1.1. Setting of the problem. We consider the nonlinear focusing damped Klein- Gordon equation

PROPERTY FOR THE NONLINEAR DAMPED KLEIN-GORDON EQUATION

RAPHAËL CÔTE AND XU YUAN

Abstract. We consider the nonlinear damped Klein-Gordon equation

∂ttu+ 2α∂tu−∆u+u− |u|^p−1u= 0 on [0,∞)×R^N

withα >0,2⩽N⩽5and energy subcritical exponentsp >2. We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times.

We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degen- erate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate t⁻¹ to the excited state.

1. Introduction

1.1. Setting of the problem. We consider the nonlinear focusing damped Klein- Gordon equation

∂

_tt

u + 2α∂

_t

u − ∆u + u − f (u) = 0 (t, x) ∈ [0, ∞ ) × R

^N

, (1.1) where f (u) = | u |

^p−1

u, α > 0, 2 ⩽ N ⩽ 5 and the exponent p satisfies

2 < p < p

^∗

(N ) with p

^∗

(N) =

 



∞ if N = 2, N + 2

N − 2 if N = 3, 4, 5.

It follows from [4, Theorem 2.3] that the Cauchy problem for (1.1) is locally well- posed in the energy space: for any initial data (u

0

, v

0

) ∈ H

¹

( R

^N

) × L

²

( R

^N

), there exists a unique (in some class) maximal solution u ∈ C ([0, T

max

), H

¹

( R

^N

)) ∩ C

¹

([0, T

max

), L

²

( R

^N

)) of (1.1). Moreover, if the maximal time of existence T

max

is finite, then lim

t↑T_max

k ⃗ u(t) k

H¹×L²

= ∞ .

Setting F (u) =

_p+1¹

| u |

^p+1

and E(⃗ u) = 1

2 Z

R^N

|∇ u |

²

+ u

²

+ (∂

t

u)

²

− 2F (u) dx, for any H

¹

× L

²

solution ⃗ u = (u, ∂

_t

u) of (1.1), it holds

E(⃗ u(t

2

)) − E(⃗ u(t

1

)) = − 2α Z

t₂

t1

k ∂

t

u(t) k

²L²

dt. (1.2) One can easily construct finite time blow-up solutions by adequately truncating a constant in space solution, whose initial data lead to finite time blow-up for the inferred ODE y

^′′

+ 2αy

^′

+ y − f (y) = 0 (and using finite speed of propagation). On the other hand, solutions to (1.1) which are globally defined for positive time, that is for which T

_max

= + ∞, are believed to possess much more structure, in the spirit of a soliton resolution: it roughly asserts that any global solution (maybe under a

2010Mathematics Subject Classification. 35L71 (primary), 35B40, 37K40.

1

genericity condition) splits for large times into a sum of decoupled rigid nonlinear objects, which should be here stationary solutions, especially in view of decay of energy (1.2).

Let us first recall from [6] (see also references therein) some features on stationary solution, namely a solution to the elliptic equation

− ∆q + q − f (q) = 0, q ∈ H

¹

( R

^N

). (1.3) We call the solutions of (1.3) bound states, and denote B the set of bound states:

B = { q : q is a nontrivial solution of (1.3) } .

Standard elliptic arguments (see e.g. [19] or [6, Theorem 8.1.1]) show that if q ∈ B , then q is of class C

²

( R

^N

) and has exponential decay as | x | → + ∞ , as well as its first and second-order derivatives.

Let

W (v) = 1 2

Z

R^N

|∇ v |

²

+ v

²

− 2F (v) dx, for v ∈ H

¹

.

We call the solutions of (1.3) which minimize the functional W by ground states;

the set of ground states is denoted by G

G = { q

0

∈ B : ∀ q ∈ B , W (q

0

) ⩽ W (q) } .

Ground states are well studied objects. They are unique up to space translation (for rather general nonlinearities): there exists a radial positive function q

0

of class C

²

, exponentially decreasing, along with its first and second-order derivatives, such that

G = { q

₀

(x − x

₀

) : x

₀

∈ R

^N

} .

We refer to Berestycki-Lions [2], Gidas-Ni-Nirenberg [18], Kwong [22], Serrin-Tang [26] (however, a positive bound state may not be a ground state, see [12]). It is well-known (see e.g. Grillakis-Shatah-Strauss [20]) that the ground state q

0

is unstable in the energy space. This result was also known in the physics literature as the Derrick’s Theorem [15].

In dimension 1, B = G (due to ODE arguments). In contrast, for any N ⩾ 2, G ⊊ B : see [6, Remark 8.1.16]. Functions q ∈ B \ G are referred to as excited states.

As a matter of fact, much less is known about excited states.

Here are some references on the construction of excited states. Berestycki-Lions [3] showed the existence of infinitely many radial nodal (i.e. sign changing) so- lutions (see also [21, 24] and the references therein). For the massless version of equation (1.3), the existence of excited states that are nonradial sign-changing and with arbitrary large energy was first proved in Ding [16] by variational argument.

Later, del Pino-Musso-Pacard-Pistoia [13] constructed more explicit solutions to the massless equation (1.3) with a centered soliton crowned with negative spikes (rescaled solitons) at the vertices of a regular polygon of radius 1. Then, following similar general strategy in [13], they constructed sign changing, non radial solu- tions to (1.3) on the sphere S

^N

(N ⩾ 4) whose energy is concentrated along special submanifolds of S

^N

in [14].

