Instability of the solitary waves for the generalized Boussinesq equations

Instability of the Solitary Waves for the Generalized Boussinesq Equations

Article  in  SIAM Journal on Mathematical Analysis · January 2020

DOI: 10.1137/18M1199198

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INSTABILITY OF THE SOLITARY WAVES FOR THE GENERALIZED BOUSSINESQ EQUATIONS

BING LI, MASAHITO OHTA, YIFEI WU, AND JUN XUE*

Abstract. In this work, we consider the generalized Boussinesq equation

∂²_tu−∂_x²u+∂_x²(∂_x²u+|u|^pu) = 0, (t, x)∈R×R,

with 0 < p < ∞. This equation has the traveling wave solutions φω(x−ωt), with the frequencyω∈(−1,1) andφω satisfying

−∂xxφω+ (1−ω²)φω−φ^p+1_ω = 0.

Bona and Sachs [2] proved that the traveling wave φω(x−ωt) is orbitally stable when 0< p <4, ^p₄ < ω²<1. Liu [9] proved the orbital instability under the conditions 0< p <4, ω²< ^p₄ or p≥4, ω²<1. In this paper, we prove the orbital instability in the degenerate case 0< p <4, ω²= ^p₄ .

1. Introduction

In this paper, we consider the stability theory of the generalized Boussinesq equation

∂_t²u−∂_x²u+∂_x²(∂_x²u+|u|^pu) = 0, (t, x)∈R×R, (1.1) with the initial data

u(0, x) =u0(x), ut(0, x) = u1(x). (1.2) Here 0< p <∞.

The Boussinesq equation was originally derived by Boussinesq [3]. It arises from studying an approximation to the evolution of the free surface of a water wave.

Equation (1.1) has the solitary wave solutionu(x, t) =φω(x−ωt), whereφωis the ground state solution of the following elliptic equation

−∂xxφω+ (1−ω²)φω−φ^p+1_ω = 0, |ω|<1. (1.3) The ground state solutionφ_ω is an even function and it has the property of exponential decay, that is, |φ_ω| ≤C₁e^−C2|x| for some C₁, C₂ >0 and |∂_xφ_ω| ≤C₃e^−C4|x| for some C₃, C₄ >0.

2010Mathematics Subject Classification. Primary 35B35; Secondary 35L70.

Key words and phrases. generalized Boussinesq equation, instability, critical frequency, traveling wave.

Funding: The work of the first and third authors was partially supported by National Natural Science Foundation of China grants 11771325 and 11571118. The work of the second author was partially supported by JSPS KAKENHI grant 15K04968. The work of the third author was also partially supported by National Youth Topnotch Talent Support Program in China. The work of the fourth author was partially supported by Research Council of Norway (No. 250070).

* Corresponding author.

SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Pages 3192–3221. This paper is the final version.

1

Equation (1.1) has the equivalent system form u_t=v_x,

vt= (−uxx+u− |u|^pu)x. (1.4) Then the system (1.4) has the following solitary wave solution

u v

(t, x) =

φ_ω(x−ωt)

−ωφ_ω(x−ωt)

.

For the H¹ ×L²-solution (u, v)^T of (1.1)–(1.2), the momentum Q and the energy E are conserved under the flow, where

Q u

v

= Z

R

uvdx; (1.5)

E u

v

= 1 2

Z

R

(|u_x|²+|u|²+|v|²) dx− 1 p+ 2

Z

R

|u|^p+2dx. (1.6) There are several related results for the generalized Boussinesq equation. For a local existence result, Liu [9] proved that the system (1.4) is locally well-posed inH¹(R)×L²(R).

For the stability theories, Bona and Sachs [2] proved that when 0 < p <4, ^p₄ < ω² <1, the solitary wave solution is orbitally stable. Liu [9] proved the orbital instability if 0< p <4 and ω² < ^p₄ or p≥ 4 and ω² <1. Liu [10] proved that when the wave speed ω = 0, the solitary wave solution is strongly unstable by blow-up. Liu, Ohta, and Todorova [11] showed that when 0< p <∞ and 0<2(p+ 2)ω² < p, the solitary wave solution is strongly unstable by blow-up. For the abstract Hamiltonian systems, we refer the readers to Grillakis, Shatah, and Strauss [5, 6] for the stability/instability theories, in which the Vakhitov-Kolokolov stability criteria of the solitary waves were confirmed except the degenerate cases. In the degenerate cases, it was also proved by Comech and Pelinovsky [4] (see also [14]) that the solitary wave solution is orbitally unstable under some regularity restrictions in the nonlinearity (for example, p should be suitably large in our cases). In this paper, we consider the stability theory on the solitary wave solutions of the generalized Boussinesq equation and aim to show its instability in the degenerate cases without any regularity restriction. It is worth noting that none of the above two frameworks of Grillakis, Shatah and Strauss [5, 6] and Comech and Pelinovsky [4] are available in our cases, either because of the degeneration or because of insufficient regularity of the nonlinearity.

