Multi solitary waves to stochastic nonlinear Schrödinger equations

Abstract

In this paper, we present a pathwise construction of multi-soliton solutions for focusing stochastic nonlinear Schrödinger equations with linear multiplicative noise, in both the $L^{2}$-critical and subcritical cases. The constructed multi-solitons behave asymptotically as a sum of K solitary waves, where K is any given finite number. Moreover, the convergence rate of the remainders can be of either exponential or polynomial type, which reflects the effects of the noise in the system on the asymptotical behavior of the solutions. The major difficulty in our construction of stochastic multi-solitons is the absence of pseudo-conformal invariance. Unlike in the deterministic case (Merle in Commun Math Phys 129:223–240, 1990; Röckner et al. in Multi-bubble Bourgain–Wang solutions to nonlinear Schrödinger equation, ar**v: 2110.04107, 2021), the existence of stochastic multi-solitons cannot be obtained from that of stochastic multi-bubble blow-up solutions in Röckner et al. (Multi-bubble Bourgain–Wang solutions to nonlinear Schrödinger equation, ar**v:2110.04107, 2021), Su and Zhang (On the multi-bubble blow-up solutions to rough nonlinear Schrödinger equations, ar**v:2012.14037v1, 2020). Our proof is mainly based on the rescaling approach in Herr et al. (Commun Math Phys 368:843–884, 2019), relying on two types of Doss–Sussman transforms, and on the modulation method in Côte and Friederich (Commun Partial Differ Equ 46:2325–2385, 2021), Martel and Merle (Ann Inst H Poincaré Anal Non Linéaire 23:849–864, 2006), in which the crucial ingredient is the monotonicity of the Lyapunov type functional constructed by Martel et al. (Duke Math J 133:405-466, 2006). In our stochastic case, this functional depends on the Brownian paths in the noise.

Appendix

1.1 Linearized operator

Let $L=(L_+,L_-)$ be the linearized operator around the ground state state defined by

$$\begin{aligned} L_{+}:= -\Delta + I -(1+p)Q^{p}, \ \ L_{-}:= -\Delta +I -Q^{p}. \end{aligned}$$

(6.1)

For any complex valued $H^1$ function, set $f := f_1 + i f_2$ in terms of the real and imaginary parts and

$$\begin{aligned} (Lf,f) :=\int f_1L_+f_1dx+\int f_2L_-f_2dx. \end{aligned}$$

(6.2)

The crucial coercivity property of linearized operators in the subcritical and critical case are summarized below.

Lemma 6.1

([61], see also [47, Lemma 2.2]) Let $1< p< 1+\frac{4}{d}$. Then, there exists $C>0$ such that

$$\begin{aligned} (Lf,f)\ge C\Vert f\Vert _{H^1}^2 -\frac{1}{C}\left( \left( \int Qf_1dx\right) ^2+\left( \int Qf_2dx\right) ^2+\left( \int \nabla Qf_1dx\right) ^2\right) , \end{aligned}$$

(6.3)

where $f=f_1+if_2$.

Lemma 6.2

( [16, Proposition 3.17]) Let $p=1 + \frac{4}{d}$. Then, there exists $C>0$ such that

$$\begin{aligned} (Lf,f)&\ge C\Vert f\Vert _{H^1}^2-\frac{1}{C}\nonumber \\&\quad \times \left( \left( \int Qf_1dx\right) ^2+\left( \int Qf_2dx\right) ^2+\left( \int \nabla Qf_1dx\right) ^2+\left( \int x\cdot \nabla Qf_1dx\right) ^2\right) , \end{aligned}$$

(6.4)

where $f=f_1+if_2$.

1.2 Decoupling lemma

Lemma 6.3

(Decoupling lemma) Let $\delta _0$ be as in (1.5). For every $1\le k\le K$, let

$$\begin{aligned} G_{i,k}(t,x) = w_k^{-\frac{2}{p-1}} g_{i}(\frac{x-v_k t - {\alpha }_k}{w_k}),\ \ i=1,2, \end{aligned}$$

(6.5)

where $1\le p\le 1+ \frac{4}{d}$, $g_i \in C_b^2$ decays exponentially fast at infinity, i.e., for some $C_1>0$,

$$\begin{aligned} |g_i(y)| \le C_1 e^{-\delta _0|y|},\ \ y\in {{\mathbb {R}}}^d,\ i=1,2, \end{aligned}$$

(6.6)

the parameters $w_k>0$, $v_k, {\alpha }_k\in {{\mathbb {R}}}^d$, satisfying that

$$\begin{aligned} w_k^{-1}+ w_k+|v_k|+|{\alpha }_k| \le C_2. \end{aligned}$$

(6.7)

Then, if $v_j \not = v_k$, $j\not =k$, we have that for any $p_1, p_2 >0$,

$$\begin{aligned} \int | G_{1,j}(t)|^{p_1}|G_{2,k}(t)|^{p_2}dx \le Ce^{-\delta |v_j-v_k| t}, \end{aligned}$$

(6.8)

where $C,\delta _2>0$ depend on $\delta _0, C_i,p_i$, $i=1,2$.