We can now go back to (1.1), and recall some previous results related to the long time dynamics of global solutions.

Under some conditions on N and p, results in [17, 23] state that for any sequence of

time, any global bounded solution of (1.1) converges to a sum of decoupled bound

states after extraction of a subsequence of times. Also in [17], Feireisl constructed

global solutions that behave as sum of an even number of ground states (i.e. multi-

solitons).

In [4], for dimension N ≥ 2, Burq, Raugel and Schlag proved the convergence of any global radial solution to one (radial) bound state, for the whole sequence of time.

In [10], it is given a complete description of 2-soliton solutions (that is, solutions which, on at least a sequence of time, behave as the sum of two decoupled ground states), in dimension N ⩽ 5. Building on the tools developed there, [9] gave a complete description of global solutions in dimension N = 1, that is, the soliton resolution in that case.

We aim at considering the behavior of solution without conditions on symmetry (like radiality). A complete description seems out of reach, because of the lack of understanding of the dynamics around general excited states, and because the system of centers of mass of the involved bound states may have itself a very intricate dynamics.

1.2. Main results. In this paper, we are instead interested in understanding the behavior of solutions to (1.1) for which only one nonlinear object appears for large times, at least for a sequence of time. More precisely, we define packed solutions as follows.

Definition 1.1. A maximal solution ⃗ u = (u, ∂

t

u) ∈ C ([0, T

max

), H

¹

× L

²

) of (1.1) is called a packed solution if there exist (W

0

, W

1

) ∈ H

¹

× L

²

, and a time sequence t

n

→ T

max

and a position sequence y

n

∈ R

^N

such that

n

lim

→∞

{k u(t

n

) − W

0

( · − y

n

) k

H¹

+ k ∂

t

u(t

n

) − W

1

( · − y

n

) k

L²

} = 0. (1.4) We say that W ⃗ = (W

0

, W

1

) is a cluster point for ⃗ u at (t

n

, y

n

)

n

.

Observe that any cluster point (W

₀

, W

₁

) is actually a bound state (q, 0). More precisely, the following Proposition holds true.

Proposition 1.2. Let ⃗ u = (u, ∂

t

u) be a packed solution of (1.1). Then ⃗ u ∈ C ([0, + ∞ ), H

¹

× L

²

) is globally defined for positive times, and if (W

0

, W

1

) ∈ H

¹

× L

²

is a cluster point for ⃗ u at (t

n

, y

n

)

n

, then W

0

= q is a bound state of (1.3) and W

1

= 0. Furthermore, the energy is bounded below, ∂

t

u ∈ L

²

([0, + ∞ ), L

²

) and for all t ⩾ 0,

E(⃗ u(t)) − E(q, 0) = 2α Z

+∞

t

k ∂

_t

u(s) k

²L²

ds. (1.5) Notice that it is unclear whether a packed solution is globally bounded in H

¹

× L

²

(recall that from arguments of [5] – see also [4] and [9] – if p ⩽

_N^N₋₂

, then any global solution to (1.1) is globally bounded in H

¹

× L

²

, but this is not known for higher powers of p).

It turns out that the description of the convergence depends deeply on the bound state. More specifically, consider the linearized operator L

q

of the energy around a bound state q:

L

q

= − ∆ + 1 − f

^′

(q), hL

q

v, v i = Z

R^N

|∇ v |

²

+ v

²

− f

^′

(q)v

²

dx. (1.6) Due to the invariances of equations, L

q

always has a important kernel: denote the Ω

ij

are the angular derivatives that are

Ω

_ij

= x

_i

∂

_xj

− x

_j

∂

_xi

for 1 ⩽ i < j ⩽ N, (1.7) and consider the vector space Z

q

spanned by the infinitesimal generator of the invariance of the equation on q:

Z

q

= Span { ∂

xn

q, n = 1, · · · , N; Ω

ij

q, 1 ⩽ i < j ⩽ N } . (1.8)

One always has Z

q

⊂ ker L

q

. Then we define non-degenerate and degenerate state.

Definition 1.3. Let q ∈ B.

(i) q is called a non-degenerate state if Z

q

= ker L

q

. (ii) q is called a degenerate state if Z

q

⊊ ker L

q

.

The most relevant example is of course the ground state q

0

which is non-degenerate.

We will comment further on degenerate excited state in the comment paragraph below. For now, let us simply mention one way to understant degeneracy (we denote

^′

for Gateau differentials). The condition that q is a bound state writes E

^′

(q) = 0. Then hL

q

v, w i = E

^′′

(q) · (v, w), so that the condition that q is degenerate, is equivalent to the fact that for some ϕ / ∈ Z

q

, the linear form E

^′′

(q) · (ϕ, · ) = 0.

Our first result is that if one cluster point of a packed solution is a non-degenerate state, then the convergence holds of all positive time, and occurs with an exponential rate. More precisely, we have the following.

Theorem 1.4. Let ⃗ u = (u, ∂

t

u) be a packed solution of (1.1), with cluster point q ∈ H

¹

at (t

n

, y

n

)

n

. If q is a non-degenerate state, we have convergence holding for all time and exponential decay, i.e. there exist µ > 0 and z

_∞

∈ R

^N

such that

∀ t ⩾ 0, k u(t) − q( · − z

_∞

) k

H¹

+ k ∂

t

u(t) k

L²

≲ e

^−µt

.

Next, we consider degree-1 excited states where ker L

q

has one extra dimension not related to the geometric invariances of the equation (1.1) and which also involves a condition on the third-order Gateau differentials of E, according to the next definition.