Before starting our theorem, we give some definitions. Let v₀ = Rx

−∞u₁(y) dy, ~u = (u, v)^T,~u₀ = (u₀, v₀)^T, and −→

Φ_ω = (φ_ω,−ωφ_ω)^T. For ε >0, we denote the set U_ε −→ Φ_ω

as U_ε −→

Φ_ω

={~u∈H¹(R)×L²(R) : inf

y∈R

k~u−−→

Φ_ω(· −y)k_H¹_×L² < ε}. (1.7) Definition 1.1. We say that the solitary wave solutionφ_ω(x−ωt)of (1.1)is orbitally stable if for any ε >0, there exists δ >0 such that if k~u0 −−→

Φωk_H¹×L² < δ, then the solution ~u(t) of (1.1) with ~u(0) = ~u₀ exists for all t ∈ R, and ~u(t) ∈ U_ε −→

Φ_ω

for all t ∈ R. Otherwise, φω(x−ωt) is said to be orbitally unstable.

Then the main result in the present paper is the following.

Theorem 1.2. Let 0 < p < 4, ω ∈ (−1,1) and φ_ω be the solution of (1.3). If |ω| = p_p

4, then the solitary wave solution φ_ω(x−ωt) is orbitally unstable.

The main method that we use in the present paper is from [19], in which the instability of the standing wave solutions of the Klein-Gordon equation in the degenerate cases was proved. Instead of construction of the Lyapunov functional, the argument in [19] is to use the monotonicity of the virial quantity to control the modulations. However, the details of this argument depend sensitively on the problem, and the key ingredients of our proof are the following.

(1) The nonstandard modulation and coercivity properties are given. More precisely, define the functionalS_ω as

S_ω(~u) = E(~u) +ωQ(~u).

Inspired by [12, 13, 18], we establish the following nonstandard coercivity properties. We prove the existence of suitable directions −→

Γ_ω,−→

Ψ_ω ∈ H¹(R)×L²(R) such that the following coercivity properties hold. Suppose that ~η ∈H¹(R)×L²(R) satisfies

D

~ η,−→

Γ_ωE

=D

~ η,−→

Ψ_ωE

= 0;

then

D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

&

~η

2 H¹×L². The choices of −→

Γ_ω,−→

Ψ_ω play important roles in our estimation. −→

Ψ_ω can be regarded as the negative direction, which satisfies

S_ω⁰⁰ −→ Φ_ω−→

Ψ_ω,−→ Ψ_ω

< 0. However, we remark that −→ Γ_ω ∈/ Ker(S_ω⁰⁰(−→

Φ_ω)), which is much different from the standard. Moreover, by suitably setting the translation and scaling parametersy, λ, we can establish the modulation by writing

~ u=

~

η+−−→

Φ_λ(t)

(· −y(t))

such that ~η verifies similar orthogonal conditions above (by replacing −→ Γ_ω,−→

Ψ_ω with −→ Γ_λ,−→

Ψ_λ, respectively).

(2) A subtle control on the modulated translation parameters is obtained. Instead of the rough control of the modulation parameter y as ˙y−λ = O(k~ηk_H¹×L²), we obtain the following finer estimate:

˙

y−λ=kφλk⁻²_L2

h Q −→

Φλ

−Q −→ Φω

i

− kφλk⁻²_L2

h

Q(~u0)−Q −→ Φω

i

+O(k~ηk²_H1×L²).

The subtle estimate benefits from the choices of −→ Γ_ω,−→

Ψ_ω in the first step and the dynamic of the solution. This estimate has great effects when we set up the structure of virial identity I⁰(t) in the following.

(3) The monotonicity of the virial quantity is constructed. The key ingredient here is to suitably define a quantity I(t) and obtain its monotonicity. To this end, the crucial issue is to prove the following structure ofI⁰(t) as

I⁰(t) = ρ(~u₀) +h(λ) +R(~u), where

ρ(~u₀)≥C₁a, C₁ >0;

h(λ)≥C2(λ−ω)²+C3a(λ−ω)²+o(λ−ω)², C2 >0, C3 >0,

and R(~u) is an easy remainder term which can be dominated by ρ and h. Here a is the difference between the initial data and the soliton. The obstacles in the proof come from

nonconservation terms among I⁰(t) and how to eliminate the first-order terms about ~η and λ. These make much technical complexity. By a delicate analysis and the utilization of the estimates above, we overcome all difficulties and finally obtain the monotonicity of I(t).

The rest of the paper is organized as follows. In Section 2, we give some preliminaries.

In Section 3, we show the coercivity property of the Hessian S_ω⁰⁰ −→ Φ_ω

. In Section 4, we show the existence of modulation parameters. In Section 5, we control the modulation parameters obtained in Section 4. In Section 6, we show the localized virial identities. Finally, we prove the main theorem in Section 7.

2. Preliminary 2.1. Notations. Forf, g ∈L²(R) = L²(R,R), we define

hf, gi= Z

R

f(x)g(x) dx

and regard L²(R) as a real Hilbert space. Similarly, forf , ~~ g ∈ L²(R)2

= L²(R,R)2

, we define

hf , ~~ gi= Z

R

f~(x)^T ·~g(x) dx.

For a function f(x), its L^q-norm kfk_L^q = Z

R

|f(x)|^qdx¹_q

and its H¹-norm kfk_H¹ = (kfk²_L2 +k∂_xfk²_L2)¹². For f~= (f, g)^T, its H¹×L²-norm kfk~ _H¹_×L² = (kfk²_H1 +kgk²_L2)¹².