Proof

We use (6.5) and the change of variables to compute

$$\begin{aligned}&\int | G_{1,j}(t)|^{p_1}|G_{2,k}(t)|^{p_2}dx\nonumber \\&\quad = w_j^{d-\frac{2p_1}{p-1}} w_k^{-\frac{2p_2}{p-1}} \int |g_1|^{p_1}(y) |g_2|^{p_2}\left( \frac{w_j y + (v_j-v_k) t +({\alpha }_j- {\alpha }_k)}{w_k}\right) dy \nonumber \\&\quad = w_j^{d-\frac{2p_1}{p-1}} w_k^{-\frac{2p_2}{p-1}} \left( \int _{\Omega } + \int _{\Omega ^c}\right) |g_1|^{p_1}(y) |g_2|^{p_2} \left( \frac{w_j y + (v_j-v_k) t +({\alpha }_j- {\alpha }_k)}{w_k}\right) dy \nonumber \\&\quad =: I_1 + I_2. \end{aligned}$$

(6.9)

where $\Omega := \{y \in {{\mathbb {R}}}^d: |y|\le \frac{1}{2w_j} |v_j - v_k|t\}$ and $\Omega ^c= {{\mathbb {R}}}^d\setminus \Omega$. On one hand, by (6.7), for t large enough,

$$\begin{aligned} \big | {w_j y + (v_j-v_k) t +({\alpha }_j- {\alpha }_k)} \big | \ge \frac{1}{2} |v_j -v_k|t - |{\alpha }_j- {\alpha }_k| \ge \frac{1}{4} |v_j - v_k| t, \ \ y\in \Omega , \end{aligned}$$

which along with the exponential decay (6.6) and $w_j, w_k \ge C_2^{-1}>0$ yields that

$$\begin{aligned} I_1 \le C e^{-\frac{\delta _1 p_2}{4w_k} |v_j-v_k|t} \int g_1^{p_1}(y) dy \le C e^{-\delta ' |v_j - v_k|t}, \end{aligned}$$

(6.10)

where $C,\delta '>0$ depend on $\delta _0, C_i, p_i$, $i=1,2$.

On the other hand, using (6.6) again we infer that

$$\begin{aligned} |g_1(y)| \le C_1 e^{-\frac{\delta _0}{2w_j}|v_j-v_k|t} , \ \ y\in \Omega ^c, \end{aligned}$$

and thus

$$\begin{aligned} I_2 \le C e^{-\frac{\delta _0 p_1}{2w_j} |v_j-v_k|t} \int _{\Omega ^c} g_2^{p_2}\left( \frac{w_j y + (v_j-v_k) t +({\alpha }_j- {\alpha }_k)}{w_k}\right) dy \le C e^{-\delta '' |v_j - v_k|t},\nonumber \\ \end{aligned}$$

(6.11)

where where $C,\delta _2>0$ depend on $\delta _0, C_i, p_i$, $i=1,2$.

Therefore, plugging (6.10) and (6.11) into (6.9) we obtain (6.8) and finish the proof. $\square$

1.3 Proof of Proposition 3.1

Below we present the proof of the geometrical decomposition in Proposition 3.1 in a fashion close to that of [12]. Given any $L>0$, $w_k^0\in {\mathbb {R}}^+$, $x_k^0, v_k\in {\mathbb {R}}^d$, ${\theta }_k^0\in {\mathbb {R}}$, $1\le k\le K$, set

$$\begin{aligned} R_L(x) : =\sum _{k=1}^{K}R_{k,L}(x) =\sum _{k=1}^{K}(w_k^0)^{-\frac{2}{p-1}}Q\left( \frac{x-v_kL-x_k^0}{w_{k}^0}\right) e^{i\left( \frac{1}{2}v_k\cdot x-\frac{1}{4}|v_k|^2L+(w_{k}^0)^{-2}L+{\theta }_{k}^0\right) }. \end{aligned}$$

(6.12)

Note that, if $L=t$, then $R_{k,L}=R_k$ with $R_k$ given by (1.12).

