Spatial decay of multi-solitons of the generalized Korteweg-de Vries and nonlinear Schrödinger equations

Abstract

We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de Vries equations. We obtain that, uniformly in time, these solutions and their derivatives decay exponentially in space on the left of and in the solitons region, and prove rapid decay on the right of the solitons. We also prove the corresponding result for multi-solitons of the nonlinear Schrödinger equations, that is, exponential decay in the solitons region and rapid decay outside.

A. Appendix

1.1 A.1 Growth of the $H^{s}$ norms of the 1-D solitons

The purpose of this appendix is to make the constants more explicit as stated in Section 3. We restrict to the space dimension 1 and monomial nonlinearity.

Proposition A.1

For all $\mu >\sqrt{p}$, there exists $s_0$ such that for all $s\geqslant s_0$,

$$\begin{aligned} \Vert Q\Vert _{H^s}\leqslant 2^{\mu ^s}. \end{aligned}$$

(A.1)

Proof

Differentiating the fundamental equation satisfied by Q, that is $Q''+Q^p=Q$, we obtain the following induction formula:

$$\begin{aligned} \forall s\in \mathbb {N},\quad Q^{(s+2)}=Q^{(s)}-\sum _{i_1+\dots +i_p=s}\genfrac(){0.0pt}0{s}{i_1,\dots ,i_p}Q^{(i_1)}\dots Q^{(i_p)}. \end{aligned}$$

Let us observe that for $i_1,\dots ,i_p\in \mathbb {N}$ such that $i_1+\dots +i_p=s$,

• if there exists $j\in \{1,\dots ,p\}$ such that $i_j=s$, then

  $$\begin{aligned} \int _\mathbb {R}\left( Q^{(i_1)}\dots Q^{(i_p)}\right) ^2\,dx&=\int _\mathbb {R}\left( Q^{(s)}\right) ^2 Q^{2(p-1)}\,dx \\&\leqslant \Vert Q\Vert _{L^\infty }^{2(p-1)}\int _\mathbb {R}\left( Q^{(s)}\right) ^2 \,dx \leqslant C\Vert Q\Vert _{H^1}^{2(p-1)}\Vert Q\Vert _{H^s}^2 \end{aligned}$$

  (C being a constant depending only on p);

• if for all $j\in \{1,\dots ,p\}$ such that $i_j\leqslant s-1$, then

  $$\begin{aligned} \int _\mathbb {R}\left( Q^{(i_1)}\dots Q^{(i_p)}\right) ^2\,dx&\leqslant \prod _{k=2}^p\Vert Q^{(i_k)}\Vert ^2_{L^\infty }\int _\mathbb {R}\left( Q^{(i_1)}\right) ^2 \,dx\\&\leqslant C\prod _{k=2}^p\Vert Q\Vert _{H^{i_{k}+1}}^{2}\Vert Q\Vert _{H^{i_1}}^2 \leqslant C\Vert Q\Vert _{H^s}^{2(p-1)}\Vert Q\Vert _{H^{s-1}}^2, \end{aligned}$$

  (C being again a constant depending only on p, which can change from one line to the other).

Thus for $s\in \mathbb {N}^*$,

$$\begin{aligned} \Vert Q^{(s+2)}\Vert _{L^2}\leqslant C\left( \Vert Q^{(s)}\Vert _{L^2}+p^s\Vert Q\Vert _{H^s}^p\right) , \end{aligned}$$

which implies

$$\begin{aligned} \Vert Q\Vert _{H^{s+2}}\leqslant Cp^s\Vert Q\Vert _{H^s}^p, \end{aligned}$$

with a constant C depending only on p.

We finally obtain

$$\begin{aligned} \Vert Q\Vert _{H^s}&\leqslant C^{\frac{p^{\lfloor \frac{s}{2}\rfloor }-1}{p-1}}p^{p^{\lfloor \frac{s}{2}\rfloor } s}\Vert Q\Vert _{H^1} \leqslant 2^{\mu ^s}, \end{aligned}$$

with $\mu >\sqrt{p}$ and s sufficiently large. $\square$

Remark 4.5

As a corollary of Proposition A.1, we also obtain the existence of a constant C depending on p and the soliton parameters such that for all $s\in \mathbb {N}$,

$$\begin{aligned} \left\| \left( \sum _{j=1}^NR_j(t)\right) ^p-\sum _{j=1}^NR_j(t)^p\right\| _{H^s}\leqslant Cp^s\max _{j=1,\dots ,N}\Vert R_j\Vert _{H^s}^pe^{-2\theta t}. \end{aligned}$$

(A.2)

1.2 A.2 Proof of estimate (3.3)

The goal is here to make explicit the constants appearing in the computations done by Martel [16, Section 3.4] in the proof of smoothness of the multi-solitons.