Definition 1.5. Let q be a degenerate excited state. q is called a degree-1 excited state if there exists ϕ ∈ H

¹

such that

ker L

q

= Z

q

⊕ Span { ϕ } and E

^′′′

(q) · (ϕ, ϕ, ϕ) 6 = 0. (1.9) Again, we will comment on this definition in the paragraph below, the main point being that degree-1 excited states are somehow the simplest degenerate bounds states.

Our second result is concerned with cluster points which are degree-1 excited states.

Theorem 1.6. Let ⃗ u = (u, ∂

t

u) be a packed solution of (1.1), with cluster point q ∈ H

¹

at (t

n

, y

n

)

n

. If q is a degree-1 excited state, then the convergence ⃗ u(t) → (q, 0) holds for all time as t → + ∞ , and the rate of convergence has algebraic decay, i.e.

there exists z

_∞

∈ R

^N

such that

∀ t > 0, k u(t) − q( · − z

_∞

) k

H¹

+ k ∂

t

u(t) k

L²

≲ t

⁻¹

.

Last, we show that the convergence rate in Theorem 1.6 can be sharp: we provide an example where the solution converges exactly at the rate t

⁻¹

to the degree-1 excited state.

Theorem 1.7. Let q be a degree-1 excited state. Then, there exists a global solution

⃗

u = (u, ∂

t

u) ∈ C ([0, + ∞ ), H

¹

× L

²

) of (1.1) such that

k u(t) − q k

H¹

+ k ∂

t

u(t) k

L²

∼ t

⁻¹

as t → + ∞ .

1.3. Comments. Let us first observe that Theorem 1.4 (and its proof) holds also in

dimension 1, but of course, they are in that case a direct consequence of the complete

description [9] of global solutions in 1D (as mentioned above, excited states only

exist for N ⩾ 2). This is the only reason why we restrict to dimension N ⩾ 2. The

restrictions to N ⩽ 5 and 2 < p <

^N+2_N−2

are to ensure a nice local well posedness

theory, and sufficient smoothness on the non-linearity so that Taylor expansion

make sense up to order 2. In this perspective let us remind that our analysis

encompasses the most physically relevant nonlinearity, the cubic one f (u) = u

³

.

Regarding Theorem 1.4: the ground state is of course non-degenerate, but one should keep in mind that is not so easy to construct degenerate excited states. As a matter of fact, the constructions in [1, 24] (see also [13, 25] for the massless case) yield non-degenerate excited states as well. This means that the scope of Theorem 1.4 is rather large and does certainly not restrict to the ground state.

We now discuss degree-1 excited states: as we mentioned, they should be under- stood as the simplest degenerate case. Already here, very little is known, and to our knowledge, our results are the first describing precisely the dynamics in a degener- ate setting. From this point of view, the condition that dim ker L

q

= dim( Z

q

) + 1 is very natural. Regarding the extra condition E

^′′′

(q) · (ϕ, ϕ, ϕ) 6 = 0, let us note that it is generic; as we will see in Lemma 2.1, it is equivalent with E

^′′′

(q) being non identically 0 on (ker L

q

)

³

.

It is remarquable that one already observes a drastic change in the dynamic in degree-1 degeneracy, when compared to non-degeneracy. The convergence here is indeed merely polynomial in time, which is a surprise: such slow rate of convergence is usually observed due to the interaction with another nonlinear object (as in [10, 9]), and this is not the case here. As it is seen in the proofs, the derivation of the main bootstrap regime is noticeably more involved in degree-1 degeneracy, and relies on the very specific algebra of the main ODE system at leading order (see Section 3).

One setting where excited states are better understood is the case of radial func- tions. Among these, radial bound states q are either non-degenerate or satisfy the first condition in the degree-1 degeneracy definition (1.9): indeed, among radial functions, the geometric kernel Z

q

is trivial and dim ker

rad

L

q

⩽ 1, see for example [4, Section 2.3].

All the arguments in the proofs below can taken word for word to the radial setting, and so our results hold for any packed radial solution converging to a radial bound state q which is either non-degenerate (Theorem 1.4) or such that E

^′′′

(q) |

(kerLq)³

6 = 0 (Theorems 1.6 and 1.7).

2. Preliminaries 2.1. Proof of Proposition 1.2.

Proof of Proposition 1.2. Denote W ⃗ (t) = (W (t), ∂

t

W (t)) the solution to (1.1) with initial data W ⃗ (0) = (W

₀

, W

₁

) and ⃗ u

_n

(t, x) = ⃗ u(t

_n

+ t, x + y

_n

) the solution to (1.1) with initial data ⃗ u

n

(0, x) = ⃗ u(t

n

, x + y

n

). We can assume that W ⃗ ∈ C ([0, T

0

], H

¹

× L

²

) for some T

0

> 0.

As the (1.1) flow is continuous in H

¹

× L

²

and ⃗ u

n

(0) → W ⃗ (0), we infer that ⃗ u

n

is defined on [0, T

0

] for n large enough and that

⃗

u

n

→ W ⃗ in C ([0, T

0

], H

¹

× L

²

). (2.1) This immediately prove that ⃗ u is globally defined for positive times. Indeed, if T

_max

< + ∞ , then for large enough n, T

_max

⩾ t

_n

+T

₀

→ T

_max

+ T

₀

, a contradiction:

hence T

_max

= + ∞ .

