On existence and uniqueness of asymptotic N-soliton-like solutions of the nonlinear Klein–Gordon equation

Abstract

We are interested in solutions of the nonlinear Klein–Gordon equation (NLKG) in $\mathbb {R}^{1+d}$, $d\ge 1$, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schrödinger equations, we obtain an N-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of N given (unstable) solitons. For $N=1$, this family completely describes the set of solutions converging to the soliton considered; for $N\ge 2$, we prove uniqueness in a class with explicit algebraic rate of convergence.

Appendix

1.1 Extension of the proofs to higher dimensions

The main parts of the proofs remain obviously unchanged. Essentially three notable adaptations are to be made, passing from the one-dimensional case to higher dimensions.

In a first instance, one has to be careful about how establishing several estimates. Although all estimates we have proved in the previous sections (in dimension 1) are identical for general d, the way we establish them when $d\ge 2$ can be altered.

For example, we point out that it is no longer possible to use the Sobolev embedding $H^1\hookrightarrow L^\infty$ when $d\ge 2$. Particularly, multi-solitons in dimension $d\ge 2$ do not necessarily take values in $L^\infty (\mathbb {R}^d)$ and in order to estimate a quantity like

$$\begin{aligned}f(u)-f(\varphi )-f'(\varphi )(u-\varphi )\end{aligned}$$

(as for proving Claim 3.6), we would proceed as follows: by (H’1), we deduce that $|f''(r)|=p(p-1)|r|^{p-2}$, thus applying Taylor formula, we have for fixed time $t\in \mathbb {R}$ and position $x\in \mathbb {R}^d$,

$$\begin{aligned} \left| f(u)-f(\varphi )-f'(\varphi )(u-\varphi )\right| (t,x)&\le \frac{|u-\varphi |^2(t,x)}{2}\sup _{r\in [\varphi (t,x),u(t,x)]}|f''(r)|\\&\le C|u-\varphi |^2(t,x)\left( |\varphi |^{p-2}(t,x)+|u|^{p-2}(t,x)\right) . \end{aligned}$$

Now, for all $\psi \in L^\infty (\mathbb {R}^d)$ and for all $z\in H^1(\mathbb {R}^d)$, Hölder inequality yields

$$\begin{aligned}\left| \int _{\mathbb {R}^d}|\psi ||u(t)-\varphi (t)|^{2}|z|^{p-2}\;dx\right| \le C\Vert \psi \Vert _{L^\infty }\left( \int _{\mathbb {R}^d}|u(t)-\varphi (t)|^{p}\;dx\right) ^{\frac{2}{p}}\left( \int _{\mathbb {R}^d}|z|^{p}\;dx\right) ^{\frac{p-2}{p}}.\end{aligned}$$

Finally, replacing z by $\varphi (t)$ and $u(t)-\varphi (t)$, we obtain an estimate of

$$\begin{aligned}\int _{\mathbb {R}^d}\psi \left( f(u)-f(\varphi )-f'(\varphi )(u-\varphi )\right) \;dx\end{aligned}$$

in terms of $\Vert u-\varphi \Vert _{H^1}$, $\Vert u\Vert _{H^1}$, and $\Vert \varphi \Vert _{H^1}$ due to the Sobolev embedding $H^1(\mathbb {R}^d)\hookrightarrow L^p(\mathbb {R}^d)$.

Secondly, in view of Proposition 2.1, one has to take into account, for all $i=1,\dots ,N$ the d directions which generate the kernel of the operator $\mathscr {H}_{\beta _i}$ when we practice modulation in dimension d. For instance, in Lemma 4.1, we would define E as follows:

$$\begin{aligned} E:=Z-\sum _{i=1}^Na_i\cdot \nabla R_i-\sum _{i=1}^Nb_iY_{+,i},\end{aligned}$$

with $a_i(t)\in \mathbb {R}^d$ and $b_i(t)\in \mathbb {R}$ such that for all $i=1,\dots ,N$ and for all $j=1,\dots ,d$:

$$\begin{aligned} \left\langle E(t),\partial _{x_j}R_i(t)\right\rangle&=0 \end{aligned}$$

(7.1)

$$\begin{aligned} \left\langle E(t),Z_{-,i}(t)\right\rangle&=0. \end{aligned}$$

(7.2)

But each time this extension does not affect the sequel; in other words, the estimates we are supposed to obtain afterwards and their proofs are the same.

