can be applied to other critical lengths L𝒩2π for which the dimension of M is larger than 2.

Method 1: Normal form.

Let z:=m1+im2. Then

m1=z+z¯2,m2=z-z¯2i,

and it follows from (3.10) and (3.29) that, as |z|0,

z˙=(iq)z+P2(z,z¯)+P3(z,z¯)+o(|z|3),(3.34)

where Pj(z,z¯) are polynomials in z,z¯ of degree j. To be more precise, we have

P2(z,z¯):=(3c1m22+c1m1m2)+i(-c1m12-3c1m1m2)=-c12(3+i)z2+c12(3-i)zz¯(3.35)

and

P3(z,z¯):=(A1+iA2)(z+z¯2)3+(B1+iB2)(z+z¯2)2(z-z¯2i)+(C1+iC2)(z+z¯2)(z-z¯2i)2+(D1+iD2)(z-z¯2i)3.(3.36)

We can rewrite (3.34) as

z˙=(iq)z+i+j=231i!j!gijziz¯j+o(|z|3),(3.37)

and it is known from [11, pp. 45, 47] that (3.37) has the following Poincaré normal form:

ξ˙=(iq)ξ+ρξ2ξ¯+o(|ξ|3),(3.38)

where

ρ=i2q(g20g11-2|g11|2-13|g02|2)+g212.(3.39)

According to (3.35) and (3.36), through a simple computation, we have

g20=-c1(3+i),g11=c12(3-i),g02=0,(3.40)g21=14(3A1+i3A2-iB1+B2+C1+iC2-i3D1+3D2).(3.41)

Using (3.40) and (3.41), the formula of ρ provided by (3.39) gives ρ=ρ1+iρ2, with

ρ1:=18(3A1+C1+B2+3D2),ρ2:=-2c12q+18(-B1-3D1+3A2+C2).

It follows that we can derive the Poincaré normal form of the reduced equation on the local center manifold (3.38). Moreover, in Cartesian coordinates, (3.38) is

ξ˙1=-qξ2+(ρ1ξ1-ρ2ξ2)(ξ12+ξ22)+o(|ξ1|3+|ξ2|3),ξ˙2=qξ1+(ρ1ξ2+ρ2ξ1)(ξ12+ξ22)+o(|ξ1|3+|ξ2|3),

where

ξ=ξ1+iξ2.

In polar coordinates, set

r=ξ12+ξ22,θ=arctanξ2ξ1.

We have, as r0,

r˙=ρ1r3+o(r3),θ˙=q+ρ2r2+o(r2).(3.42)

Now it is clear to see from (3.42) that the origin 𝟎2 is asymptotically stable for (3.10) if ρ1<0 and is not stable if ρ1>0. From (2.1)–(2.3), (3.30)–(3.33) and Appendix A, we can obtain all the coefficients Ai,Bi,Ci,Di (i=1,2). Then, using Matlab, it follows that

ρ1:=18(3A1+C1+B2+3D2)=-0.008766<0.

A straightforward computation leads to the existence of C>0 such that, at least if r(0)[0,+) is small enough, one has for the solution to (3.42),

r(t)Cr(0)1+tr(0)2for all t[0,+),

which concludes the proof of Theorem 1.1.

Method 2: Lyapunov function.

Let us start with a formal motivation. Recall that, by (1.8) and with E defined in (1.7), we have, along the trajectories of (1.6),

E˙=-12K2,

with K:=yx(0). It is therefore natural to consider the following candidate for a Lyapunov function:

V:=E-μKK˙,

where μ>0 is small enough. Indeed, one then gets

V˙:=-12K2-μ(K˙)2-μKK¨,

and one may hope to absorb -μKK¨ with -12K2-μ(K˙)2 and get V˙<0 on G{0}, at least in a neighborhood of 0.

We follow this strategy together with the approximation of g previously found. For 𝐦=(m1,m2)Ω, let (see (3.16))

g~:=m12a+m1m2b+m22cC([0,L]),(3.43)y~:=m1φ1+m2φ2+g~C([0,L]),(3.44)

and

E~:=120Ly~2𝑑x.

