Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes

We consider the dynamics of even solutions of the one-dimensional nonlinear Klein-Gordon equation $\partial_t^2 \phi - \partial_x^2 \phi + \phi - |\phi|^{2\alpha} \phi =0$ for $\alpha>1$, in the vicinity of the unstable soliton $Q$. Our main result is that stability in the energy space $H^1(\mathbb R)\times L^2(\mathbb R)$ implies asymptotic stability in a local energy norm. In particular, there exists a Lipschitz graph of initial data leading to stable and asymptotically stable trajectories. The condition $\alpha>1$ corresponds to cases where the linearized operator around $Q$ has no resonance and no internal mode. Recall that the case $\alpha>2$ is treated in Krieger-Nakanishi-Schlag using Strichartz and other local dispersive estimates. Since these tools are not available for low power nonlinearities, our approach is based on virial type estimates and the particular structure of the linearized operator observed in Chang-Gustafson-Nakanishi-Tsai.


Main results. Consider the one-dimensional focusing nonlinear Klein-Gordon equation
where α > 0. This equation also rewrites as a first order system in time for the function φ = (φ, ∂ t φ) = (φ 1 , φ 2 ), 2α+2 |φ| 2α+2 . Note that (1) is Hamiltonian. The conservation of energy of a solution (φ, ∂ t φ) of (1) writes For initial data in the energy space H 1 × L 2 , local well-posedness, as well as global wellposedness for small solutions, is well-known (see for example [5], Theorem 6.2.2 and Proposition 6.3.3). Denote by Q the standing wave solution of (1), also called soliton, explicitly given by , The linearized operator L around Q writes For any α > 0, the first eigenvalue of L is λ 0 = −α(α + 2) = −ν 2 0 (ν 0 > 0) with corresponding normalized eigenfunction (we denote A, B = A·B). The second eigenvalue of L is 0 with eigenfunction Y 1 = c 1 Q ′ . In the case α > 1, there is no other eigenvalue in [0, 1), which means that there is no internal mode for the model (see Section 1.3). Let The functions u ± (t, x) = e ±ν 0 t Y ± (x) are solutions of the linearized problem illustrating the presence of exponentially stable and unstable modes both relevant in the dynamics of solutions in the vicinity of a soliton.
Our main result is the following conditional asymptotic stability theorem.
For the sake of completeness, we provide a description of the set of initial data leading to global solutions satisfying the stability assumption (6) (see also Theorem 4.1 in [2]).