Further, we write X .Y or Y & X to indicate X ≤CY for some constant C >0. We use the notation X ∼ Y to denote X .Y .X. We also use O(Y) to denote any quantity X such that |X|.Y and use o(Y) to denote any quantity X such thatX/Y →0 ifY →0.

Throughout the whole paper, the letter C will denote various positive constants which are of no importance in our analysis.

2.2. Some basic definitions and properties. In the rest of this paper, we consider the case of 0< p < 4, and ω_c =pp

4, ω =±ω_c. Let ~u = (u, v)^T, −→

Φ_ω = (φ_ω,−ωφ_ω)^T. Recall the conserved equalities,

Q(~u) = Z

R

uvdx, E(~u) = 1

2(kuk²_L2 +ku_xk²_L2 +kvk²_L2)− 1

p+ 2kuk^p+2_Lp+2. First, we give some basic properties on the momentum and energy.

Lemma 2.1. Let |ω|=p_p

4; then the following equality holds:

∂λQ −→ Φλ

λ=ω = 0.

Proof. Note that for λ∈(−1,1), we have Q −→ Φ_λ

=−λkφ_λk²_L2. (2.1)

By rescaling, we find

φ_λ(x) = (1−λ²)^1pφ₀√

1−λ²x

. (2.2)

This implies that

Q −→ Φ_λ

=−λ(1−λ²)^2p−12kφ₀k²_L2. By a straightforward computation, we have

∂λQ −→ Φλ

=−(1−λ²)^2p−32

1− 4 pλ²

kφ0k²_L2.

Finally, we substitute λ² = ^p₄ into the equality above and thus complete the proof.

Now we define the functional S_ω as

S_ω(~u) = E(~u) +ωQ(~u). (2.3)

Then we have

Q⁰(~u) = v

u

, (2.4)

E⁰(~u) =

−∂_xxu+u− |u|^pu v

, (2.5)

S_ω⁰(~u) =

−u_xx+u− |u|^pu+ωv v +ωu

. Note thatS_ω⁰ −→

Φω

=~0. Moreover, for the real-valued vectorf~= (f, g)^T, a direct computation shows

S_ω⁰⁰ −→ Φ_ωf~=

−∂_xxf +f−(p+ 1)φ^p_ωf +ωg g+ωf

, (2.6)

and for any vector ~ξ, ~η,

D S_ω⁰⁰ −→

Φ_ωξ, ~~ ηE

=D S_ω⁰⁰ −→

Φ_ω

~ η, ~ξE

. Moreover, taking the derivative of S_ω⁰ −→

Φ_ω

=~0 with respect to ω gives S_ω⁰⁰ −→

Φ_ω

∂_ω−→

Φ_ω =−Q⁰ −→ Φ_ω

. (2.7)

Then a consequence of Lemma 2.1 is

Corollary 2.2. Let λ∈(−1,1), |ω|=ω_c; then S_λ −→

Φ_λ

−S_λ −→ Φ_ω

=o (λ−ω)² . Proof. From the definition ofSω(~u) in (2.3), we have

Sλ

−→ Φλ

−Sλ

−→ Φω

=Sω

−→ Φλ

−Sω

−→ Φω

+ (λ−ω)

Q −→ Φλ

−Q −→ Φω

. Recall that S_ω⁰ −→

Φ_ω

=~0; then we use Taylor’s expansion to calculate S_λ −→

Φ_λ

−S_λ −→ Φ_ω

=1 2 D

S_ω⁰⁰ −→ Φ_ω−→

Φ_λ−−→ Φ_ω

,−→ Φ_λ−−→

Φ_ωE + (λ−ω)

Q −→ Φ_λ

−Q −→ Φ_ω

+o (λ−ω)²

. (2.8)

Note that −→

Φλ−−→

Φω = (λ−ω)∂ω

−→

Φω+o(λ−ω);

then we find

D S_ω⁰⁰ −→

Φ_ω−→ Φ_λ−−→

Φ_ω ,−→

Φ_λ−−→ Φ_ωE

=(λ−ω)²D S_ω⁰⁰ −→

Φ_ω

∂_ω−→ Φ_ω, ∂_ω−→

Φ_ωE

+o (λ−ω)²

=−(λ−ω)²D Q⁰ −→

Φ_ω , ∂_ω−→

Φ_ωE

+o (λ−ω)²

=−(λ−ω)²∂_λQ −→ Φ_λ

λ=ω+o (λ−ω)² ,

where we have used equality (2.7) in the second step. Using Lemma 2.1, we have

∂_λQ −→ Φ_λ

λ=ω

= 0.

Hence,

Q −→ Φ_λ

−Q −→ Φ_ω

=o λ−ω ,

and D

S_ω⁰⁰ −→ Φ_ω−→

Φ_λ−−→ Φ_ω

, −→

Φ_λ −−→ Φ_ω

E

=o (λ−ω)² .

Taking these two results into (2.8), we obtain the desired estimate.

3. Coercivity

In this section, we prove a coercivity property on the Hessian of the action S_ω⁰⁰ −→ Φ_ω

. First, we study the kernel of S_ω⁰⁰ −→

Φω

in the following lemma. The proof is standard, and it is a consequence of the result from [17].