Lemma 6.4

There exists a universal small constant $\delta _*>0$ such that the following holds. For any $0<r, L^{-1} <\delta _*$ and for any $u\in H^{-1}({\mathbb {R}}^d)$ satisfying $\Vert u-R_L\Vert _{H^{-1}}\le r$, there exist unique $C^1$ functions ${\mathcal {P}}(u)=(\widetilde{\alpha }, \widetilde{\theta }, \widetilde{w}): H^{-1}\rightarrow {{\mathbb {X}}}^K$ such that u admits the decomposition

$$\begin{aligned} u= & {} \sum _{k=1}^{K}(\widetilde{w}_k w_k^0)^{-\frac{2}{p-1}}Q\left( \frac{x-v_kL-x_k^0-\widetilde{\alpha }_k}{\widetilde{w}_k w_{k}^0}\right) e^{i\left( \frac{1}{2}v_k\cdot x-\frac{1}{4}|v_k|^2L+(w_{k}^0)^{-2}L+{\theta }_{k}^0+\widetilde{\theta }_k\right) }\nonumber \\{} & {} \quad + {\varepsilon }_L \ \ (=: \sum _{k=1}^{K}\widetilde{R}_{k,L}+{\varepsilon }_L), \end{aligned}$$

(6.13)

and the following orthogonality conditions hold: for $1\le k\le K$,

$$\begin{aligned} \begin{aligned}&\textrm{Re}\ _{H^{1}}\langle \nabla \widetilde{R}_{k,L}, {\varepsilon }_L\rangle _{H^{-1}}=0,\ \ \textrm{Im}\ _{H^{1}}\langle \widetilde{R}_{k,L}, {\varepsilon }_L \rangle _{H^{-1}}=0,\\&\textrm{Re}\ _{H^{1}}\langle \frac{d}{2}\widetilde{R}_{k,L}+y_k\cdot \nabla \widetilde{R}_{k,L} - \frac{i}{2}v_k\cdot y_k \widetilde{R}_{k,L}, {\varepsilon }_L \rangle _{H^{-1}}=0, \end{aligned} \end{aligned}$$

(6.14)

where $y_k :=x-v_kL-x_k^0-\widetilde{\alpha }_k$. Moreover, there exists a universal constant $C>0$ such that,

$$\begin{aligned} \Vert \varepsilon _L\Vert _{H^{-1}}+\sum _{k=1}^{K}(|\widetilde{\alpha }_k|+|\widetilde{\theta }_k|+|\widetilde{w}_k-1|)\le C\Vert u-R_L\Vert _{H^{-1}}. \end{aligned}$$

(6.15)

Proof

The proof proceeds in four steps.

Step 1. Set $\widetilde{\mathcal {P}}_{0,k}: =(0,0,1) \in {\mathbb {X}}$ and $\widetilde{\mathcal {P}}_0 = (\widetilde{\mathcal {P}}_{0,1}, \cdots , \widetilde{\mathcal {P}}_{0,K})\in {\mathbb {X}}^K$. Similarly, let $\widetilde{\mathcal {P}}_k:= (\widetilde{\alpha }_k, \widetilde{\theta }_k,\widetilde{w}_k) \in {\mathbb {X}}$, $\widetilde{\mathcal {P}}:= (\widetilde{\mathcal {P}}_1, \cdots , \widetilde{\mathcal {P}}_K) \in {\mathbb {X}}^K$. Let

$$\begin{aligned} \alpha _k : =\widetilde{\alpha }_k+x_k^0,\quad {\theta }_k:=\widetilde{\theta }_k+{\theta }_k^0,\quad w_k:=\widetilde{w}_kw_k^0. \end{aligned}$$

(6.16)

For any $u_0\in H^1$, let $B_\delta (u_0, \widetilde{\mathcal {P}}_0)$ denote the closed ball centered at $(u_0, \widetilde{\mathcal {P}}_0)$ of radius $\delta$, i.e.,

$$\begin{aligned} B_\delta (u_0, \widetilde{\mathcal {P}}_0) :=\{(u, \widetilde{\mathcal {P}}) \in H^{-1}\times {\mathbb {X}}^K:\ \Vert u-u_0\Vert _{H^{-1}}\le \delta ,\ \ |\widetilde{\mathcal {P}}- \widetilde{\mathcal {P}}_0|\le \delta \}, \end{aligned}$$

(6.17)

where $\delta$ is a small constant to be chosen later, and

$$\begin{aligned} |\widetilde{\mathcal {P}}-\widetilde{\mathcal {P}}_0|:= \sum \limits _{k=1}^K |\widetilde{\mathcal {P}}_k- \widetilde{\mathcal {P}}_{0,k}| =\sum _{k=1}^{K}(|\widetilde{\alpha }_k|+|\widetilde{\theta }_k|+|\widetilde{w}_k-1|). \end{aligned}$$

(6.18)