1.2.
Related results and comments on the proof. First, we comment on two articles devoted to soliton dynamics for the one-dimensional nonlinear Klein-Gordon equation (1). Using techniques based on Strichartz and other local dispersive estimates, Krieger et al. [22] have completely treated the case α > 2 in the case of even data. Indeed, they classify all solutions whose energy does not exceed too much that of the ground state Q. This includes the construction, by the fixed point argument, of a C 1 center-stable manifold around the soliton and the proof of asymptotic stability and scattering (linear behavior) around the ground state for solutions on the manifold. The method seems limited to α ≥ 2 because of the use of Strichartz estimates to control the nonlinear term, see comment in Section 3.4 of [22].
By formal and numerical methods, Bizoń et al. [4] have shown that for even solutions trapped by the soliton, the convergence rate to Q heavily depends on the power α of the nonlinearity. In the L ∞ sense, they conjecture the following trichotomy: (a) fast dispersive decay for α > 1; (b) slow decay for α = 1; (c) very slow decay for 0 < α < 1. The threshold value α = 1 corresponds to the emergence of a resonance at the linear level, while α < 1 leads to one or several internal modes (see Section 1.3). Following these observations, unifying the case α > 1 was the main motivation of the present work.
Our method does not give an explicit decay rate as t → +∞, but we notice as a by-product of the proof of Theorem 1 that, for any interval I of R, it holds This is to be compared with the results obtained in [18] on the (local) asymptotic stability of the kink for the φ 4 model under small odd perturbations. Indeed, in the latter case, the presence of an internal mode leads to a lower convergence rate since the component z(t) of the solution along the internal mode only satisfies the weaker estimate +∞ 0 |z(t)| 4 dt < ∞ (see Theorem 1.2 in [18]). Although we do not claim optimality of such results, in the case of (1) with 0 < α ≤ 1, we do not expect estimates such as in (10) to hold.
The proof of Theorem 1 is mainly based on localized virial type arguments similar to that used in [18,25,27], for example. Unlike in these works, we avoid numerical computations of certain constants related to the coercivity of the virial functional by using factorization properties of the linearized operator described in [6] (see also references [29,37], cited in [6]). A formal presentation of this approach is given in Section 4.1. We point out that the same structure was crucially used in the construction of blow-up solutions for the wave maps, Yang-Mills and O(3) σ-models in [30,31]. Note that in the present paper, we compensate the loss of two derivatives due to the change of variables to still work in the energy space.
Several other conditional asymptotic stability results or classifications in a neighborhood of the ground state for the nonlinear Klein-Gordon in higher dimensions and for the nonlinear Schrödinger equation were also obtained in [10,11,32,34], for example. We also mention [21] where for the mass supercritical Schrödinger equation in one dimension, a finite co-dimensional manifold of initial data trapped by the soliton was constructed.
Concerning the generalized Korteweg-de Vries equation and related models, studies of the dynamics of the solutions close to the soliton are presented in [9,14,15,24,26,27,28,33], in blow-up contexts or for bounded solutions. Note that the method introduced in [24,26], using the special structure of a transformed linearized problem, also has some analogy with our proof.
For global existence results in the case of semilinear and quasilinear wave equations, we refer to [12,13].
Finally, we refer to [2,3] and references therein for refined descriptions of dynamics of solutions in various settings.
1.3. Resonances and internal modes. As mentioned before, the absence of any other eigenvalue in [0, 1) for the operator L when α > 1 is important in our proof. For 0 < α ≤ 1, we continue the description of the spectrum of L. For α = 1, there is an even resonance at 1. For any 0 < α < 1, there is a third eigenvalue associated to an even eigenfunction In particular, for any 0 < α < 1, the function is solution of (5). These solutions are typical of the notion of internal modes and show that asymptotic stability (even up to the exponential instable mode) cannot be true at the linear level for such value of α. An important issue is the nature of the interaction of such internal mode with the nonlinearity. We recall that such an internal mode was treated in the context of the φ 4 equation in [18]. Pioneering results on internal modes were obtained in [36]. See other references in [18]. For α ∈ ( 1 2 , 1), there are no other eigenvalue on [0, 1). For α = 1 2 , there is an odd resonance at 1. For α ∈ ( 1 3 , 1 2 ), there is a fourth eigenvalue, associated to an odd eigenfunction. For α ∈ ( 1 4 , 1 3 ), there are five eigenvalues, three of them being associated to even eigenfunctions. In particular, there are two even internal modes. This procedure can be continued for all α > 0, showing the emergence of arbitrarily many internal modes (and sometimes resonances) as α → 0 + .
The above information is taken from Section 3 of [6].