We thus repeat the arguments developed by Martel (presented slightly differently), keeping track of the growth of the constant $\lambda _s$ (with respect to s) such that

$$\begin{aligned} \forall t\geqslant T_0, \quad \Vert z(t)\Vert _{H^s}\leqslant \lambda _se^{-\theta t}. \end{aligned}$$

Proof

We consider regularity indices $s\geqslant 5$, as that case makes the argument easier: the point being that the exponential decay rate $\theta$ does not change when we go from the estimation of $\Vert z\Vert _{H^{s-1}}$ to that of $\Vert z\Vert _{H^s}$; there is a loss for $s=2$ (treated in detail in [16]), which can be avoided for $s=3,4$ using an extra argument, see the footnote below in the proof.

The starting point is to study the variations of

$$\begin{aligned} \frac{d}{dt}\int _{\mathbb {R}}\left( \partial _x^{s}z\right) ^2\,dx=2\int _\mathbb {R}\partial _x^s\left( \left( z+R\right) ^p-\sum _{j=1}^NR_j^p\right) \partial _x^{s+1}z\,dx. \end{aligned}$$

Thus, the terms which have to be controlled are the source term (involving z only linearly)

$$\begin{aligned} 2\int _{\mathbb {R}}\partial _x^{s+1}\left( R^p-\sum _{j=1}^NR_j^p\right) \partial _x^sz\,dx. \end{aligned}$$

and

$$\begin{aligned}&-2\int _{\mathbb {R}}\partial _x^{s+1}\left( \sum _{k=1}^p\genfrac(){0.0pt}0{p}{k}z^kR^{p-k}\right) \partial _x^sz\,dx= -2\sum _{k=1}^p\genfrac(){0.0pt}0{p}{k}\sum _{i_1+\dots +i_p=s+1}\genfrac(){0.0pt}0{s+1}{i_1,\dots ,i_p}I_{k,i_1,\dots ,i_p},\\&\qquad \text {where} \quad I_{k,i_1,\dots ,i_p}=\int _\mathbb {R}\partial _x^{i_1}z\dots \partial _x^{i_k}z\partial _x^{i_{k+1}}R\dots \partial _x^{i_p}R\partial _x^sz\,dx. \end{aligned}$$

(i) For $k=1$, integrating by parts, we have

$$\begin{aligned}&\sum _{i_1+\dots +i_p=s+1}\genfrac(){0.0pt}0{s+1}{i_1,\dots ,i_p}\int _\mathbb {R}\partial _x^{i_1}z\partial _x^{i_2}R\dots \partial _x^{i_p}R\partial _x^sz\,dx \nonumber \\&\quad \quad = \int _\mathbb {R}\partial _x^{s+1}zR^{p-1}\partial _x^sz\,dx+(s+1)(p-1)\int _\mathbb {R}\partial _x^sz\partial _xRR^{p-2}\partial _x^sz\,dx \nonumber \\&\quad \qquad +\sum _{\underset{\forall j,i_j\leqslant s-1}{i_1+\dots +i_p=s+1}}\int _\mathbb {R}\partial _x^{i_1}z\partial _x^{i_2}R\dots \partial _x^{i_p}R\partial _x^sz\,dx \nonumber \\&\quad \quad = \frac{2s+1}{2}\int _\mathbb {R}\left( \partial _x^sz\right) ^2\partial _x(R^{p-1})\,dx+\sum _{\underset{\forall j,i_j\leqslant s-1}{i_1+\dots +i_p=s+1}}\int _\mathbb {R}\partial _x^{i_1}z\partial _x^{i_2}R\dots \partial _x^{i_p}R\partial _x^sz\,dx. \end{aligned}$$

(A.3)