For $1\le k\le K$, let

$$\begin{aligned}&f_{1,j}^k(u, \widetilde{\mathcal {P}}) :=\textrm{Re}\ _{H^{1}}\langle \partial _j \widetilde{R}_{k,L}, {\varepsilon }_L \rangle _{H^{-1}}, \ \ 1\le j\le d, \\&f_{2}^k(u,\widetilde{\mathcal {P}}) :=\textrm{Im}\ _{H^{1}} \langle \widetilde{R}_{k,L}, {\varepsilon }_L, \rangle _{H^{-1}}, \\&f_{3}^k(u, \widetilde{\mathcal {P}}) :=\textrm{Re}\ _{H^{1}}\langle \frac{2}{p-1}\widetilde{R}_{k,L}+y_k\cdot \nabla \widetilde{R}_{k,L} - \frac{i}{2}v_k\cdot y_k \widetilde{R}_{k,L}, {\varepsilon }_L \rangle _{H^{-1}}, \end{aligned}$$

where $y_k$ is as in (6.14). Let $F^k:= (f^k_{1,1}, \cdots , f^k_{1,d}, f^k_2, f^k_{3})$ and $\frac{\partial F^k}{\partial \widetilde{\mathcal {P}}_j}$ denote the Jacobian matrix

$$\begin{aligned} \frac{\partial F^k}{\partial \widetilde{\mathcal {P}}_j}:=\left( \begin{array}{ccc} \frac{\partial f^k_{1,1}}{\partial \widetilde{\alpha }_{j,1}} &{}\cdots \ \frac{\partial f^k_{1,1}}{\partial \widetilde{\alpha }_{j,d}},\ \frac{\partial f^k_{1,1}}{\partial \widetilde{\theta }_j}, &{} \frac{\partial f^k_{1,1}}{\partial \widetilde{w}_j} \\ \vdots &{} &{} \vdots \\ \frac{\partial f^k_{3}}{\partial \widetilde{\alpha }_{j,1}} &{} \cdots \ \frac{\partial f^k_{3}}{\partial \widetilde{\alpha }_{j,d}},\ \frac{\partial f^k_{3}}{\partial \widetilde{\theta }_j},&{} \frac{\partial f^k_{3}}{\partial \widetilde{w}_j} \\ \end{array} \right) , \ \ 1\le j, k\le K, \end{aligned}$$

(6.19)

where $\widetilde{\alpha }_j:=(\widetilde{\alpha }_{j,l}, 1\le l\le d)\in {{\mathbb {R}}}^d$. Similarly, let $F:=(F^1, \cdots , F^K)$ and $\frac{\partial F}{\partial \widetilde{\mathcal {P}}} := (\frac{\partial F^k}{\partial \widetilde{\mathcal {P}}_j})_{1\le j,k\le K}$.

Note that, by the definition (6.12) of $R_L$, $F^k(R_L, \widetilde{\mathcal {P}}_0)=0$, $1\le k\le K$. Moreover, for any $(u, \widetilde{\mathcal {P}})\in B_\delta (R_L, \widetilde{\mathcal {P}}_0)$, we have that, if $\widetilde{R}_L:= \sum _{k=1}^K \widetilde{R}_{k,L}$,

$$\begin{aligned} \Vert \varepsilon _L\Vert _{H^{-1}} \le \Vert u-R_L\Vert _{H^{-1}} + \Vert R_L - \widetilde{R}_L\Vert _{H^{-1}}. \end{aligned}$$

(6.20)

By the explicit expressions of $R_L$ and $\widetilde{R}_L$ in (6.12) and (6.13), respectively,

$$\begin{aligned} \Vert R_L - \widetilde{R}_L\Vert _{H^{-1}} \le \Vert R_L - \widetilde{R}_L\Vert _{L^2} \le C\sum _{k=1}^{K}(|\widetilde{\alpha }_k|+|\widetilde{\theta }_k|+|\widetilde{w}_k-1|)\le C |\widetilde{\mathcal {P}}- \widetilde{\mathcal {P}}_0|, \end{aligned}$$

(6.21)

where $C>0$. Thus, we get that for a universal constant $\widetilde{C}>0$,

$$\begin{aligned} \Vert \varepsilon _L\Vert _{H^{-1}}\le \widetilde{C} (\Vert u-R_L\Vert _{H^{-1}} + |\widetilde{\mathcal {P}}- \widetilde{\mathcal {P}}_0|) \le 2 \widetilde{C} \delta ,\ \ \forall (u, \widetilde{\mathcal {P}})\in B_\delta (R_L, \widetilde{\mathcal {P}}_0).\nonumber \\ \end{aligned}$$

(6.22)