As E(⃗ ⃗ u(t

n

)) → E(W ⃗

0

, W

1

) we infer from the energy dissipation identity that ∂

t

u ∈ L

²

([0, + ∞ ), L

²

). Assume that W ⃗ is not a stationnary solution. Then we can furthermore assume that ∂

_t

W 6 = 0 on [0, T

₀

] × R

^N

, so that k ∂

_t

W k

L²([0,T₀]),L²)

> 0.

In particular, from the convergence (2.1), we conclude that

k ∂

t

u k

L²([tn,tn+T0],L²)

= k ∂

t

u

n

k

L²([0,T0]),L²)

→ k ∂

t

W k

L²([0,T0]),L²)

as n → + ∞ .

Let t

^′_n

be a subsequence of t

_n

such that for all n ∈ N, t

^′_n+1

⩾ t

^′_n

+ T

₀

and k ∂

_t

u k

L²([t^′_n,t^′_n+T₀],L²)

⩾ 1

2 k ∂

_t

W k

L²([0,T₀]),L²)

. There holds

k ∂

_t

u k

²L²([0,+∞),L²)

⩾ X

n

k ∂

_t

u k

²L²([t′n,t′n+T₀],L²)

⩾ X

n

1

4 k ∂

_t

W k

²L²([0,T₀]),L²)

= + ∞ , which is a contradiction. As a consequence, W ⃗ is a stationary solution, which means that for all t ⩾ 0, ∂

_t

W (t) = 0 and W (t) = q for some bound state q. In particular, W

₀

= q and W

₁

= 0.

Furthermore, the energy dissipation identity writes for all 0 ⩽ t ⩽ t

_n

: E(⃗ u(t)) − E(⃗ u(t

n

)) = 2α

Z

t_n t

k ∂

t

u(s) k

²L²

ds.

Letting n → + ∞ , we see that the left-hand side has a limit E(⃗ u(t)) − E(q, 0), and so ∂

t

u ∈ L

²

([t, + ∞ ), L

²

). This completes the proof of Proposition 1.2. □ 2.2. Notation. Let q be a bound state. Let I

q

be a subset of { (i, j) : 1 ⩽ i < j ⩽ N } such that

{ ∂

_xn

q, 1 ⩽ n ⩽ N ; Ω

_ij

q, (i, j) ∈ I

_q

} is a basis of Z

q

. For any (i, j) ∈ I

q

and ϑ ∈ R , we recall the Givens rotation:

G

ij

(ϑ) =



 

 

 

 

 

 

1 · · · 0 · · · 0 · · · 0 .. . . .. .. . .. . .. . 0 · · · cos ϑ · · · − sin ϑ · · · 0 .. . .. . . .. .. . .. . 0 · · · sin ϑ · · · cos ϑ · · · 0

.. . .. . .. . . .. ...

0 · · · 0 · · · 0 · · · 1



 

 

 

 

 

 

, (2.2)

where cos ϑ and sin ϑ appear at the intersections ith and jth rows and columns.

That is, the non-zero elements of the Givens matrix G

_i,j

(ϑ) = (g

_nm

)

_nm

are given by:

g

nn

= 1 for n 6 = i, j, g

ii

= g

jj

= cos ϑ, and g

ij

= − g

ji

= − sin ϑ.

For K ∈ N

^∗

and r > 0, we denote by B

_R^K

(r) (respectively, S

_R^K

(r)) be the ball (respectively, the sphere) of R

^K

of center 0 and of radius r.

We denote h· , ·i the L

²

scalar product for real-valued functions u, v ∈ L

²

, h u, v i :=

Z

R^N

u(x)v(x)dx.

For vector-valued functions

⃗ u =

u

₁

u

₂

, ⃗ v =

v

₁

v

₂

, the notation h· , ·i is also the L

²

scalar product,

h ⃗ u, ⃗ v i := X

k=1,2

h u

k

, v

k

i , k ⃗ u k

²_H

:= k u

1

k

²H¹

+ k u

2

k

²L²

.

We also define p ¯ = min { 3, p } > 2 (recall that the nonlinearity power p > 2).

2.3. Spectral theory of linearized operator. In this section, we introduce some spectral properties of the linearized operator for any bound state q ∈ B .

For θ = (θ

ij

)

_(i,j)∈I

q

∈ R

^#Iq

, denote the rotation

R

_θ

= G

_i1j1

(θ

_i1j1

) · · · G

_i#Iqj#Iq

(θ

_i#Iqj#Iq

).

For (z, θ) ∈ R

^N+#Iq

, we introduce the following transformation T

(z,θ)

linked to the symmetries of (1.1): for f ∈ L

²

,

T

(z,θ)

f := f (R

_θ

( · − z)).

Observe that for all q ∈ B , Z

q

is generated by taking partial derivatives of T

(z,θ)

q with respect to (z, θ) at (z, θ) = (0, 0):

∂

_xn

q = − ∂

∂z

n

T

(z,θ)

q |

(z,θ)=(0,0)

, Ω

_ij

q = ∂

∂θ

ij

T

(z,θ)

q |

(z,θ)=(0,0)

. (2.3) First, we recall standard properties of the linearized operator L

q

.