Then, using (2.4), (2.5) and (3.26)–(3.28) (compare with (3.14)), one gets that, along the trajectories of (3.10), for 𝐦Ω and ψC3([0,L]) satisfying ψ(0)=ψ(L)=0, one has

-0L(ψ+ψ′′′)y~𝑑x+ψ(0)(m12a(0)+m1m2b(0)+m22c(0))-120Lψxy~2𝑑x+0L(m˙1φ1+m˙2φ2+g~m1m˙1+g~m2m˙2)ψ𝑑x=0L(y~t+y~x+y~xxx+y~y~x)ψ𝑑x=0L[m13(A1φ1+A2φ2-bc1+φ1a+aφ1)+m12m2(B1φ1+B2φ2+2ac1-b3c1-2cc1+φ1b+φ2a+aφ2+bφ1)+m1m22(C1φ1+C2φ2+2a3c1+bc1-2c3c1+φ1c+φ2b+bφ2+cφ1)+m23(D1φ1+D2φ2+b3c1+φ2c+cφ2)+o(|𝐦|3)]ψdxas |𝐦|0.(3.45)

Then, using (3.45) with ψ:=y~ (which, by (2.4), (2.5), (3.26)–(3.28), (3.43) and (3.44), satisfies ψ(0)=ψ(L)=0), along the trajectories of (3.10), we have from (2.6), (2.7), (3.15) and (3.29)–(3.33) that the right-hand side of (3.45) is o(|𝐦|4), and

E~˙=-12K~2+o(|𝐦|4)as |𝐦|0,

with K~:Ω defined by

K~:=a(0)m12+b(0)m1m2+c(0)m22.(3.46)

Let us emphasize that, even if “along the trajectories of (3.10)” might be misleading, E~˙ is just a function of 𝐦Ω. It is the same for V~˙, K~˙, K~¨ which appear below. Using (1.11) and (3.9), we have, along the trajectories of (3.10),

K~˙=qb(0)m12+2q(c(0)-a(0))m1m2-qb(0)m22+o(|𝐦|2).(3.47)

Using (3.9), we get the existence of C>0 such that, along the trajectories of (3.10),

|K~¨|C|𝐦|2for all 𝐦Ω.

We can now define our Lyapunov function V~. Let μ(0,14]. Let V~:Ω be defined by

V~:=E~-μK~K~˙.(3.48)

From (3.48), we have the existence of η0>0 such that, for every 𝐦2 satisfying |𝐦|<η0 and along the trajectories of (3.10),

V~˙=-12K~2-μ(K~˙)2-μK~K~¨+o(|𝐦|4)-14K~2-μ(K~˙)2+μ2(K~¨)2+o(|𝐦|4)-14K~2-μ(K~˙)2+2μ2C2|𝐦|4-μ(K~2+(K~˙)2-2μC2|𝐦|4).(3.49)

Let us assume for the moment that, for every 𝐦=(m1,m2)2,

{a(0)m12+b(0)m1m2+c(0)m22=0,qb(0)m12+2q(c(0)-a(0))m1m2-qb(0)m22=0𝐦=𝟎.(3.50)

Then, by homogeneity, there exists η1>0 such that

(a(0)m12+b(0)m1m2+c(0)m22)2+(qb(0)m12+2q(c(0)-a(0))m1m2-qb(0)m22)22η1|𝐦|4(3.51)

for all 𝐦=(m1,m2)2. From (3.46), (3.47) and (3.51), we get the existence of η2>0 satisfying

K~2+(K~˙)2η1|𝐦|4for all 𝐦2 such that |𝐦|<η2.(3.52)

From (3.49) and (3.52), we get the existence of η3>0 such that, for every μ(0,η3),

V~˙-μ2η1|𝐦|4for all 𝐦2 such that |𝐦|<η3.(3.53)

Moreover, straightforward estimates show that there exists η4>0 such that, for every μ(0,η4),

η4|𝐦|2V~1η4|𝐦|2for all 𝐦2 such that |𝐦|<η4,

which, together with (3.53), proves the existence of C>0 such that, at least if 𝐦02 is small enough, the solution to (3.10) satisfies

|𝐦(t)|C|𝐦0|1+t|𝐦0|2for all t0.