Step 2. We claim that, there exist small constants $\delta _*, c_1, c_2>0$ such that for any $0<\delta , L^{-1} \le \delta _*$,

$$\begin{aligned} 0<c_1\le \bigg |\det \frac{\partial F}{\partial \widetilde{\mathcal {P}}}(u, \widetilde{\mathcal {P}}) \bigg | \le c_2<{\infty }, \ \ \forall (u, \widetilde{\mathcal {P}})\in B_{\delta }(R_L, \widetilde{\mathcal {P}}_0). \end{aligned}$$

(6.23)

To this end, we compute that for $1\le j,k\le d$,

$$\begin{aligned} \begin{aligned}&\partial _{\widetilde{\alpha }_{k,j}} f^k_{1,j} = - w_k^{-2} \Vert \partial _j Q_{w_k}\Vert _{L^2}^2+{\mathcal {O}}(\Vert \varepsilon _L\Vert _{H^{-1}}), \\&\partial _{\widetilde{\theta }_k} f_{1,j}^k = - \frac{ v_{k,j}}{2}\Vert Q_{w_k}\Vert _{L^2}^2+{\mathcal {O}}(\Vert \varepsilon _L\Vert _{H^{-1}}) , \\&\partial _{\widetilde{\theta }_k} f^k_{2} = \Vert Q_{w_k}\Vert _{L^2}^2+{\mathcal {O}}(\Vert \varepsilon _L\Vert _{H^{-1}}), \ \ \partial _{\widetilde{w}_k} f_3^k = w_k ^{-1} \Vert \Lambda Q_{w_k}\Vert _{L^2}^2+{\mathcal {O}}(\Vert \varepsilon _L\Vert _{H^{-1}}). \end{aligned} \end{aligned}$$

(6.24)

Moreover, by the exponential decay of Q, we infer that, there exists $\delta >0$ such that the other terms in the Jacobian matrices are of the order ${\mathcal {O}}(\Vert {\varepsilon }\Vert _{H^{-1}} + e^{-\delta L})$. This yields that

$$\begin{aligned} \bigg |\det \left( \frac{\partial F}{\partial \widetilde{\mathcal {P}}} \right) \bigg |&= \prod \limits _{k=1}^K \left( (w_k^0)^{-2d} \widetilde{w}_k^{-2d-1} \Vert Q_{w_k}\Vert _{L^2}^2 \Vert \Lambda Q_{w_k}\Vert _{L^2}^2 \prod \limits _{j=1}^d \Vert \partial _j Q_{w_k}\Vert _{L^2}^2 \right) \nonumber \\&\quad + {\mathcal {O}}\left( \Vert {\varepsilon }\Vert _{H^{-1}} + e^{-\delta L}\right) . \end{aligned}$$

(6.25)

Taking into account $|\widetilde{\mathcal {P}}- \widetilde{\mathcal {P}}_0| \le \delta$ we obtain (6.23), as claimed.

Step 3. In this step, we claim that there exists a universal constant $C_*(\ge 1)$ such that, for any $0<\delta , L^{-1} \le \delta _*$ and any $(u_1, \widetilde{\mathcal {P}}(u_1)), (u_2, \widetilde{\mathcal {P}}(u_2))\in B_\delta (R_L, \widetilde{\mathcal {P}}_0 )$, if $F(u_1, \widetilde{\mathcal {P}}(u_1))= F(u_2, \widetilde{\mathcal {P}}(u_2))=0$, then

$$\begin{aligned} |\widetilde{\mathcal {P}}(u_1)-\widetilde{\mathcal {P}}(u_2)|\le C_* \Vert u_1-u_2\Vert _{H^{-1}}. \end{aligned}$$

(6.26)

To this end, we infer that

$$\begin{aligned} F(u_1, \widetilde{\mathcal {P}}(u_1))-F(u_1, \widetilde{\mathcal {P}}(u_2))= F(u_2, \widetilde{\mathcal {P}}(u_2) ) - F(u_1, \widetilde{\mathcal {P}}(u_2)). \end{aligned}$$

(6.27)

By the differential mean value theorem,

$$\begin{aligned} \left( \frac{\partial F}{\partial \widetilde{\mathcal {P}}}(u_1, \widetilde{\mathcal {P}}_{r})\right) (\widetilde{\mathcal {P}}(u_1)-\widetilde{\mathcal {P}}(u_2))^t = (F(u_2, \widetilde{\mathcal {P}}(u_2))-F(u_1,\widetilde{\mathcal {P}}(u_2)))^t, \end{aligned}$$