Lemma 2.1. (i) Spectral properties. The self-adjoint operator L

q

has essential spectrum [1, + ∞ ), a finite number K ⩾ 1 of negative eigenvalues and its kernel is of finite dimension M with M ⩾ N . Let (Y

k

)

k=1,···,K

be an L

²

orthogonal family of eigenfunctions of L

q

with negative eigenvalues ( − λ

²_k

)

_k=1,···,K

, i.e.

h Y

k

, Y

k^′

i = δ

kk^′

and L

q

Y

k

= − λ

²_k

Y

k

, λ

k

> 0. (2.4) (ii) Coercivity. Denote Π

_q

the L

²

-orthogonal projection on ker L

q

. There exists c > 0 such that for all η ∈ H

¹

,

hL

q

η, η i ⩾ c k η k

²H¹

− c

⁻¹

k Π

q

η k

²L²

+ X

K k=1

h η, Y

k

i

²

. (2.5)

(iii) Cancellation. We have, for all ψ

1

, ψ

2

∈ ker L

q

and ψ

3

∈ Z

q

h f

^′′

(q)ψ

1

ψ

2

, ψ

3

i = 0. (2.6) Proof. Proof of (i) and (ii). See the proof of [7, Lemma 1].

Proof of (iii). Without loss of generality, we first consider ψ

3

= Ω

ij

q for (i, j) ∈ I

q

. For any ψ

₁

∈ ker L

q

, we have

− ∆ψ

1

+ ψ

1

− f

^′

(q)ψ

1

= 0.

Consider the transformation T

(z,θ)

with (z, θ) = (0, θ) for the above identity,

− ∆( T

(z,θ)

ψ

1

) + T

(z,θ)

ψ

1

− f

^′

( T

(z,θ)

q)( T

(z,θ)

ψ

1

) = 0.

Note that, from p > 2, we can take the derivative of above identity with respect to θ

_ij

, and then let θ = 0. It follows that

f

^′′

(q)ψ

1

ψ

3

= − ∆ ˜ ψ

1

+ ˜ ψ

1

− f

^′

(q) ˜ ψ

1

= L

q

ψ ˜

1

where ψ ˜

1

= Ω

ij

ψ

1

. Thus, by integration by parts and ψ

2

∈ ker L

q

,

h f

^′′

(q)ψ

₁

ψ

₂

, ψ

₃

i = h f

^′′

(q)ψ

₁

ψ

₃

, ψ

₂

i = hL

q

ψ ˜

₁

, ψ

₂

i = h ψ ˜

₁

, L

q

ψ

₂

i = 0.

Proceeding similarly for all the parameters in the transformation T

(z,θ)

, we complete

the proof of (iii). □

As a consequence of the above Lemma, in the case when q is furthermore assumed to be a degree-1 excited state, we now choose ϕ (introduced in Definition 1.5) with more rigid properties: namely we claim that there exists (a unique) ϕ ∈ H

¹

such that for all n = 1, · · · , N, (i, j) ∈ I

_q

,

h ϕ, ∂

x_n

q i = h ϕ, Ω

ij

q i = 0 and ker L

q

= Z

q

⊕ Span { ϕ } , (2.7)

and such that

− (4α k ϕ k

²L²

)

⁻¹

E

^′′′

(ϕ, ϕ, ϕ) = 1. (2.8) (2.7) essentially means that ϕ is the L

²

-orthogonal supplement of Z

q

. Moreover, due to (2.6) and the fact that E

^′′′

(q) is not identically 0 on L

³q

(in view of (1.9)), for such a ϕ one has E

^′′′

(q)(ϕ, ϕ, ϕ) 6 = 0: and so, by considering the transformations ϕ → − ϕ and ϕ → λϕ, the condition (2.8) can be met.

2.4. Modulation around a bound state. Let q ∈ B be a degree-1 degenerate bound state. Given time dependent C

¹

functions z, θ, a, with values in R

^N

, R

^#Iq

and R , we denote

Q = T

(z,θ)

q = q (R

_θ

( · − z)) , Φ(t, x) = T

(z,θ)

ϕ = ϕ (R

_θ

( · − z)) , (2.9) V (t, x) = a(t)Φ(t, x), G = f (Q + V ) − f (Q) − f

^′

(Q)V. (2.10) It will convenient to encompass both non-degenerate and degree-1 degenerate cases at once by setting a ≡ 0 if q is non-degenerate.

For all (i, j) ∈ I

q

and (i

^′

, j

^′

) ∈ I

q

, we denote as follows the derivatives:

Ψ

ij

= ∂Q

∂θ

ij

, Φ

ij

= ∂Φ

∂θ

ij

, Ψ

ⁱ_ij^′j′

= ∂Ψ

ij

∂θ

i^′j^′

. (2.11)

Finally, we introduce the exponential directions. For k = 1, · · · , K , we denote Υ

_k

= T

(z,θ)

Y

_k

, Υ

^ij_k

= ∂Υ

_k

∂θ

ij

, and

ν

_k^±

= − α ± q

α

²

+ λ

²_k

, ζ

_k^±

= α ± q

α

²

+ λ

²_k

and Z ⃗

_k^±

= ζ

_k^±

Υ

k

Υ

k

. (2.12) The importance of Z ⃗

_k^±

come from the following observation: if ⃗ v = (v

1

, v

2

) is a solution to the linearized (1.1) equation

∂

t

v

1

v

2

=

v

2

− 2αv

2

− L

q

v

1

, then with a

^±_k

:= h ⃗ v, ⃗ Z

_k^±

i , there hold

_dt^d

a

^±_k

= ν

_k^±

a

^±_k

.

Observe that all the function introduced are at least of class C

¹

and have pointwise exponential decay.