Lemma 3.1. The kernel of S_ω⁰⁰ −→ Φ_ω

satisfies that Ker

S_ω⁰⁰ −→ Φ_ω

=

C∂_x−→

Φ_ω :C ∈R . Proof. First, we need to show the relationship “⊃”. For any f~∈

C∂_x−→

Φ_ω : C ∈ R , using (1.3), we have

S_ω⁰⁰ −→

Φ_ωf~=S_ω⁰⁰ −→ Φ_ω

C∂_x−→ Φ_ω

=C

∂_x −∂_xxφ_ω+ (1−ω²)φ_ω−φ^p+1_ω

−ωφ⁰_ω+ωφ⁰_ω

=~0. (3.1) Then (3.1) implies thatf~is in the kernel of S_ω⁰⁰ −→

Φω

, and we have the conclusion Ker

S_ω⁰⁰ −→ Φ_ω

⊃

C∂_x−→

Φ_ω :C ∈R .

Second, we prove the reverse relationship “⊂”. For any f~ ∈ Ker S_ω⁰⁰(−→

Φ_ω)

, by the expression of S_ω⁰⁰ −→

Φ_ω

in (2.6), we have

(−∂_xxf+ (1−ω²)f−(p+ 1)φ^p_ωf = 0,

g+ωf = 0. (3.2)

By the work of Weinstein [17], the only solutions to (3.2) are f =C∂xφω,

g =−Cω∂_xφ_ω, C ∈R. This implies that f~∈

C∂_x−→

Φ_ω :C ∈R , and we have Ker

S_ω⁰⁰ −→ Φ_ω

⊂

C∂_x−→

Φ_ω :C ∈R . Finally, combining the two relationship gives us

Ker S_ω⁰⁰ −→

Φ_ω

=

C∂_x−→

Φ_ω :C ∈R .

This gives the proof of the lemma.

The second lemma is the uniqueness of the negative eigenvalue of S_ω⁰⁰ −→ Φ_ω

. Lemma 3.2. S_ω⁰⁰(−→

Φ_ω) exists only one negative eigenvalue.

Proof. It is known that the operator−∂_xx+ (1−ω²)−(p+ 1)φ^p_ω has only one negative eigen- value (see [17]), and we denote it by λ−1. Then there exists a unique associated eigenvector ζ ∈H¹(R) such that

−∂_xxζ+ (1−ω²)ζ−(p+ 1)φ^p_ωζ =λ−1ζ. (3.3) Using the expression of S_ω⁰⁰(−→

Φ_ω) in (2.6), we have D

S_ω⁰⁰(−→ Φ_ω)−→

Φ_ω,−→ Φ_ωE

= Z

R

(−∂_xxφ_ω+φ_ω−(p+ 1)φ_ω^p+1−ω²φ_ω,−ωφ_ω+ωφ_ω)·

φω

−ωφω

dx

=−pkφ_ωk^p+2_Lp+2 <0.

This implies thatS_ω⁰⁰(−→

Φω) has at least one negative eigenvalue, say, µ0. Assume its associated eigenvector ~η0 = (ξ0, η0)^T, that is,

S_ω⁰⁰(−→

Φ_ω)~η₀ =µ₀~η₀. Using the expression of S_ω⁰⁰(−→

Φ_ω) in (2.6) again, the last equality yields −∂_xxξ₀+ξ₀ −(p+ 1)φ^p_ωξ₀+ωη₀ =µ₀ξ₀,

η₀+ωξ₀ =µ₀η₀. From the second equality, we haveη₀ =−_1−µ^ω

0ξ₀. Then we substitute it into the first equality to get

−∂_xxξ₀+ (1−ω²)ξ₀−(p+ 1)φ^p_ωξ₀ =µ₀ ω²

1−µ₀ + 1 ξ₀.

Hence, by (3.3), (µ₀, ~η₀) is exactly the pair satisfying µ₀ = 1

2 λ−1+ω²+ 1− q

λ²₋₁+ 2(ω²−1)λ−1+ (ω²+ 1)²

!

, ~η₀ =



 ζ ωζ µ0−1



. (3.4) This implies that S_ω⁰⁰(−→

Φ_ω) has exactly one simple negative eigenvalue. This completes the

proof of Lemma 3.2.

The next lemma gives one of the negative direction of S_ω⁰⁰ −→ Φ_ω

. Lemma 3.3. Let

ψ~_ω = 1 2ω

∂_ωφ_ω

−ω∂_ωφ_ω

, −→ Ψ_ω =

φ_ω 0

. Then

S_ω⁰⁰ −→

Φ_ωψ~_ω =−→

Ψ_ω. (3.5)

Moreover, we have

D S_ω⁰⁰(−→

Φ_ω)ψ~_ω, ~ψ_ωE

<0.

Proof. Taking the derivative of (1.3) with respect to ω, we have

−∂xx(∂ωφω) + (1−ω²)∂ωφω−(p+ 1)φ^p_ω∂ωφω = 2ωφω. (3.6) Using the expression of S_ω⁰⁰ −→

Φ_ω

in (2.6), we have S_ω⁰⁰ −→

Φ_ωψ~_ω = 1 2ω

−∂xx(∂ωφω) + (1−ω²)∂ωφω−(p+ 1)φ^p_ω∂ωφω

0

. This combined with (3.6) gives

S_ω⁰⁰ −→ Φ_ωψ~_ω =

φ_ω 0

=−→

Ψ_ω. (3.7)

Now we show D S_ω⁰⁰ −→

Φ_ωψ~_ω, ~ψ_ωE

<0. From (3.7), we have D

S_ω⁰⁰ −→

Φ_ωψ~_ω, ~ψ_ωE

=D−→ Ψ_ω, ~ψ_ωE

= Z

R

(φ_ω,0)· 1 2ω

∂_ωφ_ω

−ω∂_ωφ_ω

dx

= 1 2ω

Z

R

φ_ω∂_ωφ_ωdx= 1

4ω∂_ωkφ_ωk²_L2. (3.8) Note that, by (2.2),

kφ_ωk²_L2 = (1−ω²)^2p−12kφ₀k²_L2. Hence,

∂_ωkφ_ωk²_L2 =− 4

p −1 ω

1−ω²kφ_ωk²_L2 <0.