(6.28)

where $\widetilde{\mathcal {P}}_r=r\widetilde{\mathcal {P}}(u_1)+(1-r)\widetilde{\mathcal {P}}(u_2)$ for some $0<r<1$, and the superscript t means the transpose of matrices. Since the Jacobian matrix $\frac{\partial F}{\partial \widetilde{\mathcal {P}}}(u_1, \widetilde{\mathcal {P}}_r)$ is invertible by (6.23), this leads to

$$\begin{aligned} (\widetilde{\mathcal {P}}(u_1)-\widetilde{\mathcal {P}}(u_2))^t= \left( \frac{\partial F}{\partial \widetilde{\mathcal {P}}}(u_1,\widetilde{\mathcal {P}}_r) \right) ^{-1} (F(u_2, \widetilde{\mathcal {P}}(u_2)) - F(v_1, \widetilde{\mathcal {P}}(u_2)))^t. \end{aligned}$$

(6.29)

Note that, by (6.24), there exists a universal constant $C>0$ such that

$$\begin{aligned} \bigg \Vert \bigg (\frac{\partial F}{\partial \widetilde{\mathcal {P}}}(u_1, \widetilde{\mathcal {P}}_r)\bigg )^{-1} \bigg \Vert \le C, \end{aligned}$$

(6.30)

where $\Vert \cdot \Vert$ denotes the Hilbert-Schmidt norm of matrices. Moreover, by (1.5),

$$\begin{aligned}&|F(u_2, \widetilde{\mathcal {P}}(u_2)) - F(u_1, \widetilde{\mathcal {P}}(u_2))|\le C \Vert u_2-u_1\Vert _{H^{-1}}. \end{aligned}$$

(6.31)

Thus, we infer from (6.29), (6.30) and (6.31) that (6.26) holds, as claimed.

Step 4. Let $\delta _*, C_*$ be the universal constants as in Step 1 and Step 2, respectively, and set

$$\begin{aligned} B:=\{ v\in B_{\frac{\delta _*}{C_*}}(R_L): \exists \widetilde{\mathcal {P}}\in B_{\delta _*}(\widetilde{\mathcal {P}}_0), \ such\ that\ F(v,\widetilde{\mathcal {P}}) =0\}. \end{aligned}$$

(6.32)

Since $B_{\frac{\delta _*}{C_*}}(R_L)$ is connected and $R_L \in B$, in order to prove that

$$\begin{aligned} B = B_{\frac{\delta _*}{C_*}}(R_L). \end{aligned}$$

(6.33)

we only need to show that B is both open and closed in $B_{\frac{\delta _*}{C_*}}(R_L)$.

To this end, For any $u\in B$, by definition there exists $\widetilde{\mathcal {P}}(u) \in B_{\delta _*}(\widetilde{\mathcal {P}}_0)$ such that $F(u,\widetilde{\mathcal {P}}(u))=0$. Taking into account the non-degeneracy of the Jacobian matrix at $(u, \widetilde{\mathcal {P}}(u))$ due to (6.23), we can apply the implicit function theorem to get a small open neighborhood ${\mathcal {U}}(u)$ of u in $B_{\frac{\delta _*}{C_*}}(R_L)$ such that ${\mathcal {U}}(u) \subseteq B$. This yield that B is open in $B_{\frac{\delta _*}{C_*}}(R_L)$.

Moreover, for any sequence $\{u_n\} \subseteq B$ such that $u_n \rightarrow u_*$ in $H^{-1}$ for some $u_*\in B_{\frac{\delta _*}{C_*}}(R_L)$, by definition there exist modulation parameters $\widetilde{\mathcal {P}}(u_n)\in B_{\delta _*}(\widetilde{\mathcal {P}}_0)$ such that $F(u_n, \widetilde{\mathcal {P}}(u_n))=0$, $n \ge 1$. In particular, $\{\widetilde{\mathcal {P}}(v_n)\} \subseteq {\mathbb {X}}^K$ is uniformly bounded and so, along a subsequence (still denoted by $\{n\}$), $\widetilde{\mathcal {P}}(v_{n}) \rightarrow \widetilde{\mathcal {P}}_*\ (\in B_{\delta _*}(\widetilde{\mathcal {P}}_0))$ for some $\widetilde{\mathcal {P}}_* \in {\mathbb {X}}^K$.

Then, let $\widetilde{R}_{k,L, \widetilde{\mathcal {P}}(u_n)}$ and $\widetilde{R}_{k,L,\widetilde{\mathcal {P}}_*}$ be the k-th soliton profiles corresponding to $\widetilde{\mathcal {P}}(u_n)$ and $\widetilde{\mathcal {P}}_*$, respectively. By the above convergence of $u_n$ and $\widetilde{\mathcal {P}}(u_n)$ we infer that $u_{n} - \sum _{k=1}^K \widetilde{R}_{k,L, \widetilde{\mathcal {P}}(u_{n})} \rightarrow u_* - \sum _{k=1}^K \widetilde{R}_{k,L,\widetilde{\mathcal {P}}_*}$ in $H^{-1}$. Taking $n\rightarrow {\infty }$ and using the fact that $F(u_{n}, \widetilde{\mathcal {P}}(u_{{n}})) =0$ we obtain $F(u_*, \widetilde{\mathcal {P}}_*) = 0$, and so $u_*\in B$. Hence, B is also closed in $B_{\frac{\delta _*}{C_*}}(R_L)$.