By direct computation and the definition of Givens rotations in (2.2), we have, for all n = 1, · · · , N and (i, j) ∈ I

_q

,

Ψ

ij

∈ Span { (Ω

ij

q) (R

θ

( · − z)) : (i, j) ∈ I

q

} ,

∂

_xn

Q ∈ Span { (∂

_x1

q) (R

_θ

( · − z)) , · · · , (∂

_xN

q) (R

_θ

( · − z)) } . (2.13) Moreover, by the chain rule, we have, for all (i, j) ∈ I

q

and k = 1, · · · , K ,

∂

t

Q = − z ˙ · ∇ Q + X

(i,j)∈I_q

θ ˙

ij

Ψ

ij

,

∂

_t

Φ = − z ˙ · ∇ Φ + X

(i,j)∈I_q

θ ˙

_ij

Φ

_ij

,

∂

t

Υ

k

= − z ˙ · ∇ Υ

k

+ X

(i,j)∈Iq

θ ˙

ij

Υ

^ij_k

,

∂

t

Ψ

ij

= − z ˙ · ∇ Ψ

ij

+ X

(i^′,j^′)∈Iq

θ ˙

i^′j^′

Ψ

ⁱ_ij^′j′

.

(2.14)

As a consequence of (2.3), we have the following expansions for small θ ∈ R

^#Iq

.

Lemma 2.2. For | θ | 1 small, we have, for all (i, j) ∈ I

_q

, (i

^′

, j

^′

) ∈ I

_q

, k = 1, . . . , K and n = 1, . . . , N,

Ψ

ij

= (Ω

ij

q) (R

θ

( · − z)) + O

H¹

( | θ | ), Φ

ij

= (Ω

ij

ϕ) (R

θ

( · − z)) + O

_H1

( | θ | ), Υ

^ij_k

= (Ω

ij

Y

k

) (R

θ

( · − z)) + O

_H1

( | θ | ),

∂

_xn

Q = (∂

_xn

q) (R

_θ

( · − z)) + O

_H1

( | θ | ), Ψ

ⁱ_ij^′j′

= (Ω

_i′j^′

Ω

_ij

q) (R

_θ

( · − z)) + O

_H1

( | θ | ).

(2.15)

If q is a non-degenerate bound state, we use same notations as above for degree-1 degenerate bound states, but with a = 0 and ϕ = 0.

For future reference, we state the following Taylor formulas involving the functions F and f , and omit its proof.

Lemma 2.3. For all s ∈ R , and x ∈ R

^N

, we have

| f

^′

(Q + s) − f

^′

(Q) | ≲ | s | + | s |

^p−1

, (2.16)

| f (Q + s) − f (Q) − f

^′

(Q)s | ≲ s

²

+ | s |

^p

, (2.17) f (Q + s) − f(Q) − f

^′

(Q)s − 1

2 f

^′′

(Q)s

²

≲ | s |

³

+ | s |

^p

, (2.18)

| f (Q + V + s) − f (Q + V ) − f

^′

(Q + V )s | ≲ s

²

+ | s |

^p

, (2.19)

| F (Q + V + s) − F (Q + V ) − f (Q + V )s | ≲ | Q + V |

^p−1

s

²

+ | s |

^p+1

, (2.20) F (Q + V + s) − F (Q + V ) − f (Q + V )s − 1

2 f

^′

(Q + V )s

²

≲ | s |

³

+ | s |

^p+1

, (2.21) where all the implied constants in the ≲ are uniform in the space variable of Q or V .

First, we introduce the standard modulation result around the non-degenerate state or degree-1 excited state q.

Proposition 2.4 (Properties of the modulation). There exists 0 < γ

₀

1 such that for any 0 < γ < γ

0

, T

1

⩽ T

2

, and any solution ⃗ u = (u, ∂

t

u) of (1.1) on [T

1

, T

2

] satisfying

sup

t∈[T₁,T₂]

inf

ξ∈R^N

k u(t) − q( · − ξ) k

H¹

+ k ∂

t

u(t) k

L²

< γ, (2.22) there exist unique C

¹

functions

[T

1

, T

2

] → R

^N

× R

^N

× R

^#Iq

× R

^#Iq

× R × R t 7→ (z(t), ℓ(t), θ(t), β(t), a(t), b(t)) such that, if we define φ ⃗ = (φ

₁

, φ

₂

) by

⃗ u =

u

∂

t

u

=

Q

− ℓ · ∇ Q

+ X

(i,j)∈Iq

β

ij

0 Ψ

ij

+

aΦ bΦ

+

φ

1

φ

2

, (2.23) where β = (β

ij

)

(i,j)∈I_q

, it satisfies, for all t ∈ [T

1

, T

2

],

k φ(t) ⃗ k

H

+ | θ(t) | ≲ k u(t) − q k

H¹

+ k ∂

t

u(t) k

L²

≲ γ,

| ℓ(t) | + | β(t) | + | a(t) | + | b(t) | ≲ k u(t) − q k

H¹

+ k ∂

t

u(t) k

L²

≲ γ, (2.24) and for all n = 1, · · · , N and (i, j) ∈ I

q

,

h φ

₁

, ∂

_xn

Q i = h φ

₁

, Ψ

_ij

i = h φ

₁

, Φ i = 0, (2.25)

h φ

2

, ∂

x_n

Q i = h φ

2

, Ψ

ij

i = h φ

2

, Φ i = 0. (2.26)

Proof. The proof of the decomposition result relies on a standard argument based on the Implicit function Theorem (See e.g. [11, Appendix B]) and we omit it. □ Second, we derive the equation of φ ⃗ from (1.1) and (2.23).