The first integral

$$\begin{aligned} \frac{2s+1}{2} \int _\mathbb {R}\left( \partial _x^sz\right) ^2\partial _x(R^{p-1})\,dx \end{aligned}$$

(A.4)

can not be bounded directly in a suitable way. The key idea in [16] is to add a lower order term

$$\begin{aligned} \frac{2s+1}{3}p\int _{\mathbb {R}}\left( \partial _x^{s-1}z\right) ^2R^{p-1}\,dx \end{aligned}$$

whose variation at leading order will precisely cancel (A.4). Thus we are lead to consider

$$\begin{aligned} F_s(t){:}{=}\int _{\mathbb {R}}\left( \partial _x^{s}z\right) ^2\,dx-\frac{2s+1}{3}p\int _{\mathbb {R}}\left( \partial _x^{s-1}z\right) ^2R^{p-1}\,dx. \end{aligned}$$

Going back to (A.3), we bound

$$\begin{aligned} \left| \sum _{\underset{\forall j,i_j\leqslant s-1}{i_1+\dots +i_p=s+1}}\int _\mathbb {R}\partial _x^{i_1}z\partial _x^{i_2}R\dots \partial _x^{i_p}R\partial _x^sz\,dx\right| \leqslant Cp^{s+1}\Vert z\Vert _{H^{s-1}}^2\Vert R\Vert _{H^s}^{p-1}. \end{aligned}$$

(A.5)

(ii) Let us now consider the case where $2\leqslant k\leqslant p$. If there exists $j\in \left\{ 1,\dots ,k\right\}$ such that $i_j=s+1$, then integrating by parts,

$$\begin{aligned} | I_{k,i_1,\dots ,i_p}|&= \frac{1}{2} | \int _\mathbb {R}\left( \partial _x^sz\right) ^2\partial _x\left( z^{k-1}R^{p-k}\right) \,dx | \nonumber \\&\leqslant \frac{1}{2}\left\| \partial _x\left( R^{p-k}z^{k-1}\right) \right\| _{L^\infty }\Vert z\Vert _{H^s}^2 \leqslant \frac{1}{2}\Vert R\Vert _{H^2}^{p-k}\Vert z\Vert _{H^2}^{k-1}\Vert z\Vert _{H^s}^2. \end{aligned}$$

(A.6)

If there exists $j\in \left\{ 1,\dots ,k\right\}$ such that $i_j=s$, then

$$\begin{aligned} |I_{k,i_1,\dots ,i_p}|\leqslant C\Vert z\Vert _{H^2}^{k-1}\Vert R\Vert _{H^2}^{p-k}\int _{\mathbb {R}}\left( \partial _x^sz\right) ^2\,dx. \end{aligned}$$

(A.7)

If there exists $j\in \left\{ 1,\dots ,k\right\}$ such that $i_j=s-1$, then for all $j'\in \{1,\dots ,k\}$ such that $j'\ne j$, $i_{j'}\leqslant \sum _{l=1}^pi_l-s+1\leqslant 2\leqslant s-3$ (by the choice of $s\geqslant 5$)³. Hence, integrating by parts,

$$\begin{aligned} |I_{k,i_1,\dots ,i_p}|&= \left| -\frac{1}{2}\int _\mathbb {R}\left( \partial _x^{s-1}z\right) ^2\partial _x\left( \partial _x^{i'_1}z\dots \partial _x^{i'_{k-1}}z\partial _x^{i_{k+1}}R\dots \partial _x^{i_p}R\right) \,dx \right| \nonumber \\&\leqslant \frac{1}{2}\Vert z\Vert _{H^{s-1}}^2\left\| \partial _x\left( \partial _x^{i'_1}z\dots \partial _x^{i'_{k-1}}z\partial _x^{i_{k+1}}R\dots \partial _x^{i_p}R\right) \right\| _{L^\infty } \nonumber \\&\leqslant C\Vert z\Vert _{H^{s-1}}^{k+1}\Vert R\Vert _{H^{s+1}}^{p-k}, \end{aligned}$$

(A.8)

where C is a universal constant depending only on p.