This completes the proof.

Now we prove the following coercivity property.

Proposition 3.4. Let |ω|<1. Suppose that ~η= (ξ, η)^T ∈H¹(R)×L²(R) satisfies D

~η, ∂_x−→ Φ_ωE

=D

~ η,−→

Ψ_ωE

= 0, (3.9)

where −→

Ψ_ω = (φ_ω,0)^T. Then D

S_ω⁰⁰ −→ Φω

~η, ~η E

&k~ηk²_H1×L². Proof. From the expression of S_ω⁰⁰ −→

Φ_ω

in (2.6), we can write S_ω⁰⁰ −→ Φ_ω

as S_ω⁰⁰ −→

Φ_ω

=L+V, whereL=

−∂_xx+ 1 ω

ω 1

, andV =

−(p+ 1)φ^p_ω 0

0 0

. HenceV is a compact perturbation of the self-adjoint operator L.

Step 1. Analyse the spectrum of S_ω⁰⁰ −→ Φ_ω

.

We first compute the essential spectrum of L. Note that for anyf~= (f, g)^T ∈H¹(R)× L²(R),

hL ~f , ~fi=D

−∂_xx+ 1 ω

ω 1

f g

,

f g

E

= Z

R

(−∂_xxf+f +ωg, ωf +g)· f

g

dx

=k∂_xfk²_L2 +kfk²_L2+ 2ωhf, gi+kgk²_L2

=kf~k²_H1×L² + 2ωhf, gi. (3.10) For the term 2ωhf, gi, applying H¨older’s and Young’s inequalities, we have

|2ωhf, gi| ≤ |ω|kfk~ ²_H1×L². Taking this estimate into (3.10), we have

hL ~f , ~fi ≥(1− |ω|)kf~k²_H1×L². Since |ω|<1, we get

hL ~f , ~fi&kf~k²_H1×L².

This means that there exists δ > 0 such that the essential spectrum of L is [δ,+∞). By Weyl’s Theorem,S_ω⁰⁰ −→

Φω

andLshare the same essential spectrum. So we obtain the essential spectrum of S_ω⁰⁰ −→

Φ_ω

. Recall that we have obtained the only one negative eigenvalue µ₀ of S_ω⁰⁰ −→

Φ_ω

in Lemma 3.2 and the kernel of S_ω⁰⁰ −→ Φ_ω

in Lemma 3.1. So the discrete spectrum of S_ω⁰⁰ −→

Φ_ω

is µ₀, 0, and the essential spectrum is [δ,+∞).

Step 2. Positivity.

The argument here is inspired by [1, 8]. By Lemma 3.2, we have the unique negative eigenvalueµ₀and eigenvector~η₀ ofS_ω⁰⁰ −→

Φ_ω

. For convenience, we normalize the eigenvector~η₀ such thatk~η₀k_L²_×L² = 1. Hence, for vector~η∈H¹(R)×L²(R), by the spectral decomposition theorem we can write the decomposition of ~η along the spectrum of S_ω⁰⁰ −→

Φ_ω ,

~

η =a_η~η₀+b_η∂_x−→ Φ_ω+~g_η,

where a_η, b_η ∈ R, ∂_x−→

Φ_ω ∈ Ker S_ω⁰⁰ −→

Φ_ω

and ~g_η lies in the positive eigenspace of S_ω⁰⁰ −→ Φ_ω

, that is,~g_η satisfies

h~g_η, ~η₀i=h~g_η, ∂_x−→ Φ_ωi= 0, and there exists an absolute constant σ >0 such that

D S_ω⁰⁰ −→

Φ_ω

~ g_η, ~g_ηE

≥σk~g_ηk²_L2×L². (3.11) Since ~η satisfies the orthogonality condition D

~ η, ∂_x−→

Φ_ωE

= 0 in (3.9) and

~

η₀, ∂_x−→ Φ_ω

= 0, we havebη = 0, and thus

~

η=a_η~η₀+~g_η. (3.12)

Substituting (3.12) into D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

, we get D

S_ω⁰⁰ −→ Φω

~η, ~η E

= D

S_ω⁰⁰ −→ Φω

(aη~η0+~gη), aη~η0+~gη

E

=a²_η D

S_ω⁰⁰ −→ Φω

~η0, ~η0

E

+ 2µ0aη

D

~gη, ~η0

E +

D S_ω⁰⁰ −→

Φω

~gη, ~gη

E . Due to the orthogonality property of eigenvector h~gη, ~η0i= 0, we have

D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

=a²_ηD S_ω⁰⁰ −→

Φ_ω

~ η₀, ~η₀E

+D S_ω⁰⁰ −→

Φ_ω

~g_η, ~g_ηE

=µ₀a²_ηh~η₀, ~η₀i+D S_ω⁰⁰ −→

Φ_ω

~g_η, ~g_ηE

=µ₀a²_η +D S_ω⁰⁰ −→

Φ_ω

~ g_η, ~g_ηE

. (3.13)

To ψ~_ω, by spectral decomposition theorem again, we may write ψ~_ω =a~η₀+b∂_x−→

Φ_ω+~g, wherea, b∈R,and~glies in the positive eigenspace ofS_ω⁰⁰ −→

Φ_ω

. We note thatD

ψ~_ω, ∂_x−→ Φ_ωE

= 0.