Lemma 2.5 (Equation of φ). ⃗ In the contex of Proposition 2.4, we have ( ∂

t

φ

1

= φ

2

+ Mod

1

+G

1

,

∂

_t

φ

₂

= ∆φ

₁

− φ

₁

− 2αφ

₂

+ f(Q + V + φ

₁

) − f (Q + V ) + Mod

₂

+G

₂

+ G, (2.27) where

Mod

1

:= z ˙ − ℓ

· ∇ Q − X

(i,j)∈Iq

θ ˙

ij

− β

ij

Ψ

ij

− ( ˙ a − b)Φ,

Mod

₂

:= ℓ ˙ + 2αℓ

· ∇ Q − X

(i,j)∈Iq

β ˙

_ij

+ 2αβ

_ij

Ψ

_ij

−

b ˙ + 2αb

Φ,

and G = f (Q + V ) − f (Q) − f

^′

(Q)V is defined in (2.10), G

1

:=a z ˙ · ∇ Φ − a X

(i,j)∈I_q

θ ˙

ij

Φ

ij

, G

₂

:= X

(i,j)∈I_q

θ ˙

_ij

(ℓ · ∇ Ψ

_ij

) + X

(i,j)∈I_q

β

_ij

( ˙ z · ∇ Ψ

_ij

) − (ℓ · ∇ ) ( ˙ z · ∇ ) Q

− X

(i^′,j^′)∈I_q

X

(i,j)∈I_q

θ ˙

_i′j^′

β

_ij

Ψ

ⁱ_ij^′j′

+ b z ˙ · ∇ Φ − X

(i,j)∈I_q

b θ ˙

_ij

Φ

_ij

. Proof. First, from (2.14) and (2.23),

∂

t

φ

1

=∂

t

u − ∂

t

Q − a∂

t

Φ − aΦ ˙

=φ

2

+ ˙ z − ℓ

· ∇ Q − X

(i,j)∈I_q

θ ˙

ij

− β

ij

Ψ

ij

− ( ˙ a − b)Φ

+ a z ˙ · ∇ Φ − a X

(i,j)∈Iq

θ ˙

_ij

Φ

_ij

= φ

₂

+ Mod

₁

+G

₁

. Using (2.14) and (2.23) again,

∂

t

φ

2

=∂

tt

u + ˙ ℓ · ∇ Q − X

(i,j)∈Iq

β ˙

ij

Ψ

ij

− bΦ ˙ + ℓ · ∇ ∂

t

Q − X

(i,j)∈I_q

β

ij

∂

t

Ψ

ij

− b∂

t

Φ

=∂

_tt

u + ˙ ℓ · ∇ Q − X

(i,j)∈Iq

β ˙

_ij

Ψ

_ij

− bΦ + ˙ G

₂

. From (1.1), (2.23), − ∆Q + Q − f (Q) = 0 and − ∆Φ + Φ − f

^′

(Q)Φ = 0,

∂

_tt

u =∆u − u − 2α∂

_t

u + f (u)

=∆φ

1

− φ

1

− 2αφ

2

+ f (Q + V + φ

1

) − f (Q + V ) + 2αℓ · ∇ Q − 2α X

(i,j)∈I_q

β

_ij

Ψ

_ij

− 2αbΦ + G.

Therefore,

∂

_t

φ

₂

= ∆φ

₁

− φ

₁

− 2αφ

₂

+ f (Q + V + φ

₁

) − f (Q + V ) + G

₂

+ G +

ℓ ˙ + 2αℓ

· ∇ Q − X

(i,j)∈I_q

β ˙

ij

+ 2αβ

ij

Ψ

ij

− b ˙ + 2αb

Φ. □

Third, we derive the control of geometric parameters from orthogonality condi- tions (2.25) and (2.26). Our goal here is to get an ODE system on the modulations parameters, at leading order, with bounds on the remainder terms as squares or higher powers of | a | and

N := k φ ⃗ k

H

+ | ℓ | + | β | + | b | . (2.28) Recall that we defined p ¯ = min { 3, p } > 2.

Lemma 2.6. In the context of Proposition 2.4, the following holds.

(i) Control of non-degenerate directions. We have

| z ˙ − ℓ | + θ ˙ − β ≲ N

²

+ | a |N , (2.29)

| ℓ ˙ + 2αℓ | + | β ˙ + 2αβ | ≲ N

²

+ | a |N + | a |

^p¯

. (2.30) (ii) Control of extra direction. For q be a degree-1 excited state, we have

| a ˙ − b | ≲ N

²

+ | a |N , (2.31)

| b ˙ + 2αb + 2αa

²

| ≲ N

²

+ | a |N + | a |

^p¯

. (2.32) Proof. Proof of (i). First, we differentiate the orthogonality h φ

₁

, ∂

_xn

Q i = 0 in (2.25),

0 = d

dt h φ

1

, ∂

x_n

Q i = h ∂

t

φ

1

, ∂

x_n

Q i + h φ

1

, ∂

t

∂

x_n

Q i . Using (2.25) and (2.27),

h ∂

t

φ

1

, ∂

x_n

Q i = h Mod

1

, ∂

x_n

Q i + h G

1

, ∂

x_n

Q i .