In the other cases, $i_1,\dots ,i_{k}\leqslant s-2$ and so

$$\begin{aligned} |I_{k,i_1,\dots ,i_p}|&\leqslant \, \left| -\int _\mathbb {R}\partial _x^{s-1}z\partial _x\left( \partial _x^{i_1}z\dots \partial _x^{i_k}z\partial _x^{i_{k+1}}R\dots \partial _x^{i_p}R\right) \,dx\right| \leqslant \Vert z\Vert _{H^{s-1}}^{k+1}\Vert R\Vert _{H^{s-1}}^{p-k}. \end{aligned}$$

(A.9)

(iii) For the source term, by integration by parts,

$$\begin{aligned} \left| 2\int _{\mathbb {R}}\partial _x^{s+1}\left( R^p-\sum _{j=1}^NR_j^p\right) \partial _x^sz\,dx\right|&\leqslant C\Vert z\Vert _{H^{s-1}}\left\| R^p-\sum _{j=1}^NR_j^p\right\| _{H^{s+2}}\\&\leqslant C\lambda _{s-1}e^{-\theta t}\left\| R^p-\sum _{j=1}^NR_j^p\right\| _{H^{s+2}}. \end{aligned}$$

(iv) When computing the time differential of $\displaystyle \int _{\mathbb {R}}\left( \partial _x^{s-1}z\right) ^2R^{p-1}\,dx$, as mentioned (and by construction), one term cancels (A.4) and the others are bounded as in (i) and (ii).

Gathering the bounds (A.5), (A.6), (A.7), (A.8), (A.9), it results that

$$\begin{aligned} \left| \frac{d}{dt}F_s(t)\right| \leqslant&\, C\lambda _{s-1}e^{-\theta t}\left\| R^p-\sum _{j=1}^NR_j^p\right\| _{H^{s+1}} +C\sum _{k=2}^p\genfrac(){0.0pt}0{p}{k}\Vert z\Vert _{H^s}^2p^{s+1}\Vert z\Vert _{H^2}^{k-1}\Vert R\Vert _{H^s}^{p-k} \nonumber \\&\quad +C\sum _{k=2}^p\genfrac(){0.0pt}0{p}{k}p^{s+1}\Vert z\Vert _{H^{s-1}}^{k+1}\Vert R\Vert _{H^s}^{p-k} \nonumber \\&\leqslant C\lambda _{s-1}e^{-\theta t}p^{s+2}2^{\mu ^{s+2}p}e^{-2\theta t}+C\Vert z\Vert _{H^s}^2e^{-2\theta t}p^{s+1}2^p2^{\mu ^s(p-2)} \nonumber \\&\quad +Ce^{-3\theta t}p^{s+1}2^p2^{\mu ^s(p-2)}\lambda _{s-1}^{p+1} \nonumber \\&\leqslant C\left( p^s2^{\mu ^sp}e^{-2\theta t}\Vert z\Vert _{H^s}^2+p^s2^{\mu ^sp}\lambda _{s-1}^{p+1}e^{-3\theta t}\right) . \end{aligned}$$

(A.10)

Moreover, we have by definition of $F_s$,

$$\begin{aligned} \int _{\mathbb {R}}\left( \partial _x^{s}z\right) ^2\,dx&\leqslant |F_s|+Cs\Vert R\Vert _{H^2}^{p-1}\Vert \partial _x^{s-1}z\Vert _{L^2}^2 \leqslant |F_s|+Cs\lambda _{s-1}^2e^{-2\theta t}. \end{aligned}$$

Hence,

$$\begin{aligned} \left| \frac{d}{dt}F_s(t)\right| \leqslant Cp^s2^{\mu ^sp}e^{-2\theta t}\left( |F_s(t)|+s\lambda _{s-1}^2e^{-2\theta t}\right) +Cp^s2^{\mu ^sp}\lambda _{s-1}^{p+1}e^{-3\theta t}. \end{aligned}$$

By integration we finally obtain

$$\begin{aligned} |F_s(t)|\leqslant Cp^{s}2^{\mu ^sp}\lambda _{s-1}^{p+1}e^{-2\theta t} \end{aligned}$$

and thus

$$\begin{aligned} \Vert z\Vert _{H^{s}}^2\leqslant Cp^{s}2^{\mu ^sp}\lambda _{s-1}^{p+1}e^{-2\theta t}. \end{aligned}$$