Indeed, since φ_ω is an even function, we have that ∂_ωφ_ω is even and ∂_xφ_ω is odd. Hence, we get

Dψ~_ω, ∂_x−→ Φ_ωE

= 1 +ω² 2ω

Z

R

∂_ωφ_ω∂_xφ_ωdx= 0.

Then b= 0, and thus

ψ~_ω =a~η₀+~g.

Therefore, a similar computation as above shows that D

S_ω⁰⁰ −→ Φω

ψ~ω, ~ψω

E

= D

S_ω⁰⁰ −→ Φω

(a~η0 +~g), a~η0+~g E

= D

S_ω⁰⁰ −→ Φω

(a~η0), a~η0

E +

D S_ω⁰⁰ −→

Φω

~g, ~g E

=µ0a²+ D

S_ω⁰⁰ −→ Φω

~g, ~g E

. For convenience, let −δ₀ = D

S_ω⁰⁰ −→

Φ_ωψ~_ω, ~ψ_ωE

. Then by Lemma 3.3, we know that δ₀ > 0.

Moreover, we have

−δ₀ =µ₀a²+D S_ω⁰⁰ −→

Φ_ω

~g, ~gE

. (3.14)

Using the orthogonality assumption D

~ η,−→

Ψ_ωE

= 0 in (3.9) and (3.5), we have 0 =D

~ η,−→

Ψ_ωE

=D

a_η~η₀+~g_η, S_ω⁰⁰ −→ Φ_ωψ~_ωE

=D

a_η~η₀+~g_η, S_ω⁰⁰ −→ Φ_ω

(a~η₀+~g)E

=D

a_η~η₀, S_ω⁰⁰ −→ Φ_ω

(a~η₀)E +D

~g_η, S_ω⁰⁰ −→ Φ_ω

~gE

=µ₀aa_ηh~η₀, ~η₀i+D S_ω⁰⁰ −→

Φ_ω

~g, ~g_ηE

=µ₀aa_η+D S_ω⁰⁰ −→

Φ_ω

~g, ~g_ηE . So we get the equality

0 = µ₀aa_η+D S_ω⁰⁰ −→

Φ_ω

~g, ~g_ηE . By the Cauchy-Schwarz inequality, we have

(µ₀aa_η)² =D S_ω⁰⁰ −→

Φ_ω

~g, ~g_ηE2

≤D S_ω⁰⁰ −→

Φ_ω

~g, ~gED S_ω⁰⁰ −→

Φ_ω

~g_η, ~g_ηE . This gives

(−µ₀a²)(−µ₀a²_η)≤D S_ω⁰⁰ −→

Φ_ω

~ g, ~gED

S_ω⁰⁰ −→ Φ_ω

~ g_η, ~g_ηE

. (3.15)

The last equality combining with (3.14) implies that

−µ₀a²_η ≤ D

S_ω⁰⁰ −→ Φ_ω

~ g, ~gED

S_ω⁰⁰ −→ Φ_ω

~ g_η, ~g_ηE

−µ₀a² = D

S_ω⁰⁰ −→ Φ_ω

~ g, ~gED

S_ω⁰⁰ −→ Φ_ω

~ g_η, ~g_ηE D

S_ω⁰⁰ −→ Φ_ω

~g, ~gE +δ₀

, that is,

µ₀a²_η ≥ − D

S_ω⁰⁰ −→ Φ_ω

~g, ~gED S_ω⁰⁰ −→

Φ_ω

~g_η, ~g_ηE D

S_ω⁰⁰ −→ Φ_ω

~g, ~gE +δ₀

. (3.16)

Inserting (3.16) into (3.13), we obtain D

S_ω⁰⁰ −→ Φ_ω

~ η, ~ηE

≥ 1−

D S_ω⁰⁰ −→

Φ_ω

~g, ~gE D

S_ω⁰⁰ −→ Φ_ω

~ g, ~gE

+δ₀ D

S_ω⁰⁰ −→ Φ_ω

~ g_η, ~g_ηE

= δ₀

D S_ω⁰⁰ −→

Φ_ω

~g, ~gE +δ₀

D S_ω⁰⁰ −→

Φ_ω

~g_η, ~g_ηE . Recalling that~g_η satisfies (3.11), we have

D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

≥ δ₀σ D

S_ω⁰⁰ −→ Φ_ω

~g, ~gE +δ₀

k~g_ηk²_L2×L², σ >0. (3.17) From the expression of ~η in (3.12) and the inequality (3.16), we have

k~ηk²_L2×L² =ka_η~η₀+~g_ηk²_L2×L² =a²_η+k~g_ηk²_L2×L²

≤ − D

S_ω⁰⁰ −→ Φ_ω

~g, ~gE µ₀δ₀

D S_ω⁰⁰ −→

Φω

~η, ~η E

+k~gηk²_L2×L²

.D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

. Therefore, this gives

D S_ω⁰⁰ −→

Φ_ω

~η, ~ηE

&

~η

2

L²×L². (3.18)