It only remains to prove (3.50). From Appendix A, one gets that c(0)0.01180, then (3.50) holds if m1=0. Let us now deal with the case m10. If we divide both polynomials in the two equations on the left-hand side of (3.50) by m12, then the two resulting polynomials have a common root if and only if their resultant is zero. This resultant is the determinant of the Sylvester matrix S:

S:=(c(0)b(0)a(0)00c(0)b(0)a(0)-b(0)-2(a(0)-c(0))b(0)00-b(0)-2(a(0)-c(0))b(0)).

Straightforward computations show that

det(S)=4a(0)3c(0)-a(0)2[b(0)2+8c(0)2]+a(0)[6b(0)2c(0)+4c(0)3]-b(0)2c(0)2-b(0)4.(3.54)

Correction added on 10 September 2018 after online publication: Mistakes within (3.54) have been corrected. This would not influence the later calculations since we use a software to calculate directly the value of the determinant det(S) rather than using the general formula.

From (3.54) and Appendix A (see in particular (A.7)–(A.9)), we have

det(S)-0.01970.

Hence, the two resulting polynomials do not have a common root. Thus, (3.50) is proved.

Remark 3.2.

It follows from our proof of Theorem 1.1 that the decay rate stated in (1.14) is optimal in the following sense: there exists ε>0 such that, for every y0G with y0L2(0,L)ε,

y(t,)L2(0,L)εy0L2(0,L)1+ty0L2(0,L)2.

For the Lyapunov approach, let us point out that, decreasing if necessary η3>0, one has, for every μ(0,η3),

V~˙-1η3|𝐦|4for all 𝐦2 such that |𝐦|<η3.

4 Conclusion and future works

In this article, we have proved that for the critical case of L=2π7/3, 0L2(0,L) is locally asymptotically stable for the KdV equation (1.6). First, we recalled that the equation has a two-dimensional local center manifold. Next, through a second-order power series approximation at 0M of the function g defining the local center manifold, we derived the local asymptotic stability of 0L2(0,L) on the local center manifold and obtained a polynomial decay rate for the solution to the KdV equation (1.6) on the center manifold.

Since the KdV equation (1.6) also has other (periodic) steady states than the origin (see Remark 1.2), it remains an open and interesting problem to consider the (local) stability property of these steady states for the KdV equation (1.6). Furthermore, it remains to consider all the other critical cases with a two-dimensional (local) center manifold as well as all the last remaining critical cases, i.e., when the equation has a (local) center manifold with a dimension larger than 2.

A On the solution a, b and c to equations (3.26), (3.27) and (3.28)

Set

f+(x):=a(x)+c(x),f-(x):=a(x)-c(x),(A.1)

and

{g+(x):=φ1(x)φ1(x)+φ2(x)φ2(x)+3c1φ1(x)-c1φ2(x),g-(x):=φ1(x)φ1(x)-φ2(x)φ2(x)-3c1φ1(x)-c1φ2(x),g(x):=φ1(x)φ2(x)+φ1(x)φ2(x)+c1φ1(x)-3c1φ2(x).(A.2)

First, adding each equation of (3.28) to the corresponding equation of (3.26), we have the following ODE for f+(x):

{f+′′′(x)+f+(x)+g+(x)=0,f+(0)=f+(L)=0,f+(L)=0.(A.3)

Second, subtracting each equation of (3.28) from the corresponding equation of (3.26), we obtain

{2qb(x)+f-(x)+f-′′′(x)+g-(x)=0,f-(0)=f-(L)=0,f-(L)=0,(A.4)

which gives

b(x)=-12q(f-(x)+f-′′′(x)+g-(x)).(A.5)

Substitute (A.5) into (3.27), then the following ODE for f-(x) is obtained:

{f-(6)(x)+2f-(4)(x)+f-′′(x)+4q2f-(x)+g-(x)+g-′′′(x)-2qg(x)=0,f-(0)=f-(L)=f-(L)=f-′′′(L)=0,f-(0)+f-′′′(0)=0,f-′′(L)+f-(4)(L)=0,(A.6)

where the boundary conditions follow from (2.4), (2.5), (3.27), (A.2), (A.4) and (A.5).