We can therefore take

$$\begin{aligned} \lambda _{s}\leqslant Cp^{\frac{s}{2}}2^{\frac{p}{2}\mu ^s}\lambda _{s-1}^{\frac{p+1}{2}}, \end{aligned}$$

(A.11)

which yields via an induction

$$\begin{aligned} \lambda _{s}\leqslant C^{\sum _{k=0}^{s-1}\left( \frac{p+1}{2}\right) ^k}p^{\sum _{k=0}^{s-1}\frac{s-k}{2}\left( \frac{p+1}{2}\right) ^k}\lambda _0^{\left( \frac{p+1}{2}\right) ^{s}}2^{\frac{p}{2}\sum _{k=0}^{s-1}\mu ^{s-k}\left( \frac{p+1}{2}\right) ^k}, \end{aligned}$$

and so,

$$\begin{aligned} \lambda _{s}&\leqslant C^{2\left( \frac{p+1}{2}\right) ^{s}}p^{\frac{s}{2}\left( \frac{p+1}{2}\right) ^{s-1}}2^{\left( \max \{\mu ,\frac{p+1}{2}\}\right) ^s}\lambda _0^{\left( \frac{p+1}{2}\right) ^{s}} \leqslant 2^{\mu _0^s}, \end{aligned}$$

(A.12)

with $\mu _0>\max \{\mu ,\frac{p+1}{2}\}>\max \left\{ \sqrt{p},\frac{p+1}{2}\right\}$ and s sufficiently large (depending on $\mu _0$). $\square$

Remark 4.6

Note that we can refine a bit (A.8) but the final estimate concerning $\lambda _s$ would not be better than (A.12), considering that $\mu _0$ is to be chosen strictly greater than $\max \{\sqrt{p},\frac{p+1}{2}\}$.

1.3 A.3 Details concerning Remarks 3.5 and 3.11

Let $s\in \mathbb {N}$. In order to ensure that $\int _{x>\beta t}\left( \partial _x^sz\right) ^2e^{\epsilon (x-\beta t)}\,dx$ is finite for some $\epsilon >0$, it suffices by (3.17) that

$$\begin{aligned} \sum _{n=0}^{+\infty }\int _\mathbb {R}\left( \partial _x^sz\right) ^2\epsilon ^n\varphi _{[n]}\,dx \end{aligned}$$

is finite. Thus it suffices that the series $\sum _{n\geqslant 0}C(s,\varphi _{[n]})\epsilon ^n$ converges.

This condition is satisfied under the following assumptions:

$$\begin{aligned} C(s,\varphi _{[1]}) \leqslant c_0C(s+1,\varphi ) \quad \text {and} \quad \lambda _s \leqslant \tilde{c_0}^s. \end{aligned}$$

(A.13)

Indeed, if we assume (A.13), we obtain

$$\begin{aligned} C(s,\varphi _{[n]})\leqslant c_0^n\tilde{c_0}^{2(s+n)}\leqslant \tilde{c_0}^{2s}(c_0\tilde{c_0}^2)^n, \end{aligned}$$

which guarantees the existence of $\epsilon >0$ such that the series $\sum _{n\geqslant 0}C(s,\varphi _{[n]})\epsilon ^n$ converges.

From Proposition 3.3, we also deduce, proceeding step by step, that:

$$\begin{aligned} C(s,\varphi _{[n]})&\leqslant c(\eta ,\kappa _1,\kappa _2)2^{\mu _0^s}C(s+1,\varphi )\\&\leqslant c(\eta ,\kappa _1,\kappa _2)^n2^{\mu _0^s+\dots +\mu _0^{s+n-1}}C(s+n,\varphi )\\&\leqslant c(\eta ,\kappa _1,\kappa _2)^n2^{n\mu _0^{s+n-1}}\lambda _{s+n}^2. \end{aligned}$$

Taking $\tilde{\mu _0}>\mu _0$, this shows that the integral

$$\begin{aligned} \int _{x \geqslant \beta t}\left( \partial _x^sz\right) ^2(t,x)\left( \sum _{n=0}^{+\infty }\frac{(x-\beta t)^n}{2^{\tilde{\mu _0}^{s+n}}}\right) \,dx \end{aligned}$$

is finite.