2.1.
Decomposition of a solution in a vicinity of the soliton. Let φ = (φ, ∂ t φ) be a solution of (1) satisfying (6) for some small δ > 0. We decompose (φ, ∂ t φ) as follows where we observe that φ also writes as From (6), for all t ∈ [0, ∞), it holds Moreover, using Q ′′ − Q + f (Q) = 0, LY 0 = −ν 2 0 Y 0 and (12), the systems of equations of (a 1 , a 2 ) and (u 1 , u 2 ) write   and where 2.2. Notation for virial arguments. Let ρ be the following weight function For any function w ∈ H 1 , consider the norm We consider a smooth even function χ : R → R satisfying For A > 0, we define the functions ζ A and ϕ A as follows For B > 0, we also define and we consider the function ψ defined as The notation X Y means X ≤ CY for a constant independent of A and B. These functions ζ A , ϕ A , ζ B , ϕ B and ψ B will be used in two distinct virial arguments with different scales 3. Virial argument in u and We refer to [18] for the use of such virial argument in a similar context. Here, w represents a localized version of u 1 , in the scale A (see (24)). We shall prove the following result.
Proposition 1. There exist C 1 > 0 and δ 1 > 0 such that for any 0 < δ ≤ δ 1 , the following holds. Fix A = δ −1 . Assume that for all t ≥ 0, (15) holds. Then, for all t ≥ 0, Remark 1. Note that estimate (27) does not involve any type of spectral analysis. Its purpose is to give a simple control of (∂ x w) 2 in terms of sech( x 2 )w 2 and |a 1 | 4 .
The rest of this section is devoted to the proof of Proposition 1. We compute from (25) Replacingu 1 by u 2 and integrating by parts, the first integral in the right-hand side vanishes. The expression ofu 2 in (17) rewriteṡ To treat the first line in the expression ofİ, we claim the following. Moreover and Proof. Proof of (28). By integration by parts We rewrite the above expression using the auxiliary function w. Indeed, Next, Identity (28) follows. Proof of (29)- (30). By elementary computations, we have which proves (29). Estimate (30) then follows from the definition of χ.
To treat the second line in the expression ofİ, we claim the following.
and thus, by decay estimates on Q and Y 0 , and by (15), Using integration by parts, for an implicit constant independent of A. Thus, by the Cauchy-Schwarz inequality, Second, we decompose We rewrite I 1 , I 2 , I 3 and I 4 as follows and To control the two terms that are purely nonlinear in u 1 , we need the following claim.
Proof of Claim 1. The first equality in (36) corresponds to the definition of w in (26). Next, by integration by parts and standard estimates, we have Thus, which implies (36).
In particular, (36) implies that which takes care of the last terms in I 1 and I 4 . By Taylor expansion, α ≥ 1, |a 1 | 1 and u 1 L ∞ 1, we have Similarly, using also (35) and A ≥ 4, we find the following estimates
In particular, let (u 1 , u 2 ) be a solution of (5), and setũ 1 = U u 1 ,ũ 2 = U u 2 . Then, The key point for our analysis is that for α > 1, the potential in L 0 is positive. This property happens to be the only spectral information needed for the proof of Theorem 1.
Observe that U Y 0 = 0, U Q ′ = −αQ and SQ = 0, which means that the prior decomposition of the solution (φ, ∂ t φ) as in Section 2.1 and a coercivity argument as in Section 5 are necessary to avoid loosing information through the transformation. (Here, we work with even functions and so only the direction Y 0 is relevant.) 4.2. Transformed problem. With respect to the above heuristic, we need to localize and regularize the functions involved. For γ > 0 small to be defined later, set where χ B is defined in (23). We refer to Section 5 for coercivity results relating u 1 and v 1 . The introduction of the operator (1 − γ∂ 2 x ) −1 with a small constant γ is needed to compensate the loss of two derivatives due to the operator SU , without destroying the special algebra described heuristically. Now, we explain the role of the localization term χ B in the definitions of v 1 and v 2 . Note that Proposition 1 provides an estimate on the function w, which is a localized version of u (see (26)). To use this information, the functions v 1 and v 2 also need to contain a certain localization.
We deduce the following system for (v 1 , v 2 ) from the one for (u 1 , u 2 ) in (17) v First, we note that Therefore, we have obtained the following system for ( For this transformed system we construct a second virial functional, where the spectral analysis reduces to the fact that the potential in L 0 is positive.

4.3.
Virial functional for the transformed problem. We set and (see (22) and (23)) Here, z represents a localized version of the function v 1 . The scale of localization B is intermediate between the one involved in the definition of w from u 1 (see (24) and (26)) and the weight function ρ defined in (19) (similar to a localization at the soliton scale).