A third change to be done concerns the way we define the different Lyapunov functionals which are studied throughout the article. To deal with dimensions greater or equal than 2, we reduce the problem to the case of a one-dimensional variable. For instance, let us explain how to generalize Step 4 in subsection 3.2.2 to all dimensions. The subset

$$\begin{aligned}\mathcal {M}:=\bigcup _{i\ne j}\left\{ \ell \in \mathbb {R}^d|\;\ell \cdot (\beta _j-\beta _i)=0\right\} \end{aligned}$$

of $\mathbb {R}^d$ is of zero Lebesgue measure. Hence, there exists $\ell \in \mathbb {R}^d$ such that for all $i\ne j$,

$$\begin{aligned}\ell \cdot (\beta _j-\beta _i)\ne 0.\end{aligned}$$

In particular $\ell \ne 0$ and, even if it means considering $\frac{\ell }{|\ell |}$, we can assume that $|\ell |=1$, so that $\forall \;i=1,\dots ,N$, $|\ell \cdot \beta _i|<1$. Now, defining $\tilde{\beta }_i:=\ell \cdot \beta _i$, and even if it means changing the permutation $\eta$, we have

$$\begin{aligned}-1<\tilde{\beta }_{\eta (1)}<\tilde{\beta }_{\eta (2)}<\dots<\tilde{\beta }_{\eta (N)}<1.\end{aligned}$$

Then, the direction described by $\ell$ is to be favored: we consider the following cut-off functions:

$$\begin{aligned}\psi _k(t)=\psi \left( \frac{1}{\sqrt{t}}\left( \ell \cdot x-\frac{\tilde{\beta }_{\eta (k)}+\tilde{\beta }_{\eta (k+1)}}{2}-\ell \cdot \frac{x_{\eta (k)}+x_{\eta (k+1)}}{2}\right) \right) .\end{aligned}$$

At this stage, the definition of the functions $\phi _k$ in terms of the $\psi _k$ is kept unchanged and the corresponding Lyapunov functional is to be written:

$$\begin{aligned}\mathcal {F}_{W}(t)=\sum _{k=1}^K\int _{\mathbb {R}^d}\left( w_1^2+(\partial _xw_{1})^2+w_2^2-f'(Q_{\eta (k)})w_1^2+2\beta _{\eta (k)}\cdot \nabla w_1w_2\right) \phi _k\;dx.\end{aligned}$$

1.2 Proof of Corollary 1.8

The proof is an immediate adaptation of that of Proposition 4.12 in [3].

Let $A>0$ and denote $t_A:=-\frac{\ln (A)}{e_\beta }$. In the sense of the $H^1\times L^2$-norm, we have:

$$\begin{aligned} U^1(t+t_A,\cdot +\beta t_A)&=R_\beta (t+t_A,\cdot +\beta t_A)+e^{-e_\beta (t+t_A)}Y_{+,\beta }(t+t_A,\cdot +\beta t_A)+\mathrm {O}\left( e^{-2e_\beta t}\right) \\&=R_\beta (t)+Ae^{-e_\beta t}Y_{+,\beta }(t)+\mathrm {O}\left( e^{-2e_\beta t}\right) . \end{aligned}$$

Then, $\Vert U^1(t+t_A,\cdot +\beta t_A)-R_\beta (t)\Vert _{H^1\times L^2}\underset{t\rightarrow +\infty }{\rightarrow }0$ so that there exist $\tilde{A}\in \mathbb {R}$ and $t_0=t_0(\tilde{A})\in \mathbb {R}$ such that for all $t\ge t_0$,

$$\begin{aligned}U^{\tilde{A}}(t)=U^1(t+t_A,\cdot +\beta t_A).\end{aligned}$$

But on the other hand,

$$\begin{aligned}U^{\tilde{A}}(t)=R_\beta (t)+\tilde{A}e^{-e_\beta t}Y_{+,\beta }(t)+\mathrm {O}\left( e^{-2e_\beta t}\right) .\end{aligned}$$

Hence,

$$\begin{aligned}(A-\tilde{A})e^{-e_\beta t}Y_{+,\beta }(t)=\mathrm {O}\left( e^{-2e_\beta t}\right) ,\end{aligned}$$

which implies $A=\tilde{A}$. Consequently, $U^A(t)=U^1(t+t_A,\cdot +\beta t_A).$

If $A<0$, we have just to repeat the above argument with $-A$ instead of A.

Lastly, let us identify $U^0$. Given that $R_\beta$ is a solution of (NLKG) which satisfies (1.5), Theorem 1.2 provides the existence of $A\in \mathbb {R}$ and of $t_0\in \mathbb {R}$ such that for all $t\ge t_0$, $U^A(t)=R_\beta (t)$.

Since $U^A$ satisfies (1.4), we deduce that

$$\begin{aligned}\left\| Ae^{-e_\beta t}Y_{+,\beta }(t)\right\| _{H^1\times L^2}\le Ce^{-2e_\beta t}.\end{aligned}$$

Thus $A=0$ and $U^0=R_\beta$ is defined for all $t\in \mathbb {R}$.

1.3 A result of analytic theory of differential equations

Lemma 7.1

Let $t_0\in \mathbb {R}$, $\mathcal {A}:[t_0,+\infty )\rightarrow \mathbb {R}$ be a $\mathscr {C}^1$ bounded function, and $\xi :[t_0,+\infty )\rightarrow \mathbb {R}^+$ be continuous and integrable.