From (2.7), (2.13), (2.15), (2.24), the expression of Mod

1

and change of variables, h Mod

1

, ∂

x_n

Q i

= h ( ˙ z − ℓ) · ∇ Q, ∂

_xn

Q i − X

(i,j)∈Iq

θ ˙

_ij

− β

_ij

h Ψ

_ij

, ∂

_xn

Q i − ( ˙ a − b) h Φ, ∂

_xn

Q i

= h ( ˙ z − ℓ) · ∇ q, ∂

_xn

q i − X

(i,j)∈I_q

θ ˙

_ij

− β

_ij

h Ω

_ij

q, ∂

_xn

q i + O(γ( | z ˙ − ℓ | + | θ ˙ − β | )).

From the expression of G

1

and (2.24),

|h G

1

, ∂

x_n

Q i| ≲ | a |

| z ˙ − ℓ | + | ℓ | + | θ ˙ − β | + | β |

≲ γ

| z ˙ − ℓ | + | θ ˙ − β |

+ | a |N . Next, using again (2.14),

∂

_t

∂

_xn

Q = ∂

_xn

∂

_t

Q = − z ˙ · ∇ ∂

_xn

Q + X

(i,j)∈I_q

θ ˙

_ij

∂

_xn

Ψ

_ij

. (2.33) Thus, from (2.24) and the Sobolev embedding Theorem,

|h φ

1

, ∂

t

∂

x_n

Q i| ≲ k φ ⃗ k

_H

| z ˙ − ℓ | + | ℓ | + | θ ˙ − β | + | β |

≲ γ

| z ˙ − ℓ | + | θ ˙ − β | + N

²

. Combining above estimates, we have

h ( ˙ z − ℓ) · ∇ q, ∂

x_n

q i − X

(i,j)∈I_q

θ ˙

ij

− β

ij

h Ω

ij

q, ∂

x_n

q i

= O

γ

| z ˙ − ℓ | + | θ ˙ − β |

+ N

²

+ | a |N

.

This gives N inequalities. Proceeding similarly for the #I

_q

orthogonality conditions h φ

1

, Ω

ij

Q i = 0 in (2.25), and using the fact that the family

{ ∂

_xn

q, 1 ⩽ n ⩽ N ; Ω

_ij

q, (i, j) ∈ I

_q

}

is linearly independent, so that its Gram matrix is invertible, we obtain

| z ˙ − ℓ | + θ ˙ − β ≲ γ | z ˙ − ℓ | + θ ˙ − β + N

²

+ | a |N , which implies (2.29), upon taking γ small enough.

Second, we differentiate the orthogonality h φ

2

, ∂

x_n

Q i = 0 in (2.26), 0 = d

dt h φ

2

, ∂

x_n

Q i = h ∂

t

φ

2

, ∂

x_n

Q i + h φ

2

, ∂

t

∂

x_n

Q i . From (2.27), we have

h ∂

t

φ

2

, ∂

xn

Q i = h ∆φ

1

− φ

1

+ f

^′

(Q)φ

1

, ∂

xn

Q i − 2α h φ

2

, ∂

xn

Q i

+ h G

2

, ∂

x_n

Q i + h R, ∂

x_n

Q i + h G, ∂

x_n

Q i + h Mod

2

, ∂

x_n

Q i , where

R = f (Q + V + φ

₁

) − f (Q + V ) − f

^′

(Q)φ

₁

.

Based on − ∆∂

_xn

q + ∂

_xn

q − f

^′

(q)∂

_xn

q = 0, integration by parts and (2.26), h ∆φ

1

− φ

1

+ f

^′

(Q)φ

1

, ∂

x_n

Q i − 2α h φ

2

, ∂

x_n

Q i = 0.

Then, by the expression of G

2

, (2.24) and (2.29),

|h G

2

, ∂

x_n

Q i| ≲

| z ˙ − ℓ | + | θ ˙ − β | + | ℓ | + | β |

( | ℓ | + | β | + | b | )

≲ N

²

( N + | a | + 1) ≲ N

²

. Next, using (2.16), (2.19) and (2.24),

| R | ≲ | f (Q + V + φ

1

) − f (Q + V ) − f

^′

(Q + V )φ

1

| + | (f

^′

(Q + V ) − f

^′

(Q))φ

1

|

≲ | φ

1

|

²

+ | φ

1

|

^p

+ ( | V | + | V |

^p−1

) | φ

1

| ≲ | φ

1

|

²

+ | φ

1

|

^p

+ | a | | Φ | + | Φ |

^p−1

| φ

1

| . It follows that

|h R, ∂

x_n

Q i| ≲ Z

R^N

| φ

1

|

²

+ | φ

1

|

^p

+ | φ

1

|| V |

dx ≲ N

²

+ | a |N .

For q be a non-degenerate state, we have a = 0 which implies G = 0. For q be a degree-1 excited state, from (2.18), we have

G − 1

2 f

^′′

(Q)V

²

≲ | V |

³

+ | V |

^p

≲ | a |

³

+ | a |

^p

| Φ |

³

+ | Φ |

^p

. It follows that

h G, ∂

x_n

Q i − h 1

2 f

^′′

(Q)V

²

, ∂

x_n

Q i

≲ | a |

³

+ | a |

^p

Z

R^N

| Φ |

³

+ | Φ |

^p

| ∂

_xn

Q | dx ≲ | a |

³

+ | a |

^p

. Note that, by (2.6), (2.13) and change of variables,

h 1

2 f

^′′

(Q)V

²

, ∂

_xn

Q i = 0.