Proposition 2.
There exist C 2 > 0 and δ 2 > 0 such that for γ small enough and for any 0 < δ ≤ δ 2 , the following holds. Fix B = δ − 1 4 . Assume that for all t ≥ 0, (15) holds. Then, for all t ≥ 0,J Remark 2. The objective of estimate (41) is to control the local norm z 2 ρ up to small error in terms of w 2 ρ and |a 1 | 3 . The rest of this section is devoted to the proof of Proposition 2. As in the computation ofİ in the proof of Proposition 1, we have from (39), First, using the definition of L 0 in (37) and integrating by parts, we have By the definition of z in (40), proceeding as in the proof of (31) in Lemma 1, we have Thus, where we have set Recalling (40), (23), (22) and integrating by parts, Therefore, setting we have obtained Second, since for x ∈ [0, +∞) → ζ B (x) is non-increasing, we have for x ≥ 0, Since Q ′ (x) ≤ 0 for x ≥ 0, we obtain, for a constant C > 0, choosing B 0 large enough. By parity, this estimate holds for any x ∈ R.
Using this lemma, and the above computations for J 1 , we concludė To control the terms J 1 , J 2 , J 3 and J 4 , we need some technical estimates.
(i) Estimates on w.
(ii) Estimates on z.
We multiply by ζ B (x) and V 0 (y) and integrate in x ∈ R and y ∈ R. Since ζ B B, |x|ζ B B 2 and |y|V 0 1, we obtain (47). Proof of (48) and (49). Note by direct computations that Thus, SU f L 2 f H 2 . Moreover, using Fourier analysis, Using (51), the definition of v 1 in (38), the definition of w in (26) and A ≫ B 2 , we obtain and then (44) implies (48). Moreover, by direct computation Thus, similarly, Using (52), we obtain By the definition of w, A ≫ B 2 and the definition of χ B and ζ A , we have and χ B u 1 sech(x) L 2 w sech(x) L 2 . Thus, estimate (44) imply (49).
Lemma 5. For any 0 < K ≤ 1 and γ > 0 small enough, for any f ∈ L 2 , where the implicit constant is independent of γ and K.

4.5.
Control of error terms. Now, we are in a position to control the error terms in (43).
Control of J 1 . By the definition of ζ B , it holds Thus, using the properties of χ in (21), we have Next, since |ϕ B | B and |(χ 2 Using (48)-(49), we conclude for this term Control of J 2 . By the Cauchy-Schwarz inequality, First, we estimate using (53) From the definition of z in (40), we have Using |χ ′ | 1, the definitions of χ B and ζ B and again the definition of z Second, we also estimate using (53)

It follows using also (48) that
Thus, as before,
The value of γ being now fixed, we do not mention anymore dependency in γ. Using standard inequalities and B large enough, we obtain, for a possibly large constant C > 0, Choosing (as specified in the statement of Proposition 2) 1 4 , and next using the assumption (15), we have . Therefore, using again (15), for δ small enough (to absorb some constants), we obtaiṅ This estimate completes the proof of Proposition 2.

Coercivity and proof of Theorem 1
In this section, the constant γ is fixed as in Proposition 2.
It holds Proof. Using the expression of S and U , we rewrite (62) as Integrating between 0 and x > 0, this yields, for some constant a, which rewrites as Integrating on [0, x], x > 0, and multiplying by Y 0 , it holds, for some constant b, Let us now estimate ũ 2 sech x 2 . First, by the Cauchy-Schwarz inequality, Second, Thus, Third, since |Q ′ | Q 1, we obtain similarly, Collecting these estimates, we obtain, for all x ≥ 0, The same holds for x ≤ 0, and thus To complete the proof, we estimate the constants a and b in (65). Using (63) and parity property, projecting (65) on Y 0 yields We conclude the proof using again (65).
The next result is a consequence of the previous general lemma, in the framework of the time-dependent functions introduced in (12), (26), (38) and (40).
6. Proof of Theorem 2 6.1. Conservation of energy. Using (3) and (4) and performing a standard computation, we expand the conservation of energy (2) for a solution (φ, ∂ t φ) written under the form (11) with the orthogonality conditions (12), to obtain 3 H 1 . Using the notation (13), we have Let δ 0 > 0 be defined by Thus, by conservation of energy, estimate (78) at some t > 0 gives Under the orthogonality conditions (12), the parity of u 1 , from the spectral analysis recalled in the Introduction (see [6]), it follows that for some µ > 0, Thus, as long as u 1 H 1 + u 2 L 2 + |b + | + |b − | δ 1/2 0 , the following energy estimate holds 6.2. Construction of the graph. By the energy estimate (80), Lemma 8 and a standard contradiction argument, we construct initial data leading to global solutions close to the ground state Q.