Employing the method of undetermined coefficients, we get the following (unique) solution to the nonhomogeneous ODE (A.3):

f+(x)=l=13C+lf+l(x)+c+11cos(121x)+c+12sin(121x)+c+21cos(321x)+c+31cos(421x)+c+32sin(421x)+c+41cos(521x)+c+42sin(521x)+c+51cos(621x)+c+61cos(921x),

where the fundamental solutions f+l(x), l=1,2,3, are

f+1(x)=1,f+2(x)=cos(x),f+3(x)=sin(x),

and the constants are

c+11=3c1Θ(121)-(121)3,c+12=-33c1Θ-(121)+(121)3,c+21=Θ21821(121)-(121)3,c+31=-2c1Θ(121)-(121)3,c+32=23c1Θ-(121)+(121)3,c+41=c1Θ(121)-(121)3,c+42=3c1Θ-(121)+(121)3,c+51=Θ21821(121)-(121)3,c+61=Θ2-1821(121)-(121)3,

and

C+l=det(A+l)det(A+),l=1,2,3.

Here,

A+=(f+1(0)f+2(0)f+3(0)f+1(L)f+2(L)f+3(L)f+1(L)f+2(L)f+3(L)),

and each A+l is the matrix formed by replacing the l-th column of A+ with a column vector -b+, where

b+=(b+1b+2b+3)𝖳,

and

b+1=c+11+c+21+c+31+c+41+c+51+c+61,b+2=c+11cos(121L)+c+12sin(121L)+c+21cos(321L)+c+31cos(421L)+c+32sin(421L)+c+41cos(521L)+c+42sin(521L)+c+51cos(621L)+c+61cos(921L),b+3=-121c+11sin(121L)+121c+12cos(121L)-321c+21sin(321L)-421c+31sin(421L)+421c+32cos(421L)-521c+41sin(521L)+521c+42cos(521L)-621c+51sin(621L)-921c+61sin(921L).

Similarly, the method of undetermined coefficients gives the following (unique) solution to the nonhomogeneous ODE system (A.6):

f-(x)=l=16C-lf-l(x)+c-11cos(121x)+c-12sin(121x)+c-21cos(221x)+c-31cos(421x)+c-32sin(421x)+c-41cos(521x)+c-42sin(521x)+c-51cos(821x)+c-61cos(1021x),

where the fundamental solutions f-l(x), l=1,,6, are

f-1(x)=eα1xcos(β1x),f-2(x)=eα1xsin(β1x),f-3(x)=e-α1xcos(β1x),f-4(x)=e-α1xsin(β1x),f-5(x)=cos(β2x),f-6(x)=sin(β2x),

with

α1=(20+57)13-7(20+57)-1327,β1=(20+57)13+7(20+57)-13221,β2=(20+57)13+7(20+57)-1321,

and the constants are

c-11=-3Θ240212+4qΘ2221+9qc1Θ(121)6-2(121)4+(121)2-4q2,c-12=-93qc1Θ(121)6-2(121)4+(121)2-4q2,c-21=3Θ218212-4qΘ23221(221)6-2(221)4+(221)2-4q2,c-31=3Θ2240212-4qΘ21221-6qc1Θ(421)6-2(421)4+(421)2-4q2,c-32=63qc1Θ(421)6-2(421)4+(421)2-4q2,c-41=-3Θ2600212+4qΘ23021-3qc1Θ(521)6-2(521)4+(521)2-4q2,c-42=-33qc1Θ(521)6-2(521)4+(521)2-4q2,c-51=3Θ22048212-4qΘ21621(821)6-2(821)4+(821)2-4q2,c-61=3Θ21250212+4qΘ2521(1021)6-2(1021)4+(1021)2-4q2,

and

C-l=det(A-l)det(A-),l=1,,6.

Here, the matrix A- is defined by

A-=(α-1α-2α-3α-4α-5α-6)

with

α-l=(f-l(0)f-l(L)f-l(L)f-l(0)+f-l′′′(0)f-l′′′(L)f-l′′(L)+f-l(4)(L))𝖳,l=1,,6.