Therefore, (6.33) is verified. The geometrical decomposition (6.13) and the orthogonality conditions in (6.14) hold. Moreover, estimate (6.15) follows from (6.22) and (6.26) by taking $u_1= u$ and $u_2 = R_L$. The proof of Lemma 6.4 is complete. $\square$

Proof of Proposition 3.1

Since $u(T)=R(T)$, by the local wellposedness theory, there exists $T^*$ close to T, such that $u(t)\in C^1([T^*,T]; H^{-1}) \bigcap C([T^*,T];H^1)$ and $\Vert u(t)-R(T)\Vert _{H^{1}} \in B_{\delta }(u(T))$ for all $t\in [T^*,T]$, where $\delta >0$ is as in Lemma 6.4.

Hence, applying Lemma 6.4 to $\{u(t)\}$ we obtain that for T large enough, there exist unique $C^1$ functions $(\alpha _k,{\theta }_k,\omega _k) \in C^1([T^*,T]; {\mathbb {X}}^K)$, $1\le k\le K$, such that for any $t\in [T^*,T]$, u(t) admits the decomposition (6.13) and the orthogonality conditions in (6.14) hold with t replacing T.

Then, taking into account $u(t)\in H^1$ and (6.13), the remainder ${\varepsilon }(t)$ is indeed in the space $H^1$. Thus, the parings between $H^{-1}$ and $H^1$ in (6.14) are exactly the $L^2$ inner products, which yields the orthogonality conditions in (3.5) for any $[T^*,T]$. Therefore, the proof is complete. $\square$

1.4 Proof of (4.45)

We set $\widetilde{S}_k:= \sum _{j=k}^K \widetilde{R}_j$, $1\le k\le K$. Then,

$$\begin{aligned} \widetilde{S}_k = \widetilde{R}_k + \widetilde{S}_{k+1},\ \ 1\le k\le K-1. \end{aligned}$$

(6.34)

Lemma 6.5

Let $0<q<{\infty }$, we have

$$\begin{aligned} \left| |\widetilde{S}_k |^q-|\widetilde{R}_k|^q\right| \le C h(\widetilde{S}_{k+1}), \end{aligned}$$

(6.35)

where $C>0$, $h(\widetilde{S}_{k+1}) = |\widetilde{S}_{k+1}|^q$ if $0<q<1$, and $h(\widetilde{S}_{k+1}) = |\widetilde{S}_{k+1}|$ if $1\le q<{\infty }$.

Proof

The case where $0<q<1$ follows from the inequality

$$\begin{aligned} (a+b)^{q}\le a^q+b^q, \ \ a,b \ge 0, \end{aligned}$$

while the case $1\le q<{\infty }$ follows from the inequality

$$\begin{aligned} ||\widetilde{S}_k |^q-|\widetilde{R}_k|^q | \le C(|\widetilde{S}_{k+1}|^{q-1} + |\widetilde{R}_{k}|^{q-1}) |\widetilde{S}_{k+1}| \end{aligned}$$

and the uniform boundedness of $\widetilde{R}_j$, $1\le j\le K$. $\square$

Lemma 6.6

There exist constants $C, \delta _2>0$ depending on $w_k^0$, $x_k^0$, $v_k$ and $\delta _0$ such that

$$\begin{aligned} |\int |\widetilde{S}_{k}|^{p+1}-|\widetilde{R}_k|^{p+1}-|\widetilde{S}_{k+1}|^{p+1}dx|\le Ce^{-\delta _2 t}. \end{aligned}$$

(6.36)

Proof

Using the expansion

$$\begin{aligned} |\widetilde{S}_k|^2 = |\widetilde{R}_k|^2 + |\widetilde{S}_{k+1}|^2 + 2 \textrm{Re} (\widetilde{R}_k \widetilde{S}_{k+1}), \end{aligned}$$