To obtain the final conclusion, we still need to estimate D

S_ω⁰⁰ −→ Φ_ω

~ η, ~ηE

&k~ηk²_H1×L². Using the expression of S_ω⁰⁰ −→

Φ_ω) in (2.6), we have hS_ω⁰⁰ −→

Φω

~η, ~ηi= Z

R

(−∂xxξ+ξ−(p+ 1)φ^p_ωξ+ωη, η+ωξ)· ξ

η

dx

=k∂_xξk²_L2 +k~ηk²_L2×L² + 2ω Z

R

ξηdx−(p+ 1) Z

R

|φ_ω|^pξ²dx.

Thus by H¨older’s and Young’s inequalities and (3.18), we get k∂_xξk²_L2 =D

S_ω⁰⁰ −→ Φ_ω

~ η, ~ηE

−2ω Z

R

ξηdx+ (p+ 1) Z

R

|φ_ω|^pξ²dx− k~ηk²_L2×L²

≤D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

+ 2|ω|kξk_L²kηk_L² + (p+ 1)kφ_ωk^p_L∞kξk²_L2

≤D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

+

|ω|+ (p+ 1)kφ_ωk^p_L∞

k~ηk²_L2×L²

.D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

+k~ηk²_L2×L² .D S_ω⁰⁰ −→

Φ_ω

~ η, ~ηE

. (3.19)

Therefore, together (3.18) and (3.19), we obtain

k~ηk²_H1×L² =k∂_xξk²_L2 +k~ηk²_L2×L² .D S_ω⁰⁰ −→

Φ_ω

~η, ~ηE .

Thus we obtain the desired result.

Applying Proposition 3.4, we obtain the following corollary, which is the nonstandard coercivity property and one of the key ingredients in our proof. Corollary 3.5 shows that we can replace the element∂x

−→

Φω in the orthogonal condition (3.20) by a suitably defined vector

−→

Γ_ω. The new orthogonal condition D

~ η,−→

Γ_ωE

= 0 has an essential effect on the estimates of the translation parameter y and λ in Section 5.

Corollary 3.5. Let |ω|<1. Suppose that ~η∈H¹(R)×L²(R) satisfies D

~ η,−→

Γω

E

= D

~ η,−→

Ψω

E

= 0, (3.20)

where −→

Γ_ω ∈H¹(R)×L²(R) and ∂_x−→ Γ_ω =−→

Ψ_ω = (φ_ω,0)^T. Then D

S_ω⁰⁰ −→ Φ_ω

~ η, ~ηE

&

~η

2

H¹×L². (3.21)

Proof. We define

ξ~=~η+b∂_x−→

Φ_ω, ~ξ∈H¹(R)×L²(R).

If we choose

b=− D

~ η, ∂_x−→

Φ_ωE k∂_x−→

Φ_ωk²_L2×L²

, then

D~ξ, ∂_x−→ Φ_ωE

= 0.

Moreover, by (3.20), we have Dξ,~ −→

Ψ_ωE

=D

~

η+b∂_x−→ Φ_ω,−→

Ψ_ωE

=D

~ η,−→

Ψ_ωE +bD

∂_x−→ Φ_ω,−→

Ψ_ωE

. (3.22)

Note that bD

∂_x−→ Φ_ω,−→

Ψ_ωE

=b Z

R

∂_xφ_ω,(−ω)∂_xφ_ω

· φ_ω

0

dx=b Z

R

∂_xφ_ωφ_ωdx= 0.

Hence, D

~ξ,−→ Ψ_ωE

= 0. Therefore, ξ~satisfies the orthogonality condition (3.9) in Proposition 3.4. Then using the conclusion of Proposition 3.4 and S_ω⁰⁰(−→

Φ_ω)∂_x−→

Φ_ω =~0, we get D

S_ω⁰⁰(−→ Φ_ω)~η, ~ηE

=D S_ω⁰⁰(−→

Φ_ω)

~ξ−b∂_x−→ Φ_ω

,

~ξ−b∂_x−→ Φ_ωE

=D S_ω⁰⁰(−→

Φ_ω)~ξ, ~ξE

−2bD S_ω⁰⁰(−→

Φ_ω)∂_x−→ Φ_ω, ~ξE

+b²D S_ω⁰⁰(−→

Φ_ω)∂_x−→ Φ_ω, ∂_x−→

Φ_ωE

=D S_ω⁰⁰(−→

Φ_ω)~ξ, ~ξE

&k~ξk²_H1×L²,

where we have used the self-adjoint property of the operatorS_ω⁰⁰(−→

Φ_ω) in the second step.