Each A-l is the matrix formed by replacing the l-th column of A- with a column vector -b-, where

b-=(b-1b-2b-3b-4b-5b-6)𝖳

and

b-1=c-11+c-21+c-31+c-41+c-51+c-61,b-2=c-11cos(121L)+c-12sin(121L)+c-21cos(221L)+c-31cos(421L)+c-32sin(421L)+c-41cos(521L)+c-42sin(521L)+c-51cos(821L)+c-61cos(1021L),b-3=-121c-11sin(121L)+121c-12cos(121L)-221c-21sin(221L)-421c-31sin(421L)+421c-32cos(421L)-521c-41sin(521L)+521c-42cos(521L)-821c-51sin(821L)-1021c-61sin(1021L),b-4=202121c-12cos(121L)+202121c-32cos(421L)-2021c-42cos(521L),

b-5=-202121c-11sin(121L)+202121c-12cos(121L)-342121c-21sin(221L)-202121c-31sin(421L)+202121c-32cos(421L)+202121c-41sin(521L)-202121c-42cos(521L)+3442121c-51sin(821L)+7902121c-61sin(1021L),b-6=-20212c-11cos(121L)-20212c-12sin(121L)-68212c-21cos(221L)-80212c-31cos(421L)-80212c-32sin(421L)+100212c-41cos(521L)+100212c-42sin(521L)+2752212c-51cos(821L)+7900212c-61cos(1021L).

Therefore, we derive from (A.1) that

a(x)=12(f+(x)+f-(x))=12[l=13C+lf+l(x)+l=16C-lf-l(x)+(c+11+c-11)cos(121x)+(c+12+c-12)sin(121x)+c-21cos(221x)+c+21cos(321x)+(c+31+c-31)cos(421x)+(c+32+c-32)sin(421x)+(c+41+c-41)cos(521x)+(c+42+c-42)sin(521x)+c+51cos(621x)+c-51cos(821x)+c+61cos(921x)+c-61cos(1021x)](A.7)

and

c(x)=12(f+(x)-f-(x))=12[l=13C+lf+l(x)-l=16C-lf-l(x)+(c+11-c-11)cos(121x)+(c+12-c-12)sin(121x)-c-21cos(221x)+c+21cos(321x)+(c+31-c-31)cos(421x)+(c+32-c-32)sin(421x)+(c+41-c-41)cos(521x)+(c+42-c-42)sin(521x)+c+51cos(621x)-c-51cos(821x)+c+61cos(921x)-c-61cos(1021x)].(A.8)

From (A.5), we obtain

b(x)=-12q(f-(x)+f-′′′(x)+g-(x))=-12q[l=16C-lf-l(x)+l=16C-lf-l′′′(x)-(202121c-11+221Θ2+3c1Θ)sin(121x)+(202121c-12+33c1Θ)cos(121x)-(342121c-21+921Θ2)sin(221x)-(202121c-31+2c1Θ)sin(421x)+(202121c-32-23c1Θ)cos(421x)+(202121c-41+3021Θ2+c1Θ)sin(521x)-(202121c-42+3c1Θ)cos(521x)-1221Θ2sin(621x)+(8×432121c-51-1621Θ2)sin(821x)+(79021c-61-521Θ2)sin(1021x)].(A.9)

Acknowledgements

We would like to thank Shengquan Xiang for useful comments on a preliminary version of this article.

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About the article

Received: 2016-04-24

Revised: 2016-07-07

Accepted: 2016-08-30

Published Online: 2016-10-14

Published in Print: 2018-11-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11401021

Award identifier / Grant number: 11301387

Funding Source: European Research Council

Award identifier / Grant number: 266907 (CPDENL)

The authors were supported by European Research Council advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7). In addition, J. Chu was supported by the National Natural Science Foundation of China (no. 11401021) and the Doctoral Program of Higher Education (no. 20130006120011). P. Shang was supported by the National Natural Science Foundation of China (no. 11301387) and the Doctoral Program of Higher Education (no. 20130072120008).


Citation Information: Advances in Nonlinear Analysis, Volume 7, Issue 4, Pages 497–515, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0097.

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