If, for some $\rho >0$,

$$\begin{aligned}\forall \;t\ge t_0,\qquad |\mathcal {A}'(t)+\rho \mathcal {A}(t)|\le \xi (t)\sup _{t'\ge t}|\mathcal {A}(t')|,\end{aligned}$$

then there exists $c>0$ such that

$$\begin{aligned}\forall \;t\ge t_0,\qquad |\mathcal {A}(t)|\le ce^{-\rho t}.\end{aligned}$$

Proof

Let us assume that

$$\begin{aligned} \forall \;t\ge t_0,\qquad |\mathcal {A}'(t)+\rho \mathcal {A}(t)|\le \xi (t)\sup _{t'\ge t}|\mathcal {A}(t')|, \end{aligned}$$

(7.3)

for some $\rho >0$. Then for all $t\ge t_0$,

$$\begin{aligned}|(e^{\rho t}\mathcal {A})'(t)|\le \xi (t)e^{\rho t}\sup _{t'\ge t}|\mathcal {A}(t')|.\end{aligned}$$

Let us consider $t\ge t_0$. For $t'\ge t$, we obtain by integration

$$\begin{aligned}|e^{\rho t'}\mathcal {A}(t')- e^{\rho t}\mathcal {A}(t)|\le \int _t^{t'}\xi (s)e^{\rho s}\sup _{u\ge s}|\mathcal {A}(u)|\;ds.\end{aligned}$$

This implies that, for $t'\ge t$,

$$\begin{aligned}e^{\rho t'}|\mathcal {A}(t')|\le e^{\rho t}|\mathcal {A}(t)|+\sup _{u\ge t}|\mathcal {A}(u)|e^{\rho t'}\int _t^{t'}\xi (s)\;ds.\end{aligned}$$

From the preceding line, we deduce that for all $t'\ge t$,

$$\begin{aligned} |\mathcal {A}(t')|\le |\mathcal {A}(t)|+\sup _{u\ge t}|\mathcal {A}(u)|\int _t^{+\infty }\xi (s)\;ds. \end{aligned}$$

(7.4)

Now we consider $t_1\ge t_0$ such that $\int _{t_1}^{+\infty }\xi (s)\;ds<\frac{1}{2}$ (which is indeed possible given that $\int _t^{+\infty }\xi (s)\;ds\rightarrow 0$ as $t\rightarrow +\infty$). By passing to the supremum on $t'$ in (7.4), we obtain for all $t\ge t_1$,

$$\begin{aligned}\sup _{t'\ge t}|\mathcal {A}(t')|\le 2|\mathcal {A}(t)|.\end{aligned}$$

Consequently, assumption (7.3) becomes

$$\begin{aligned} \forall \;t\ge t_1,\qquad |\mathcal {A}'(t)+\rho \mathcal {A}(t)|\le 2\xi (t)|\mathcal {A}(t)|. \end{aligned}$$

(7.5)

Let us define $y(t):=e^{\rho t}|\mathcal {A}(t)|$. By integration of (7.5), we obtain

$$\begin{aligned} \forall \;t\ge t_1,\qquad y(t)\le y(t_1)+\int _{t_1}^t2\xi (s)y(s)\;ds. \end{aligned}$$

(7.6)

By a standard Grönwall argument, we conclude to the existence of $C>0$ such that for all $t\ge t_1, y(t)\le C$, which implies the desired result. For the sake of completeness, let us explicit this argument.

We define $Y(t):=\exp \left( -\int _{t_1}^t2\xi (s)\;ds\right) \int _{t_1}^t2\xi (s)y(s)\;ds$ for $t\ge t_1$. The function Y is $\mathscr {C}^1$ on $[t_1,+\infty )$ and for all $t\ge t_1$,

$$\begin{aligned} Y'(t)&=2\xi (t)\exp \left( -\int _{t_1}^t2\xi (s)\;ds\right) \left[ y(t)-\int _{t_1}^t2\xi (s)y(s)\;ds\right] \\&\le 2y(t_1)\xi (t)\exp \left( -\int _{t_1}^t2\xi (s)\;ds\right) \end{aligned}$$

by (7.6). Integrating the preceding inequality and observing that $Y(t_1)=0$, we have

$$\begin{aligned}Y(t)\le \int _{t_1}^t2\xi (s)y(t_1)\exp \left( -\int _{t_1}^s2\xi (u)\;du\right) \;ds.\end{aligned}$$

We then infer

$$\begin{aligned} \int _{t_1}^t2\xi (s)y(s)\;ds=\exp \left( \int _{t_1}^t2\xi (s)\;ds\right) Y(t)\le 2y(t_1)\int _{t_1}^t\xi (s)\exp \left( \int _s^t2\xi (u)\;du\right) \;ds. \end{aligned}$$

(7.7)

Lastly, we denote $\nu :=\int _{t_1}^{+\infty }\xi (s)\;ds$; gathering (7.6) and (7.7), we obtain

$$\begin{aligned}\forall \;t\ge t_1,\qquad y(t)\le y(t_1)+2y(t_1)e^{2\nu }\nu .\end{aligned}$$

This achieves the proof of Lemma 7.1. $\square$