Now we claim that k~ξk²_H1×L² & k~ηk²_H1×L². Indeed, using the orthogonality assumption (3.20), we have

Dξ,~ −→ Γ_ωE

=D

~

η+b∂_x−→ Φ_ω,−→

Γ_ωE

=−b Z

R

(φ_ω,−ωφ_ω)· φω

0

=−bkφ_ωk²_L2. Thus, by H¨older’s inequality, we have

|b|=

Dξ,~ −→ Γ_ωE

kφ_ωk²_L2

.kξk~ _H¹_×L². (3.23)

Now from (3.23), k~ηk_H¹_×L² =

~ξ−b∂_x−→ Φ_ω

H¹×L² ≤ kξk~ _H¹_×L² +|b|

∂_x−→

Φ_ω

H¹×L² .k~ξk_H¹_×L².

This completes the proof.

4. Modulation

We now suppose for contradiction that the solitary wave solution is stable; that is, for any ε >0, there exists δ >0 such that when

k~u₀−−→

Φ_ωk_H¹×L² < δ, we have

~

u∈U_ε −→ Φ_ω

. (4.1)

Then the modulation theory shows that by choosing suitable parameters, the orthogonality conditions in Corollary 3.5 can be verified. The modulation is obtained via the standard implicit function theorem.

Proposition 4.1. (Modulation). Let |ω| = ω_c. There exists ε₀ > 0 such that for any ε∈(0, ε₀), ~u∈U_ε −→

Φ_ω

, the following properties are verified. There exist C¹-functions y :R→R, λ:R→R⁺

such that if we define ~η by

~

η(t) =~u t,·+y(t)

−−−→

Φ_λ(t), (4.2)

then ~η satisfies the following orthogonality conditions for any t∈R: D

~ η,−−→

Γ_λ(t)E

=D

~ η,−−→

Ψ_λ(t)E

= 0, (4.3)

where −→

Γ_λ ∈ H¹(R)×L²(R) and ∂_x−→ Γ_λ = −→

Ψ_λ = φ_λ

0

. Moreover, the following estimate verifies that

k~ηk_H¹×L² +|λ−ω|.ε. (4.4)

Proof. We use the implicit function theorem to prove this proposition. Here we only give the important steps of the proof and refer the reader to [17, 18, 12, 13] for the similar argument.

Define

p= (~u;λ, y), p₀ = (−→ Φ_ω;ω,0).

Letε be the parameter decided later, and define the functional pair (F₁, F₂) :U_ε −→ Φ_ω

×R× R⁺ →R² as

F₁(p) =D

~ η,−→

Γ_λE

, F₂(p) =D

~ η,−→

Ψ_λE .

We claim that there exists ε0 > 0, such that for any ε ∈ (0, ε0), there exists a unique C¹ map: U_ε −→

Φ_ω

→R⁺×R such that (F₁(p), F₂(p)) = 0.

Indeed, firstly we have

F₁(p₀) =F₂(p₀) = 0.

Second, we prove that

|J|=

∂_λF₁ ∂_yF₁

∂_λF₂ ∂_yF₂ p=p0

6= 0.

Indeed, a direct calculation gives that

∂_λF₁(p) =∂_λD

~ η,−→

Γ_λE

=∂_λD

~

u t, x+y(t)

−−−→

Φ_λ(t),−→ Γ_λE

=D

~

u t, x+y(t)

−−−→

Φ_λ(t), ∂_λ−→ Γ_λE

−D

∂_λ−−→

Φ_λ(t),−→ Γ_λE

. When p=p0, we observe that~u t, x+y(t)

−−−→

Φ_λ(t) = 0, and the first term vanishes. For the second term, we note that−→

Γ_λ is an odd vector and ∂_λ−−→

Φ_λ(t) is an even vector, so we get

∂_λF₁(p) p=p0

= 0.

A similar computation shows that

∂_yF₁(p) p=p0

=D

∂_x~u(x+y),−→ Γ_λE

p=p0

=D

∂_x−→ Φ_λ,−→

Γ_λE p=p0

=−kφ_ωk²_L2;

∂_λF₂(p) p=p0

=−D

∂_λ−→ Φ_λ,−→

Ψ_λE p=p0

=−D

∂_λφ_λ, φ_λE p=p0

=−1

2∂_λkφ_λk²_L2

p=p0

= 1

2ωkφ_ωk²_L2;

∂_yF₂(p) p=p0

=D

∂_x−→ Φ_λ,−→

Ψ_λE p=p0

= Z

R

∂_xφ_λφ_λdx p=p0

= 0.

Then we find that

∂_λF₁ ∂_yF₁

∂_λF₂ ∂_yF₂ p=p0

= 1

2ωkφ_ωk⁴_L2 6= 0.

Therefore, the implicit function theorem implies that there exists ε₀ >0 such that for any ε∈(0, ε0),~u∈Uε

−→ Φω

, there exist unique C¹-functions y :U_ε −→

Φ_ω

→R, λ :U_ε −→ Φ_ω

→R⁺, such that

D

~ η,−→

Γ_λE

=D

~ η,−→

Ψ_λE

= 0. (4.5)

Furthermore,

∂_uλ ∂_vλ

∂_uy ∂_vy

=J⁻¹

∂_uF₁ ∂_vF₁

∂_uF₂ ∂_vF₂

. This implies that

|λ−ω|.k~u−−→

Φ_ωk_H¹_×L² < ε.

This finishes the proof of the proposition.

5. Dynamic of the parameters

In this section, we control the modulation parameters y and λ. The effect of giving a precise control on modulation parameters is to obtain the structure ofI⁰(t) in Section 7. The main result is the following.