and Lemmas 6.3 and 6.5 we have

$$\begin{aligned}&\big | \int |\widetilde{S}_k|^{p+1}-|\widetilde{R}_k|^{p+1}-|\widetilde{S}_{k+1}|^{p+1}dx \big | \\&\le \int \big ||\widetilde{S}_k|^{p-1}-|\widetilde{R}_k|^{p-1}\big | |\widetilde{R}_k|^2 +\big ||\widetilde{S}_k|^{p-1}-|\widetilde{S}_{k+1}|^{p-1}\big ||\widetilde{S}_{k+1}|^2 +2|\widetilde{S}_{k}|^{p-1}|\widetilde{R}_k\widetilde{S}_{k+1}|dx \\&\le C \int h(\widetilde{S}_{k+1}) |\widetilde{R}_k|^2 + h(\widetilde{R}_k) |\widetilde{S}_{k+1}|^2 +2|\widetilde{S}_{k}|^{p-1} |\widetilde{R}_k \widetilde{S}_{k+1}|dx\\&\le Ce^{-\delta _2 t}, \end{aligned}$$

which yields (6.36). $\square$

Lemma 6.7

There exist constants $C, \delta _2>0$ depending on $w_k^0$, $x_k^0$, $v_k$ and $\delta _0$ such that

$$\begin{aligned} |\int (| \widetilde{S}_{k}|^{p-1} \widetilde{S}_{k} - | \widetilde{R}_k|^{p-1} \widetilde{R}_k- | \widetilde{S}_{k+1}|^{p-1} \widetilde{S}_{k+1}) \overline{{\varepsilon }}dx| \le Ce^{-\delta _2 t} \Vert {\varepsilon }\Vert _{L^2}. \end{aligned}$$

(6.37)

Proof

By the expansion (6.34), Lemmas 6.3 and 6.5 and Hölder’s inequality,

$$\begin{aligned}&\big |\int (| \widetilde{S}_{k}|^{p-1} \widetilde{S}_{k} -| \widetilde{R}_k|^{p-1} \widetilde{R}_k-| \widetilde{S}_{k+1}|^{p-1} \widetilde{S}_{k+1})\overline{{\varepsilon }}dx\big |\\&\le \int \left( \big || \widetilde{S}_{k}|^{p-1} -| \widetilde{R}_k|^{p-1} \big ||\widetilde{R}_k| +\big || \widetilde{S}_{k}|^{p-1} -| \widetilde{S}_{k+1}|^{p-1} \big || \widetilde{S}_{k+1}|\right) |{\varepsilon }|dx\\&\le C (\Vert h(\widetilde{S}_{k+1}) \widetilde{R}_k\Vert _{L^2}+ \Vert h(\widetilde{R}_k) \widetilde{S}_{k+1}\Vert _{L^2}) \Vert {\varepsilon }\Vert _{L^2}\\&\le Ce^{-\delta t} \Vert {\varepsilon }\Vert _{L^2}, \end{aligned}$$

which yields (6.37). $\square$

Lemma 6.8

There exist constants $C, \delta _2>0$ depending on $w_k^0$, $x_k^0$, $v_k$ and $\delta _0$ such that

$$\begin{aligned} |\int (| \widetilde{S}_{k}|^{p-1} -| \widetilde{R}_k|^{p-1}-| \widetilde{S}_{k+1}|^{p-1} ) |{\varepsilon }|^2dx|\le Ce^{-\delta _2 t} \Vert {\varepsilon }\Vert _{L^2}^2. \end{aligned}$$

(6.38)

Proof

Let $\Omega _k:=\{ x: |x-v_kt|\le \frac{1}{2} \min _{j\not =k} |v_k-v_j|t \}$. By Lemma 6.5,

$$\begin{aligned}&\bigg |\int (| \widetilde{S}_k|^{p-1} -| \widetilde{R}_k|^{p-1}-| \widetilde{S}_{k+1}|^{p-1} ) |{\varepsilon }|^2dx\bigg | \nonumber \\&\le 2\int _{\Omega _k} (h(\widetilde{S}_{k+1}) + |\widetilde{S}_{k+1}|^{p-1}) |{\varepsilon }|^2dx + 2\int _{\Omega _k^c} (h(\widetilde{R}_k) + |\widetilde{R}_k|^{p-1}) |{\varepsilon }|^2dx \nonumber \\&\le C \Vert h(\widetilde{S}_{k+1}) + |\widetilde{S}_{k+1}|^{p-1} \Vert _{L^{\infty }(\Omega _k)} \Vert {\varepsilon }\Vert _{L^2}^2 + C \Vert h(\widetilde{R}_k) + |\widetilde{R}_k|^{p-1}\Vert _{L^{\infty }(\Omega _k^c)} \Vert {\varepsilon }\Vert _{L^2}^2. \end{aligned}$$

(6.39)

Note that, for $x\in \Omega _k$, for any $j\not =k$,

$$\begin{aligned} |x-v_j t-{\alpha }_j| \ge |v_j-v_k|t - |x- v_kt| - |{\alpha }_j| \ge \frac{1}{4} |v_j - v_k|t, \end{aligned}$$

and thus by the exponential decay of Q,

$$\begin{aligned} \Vert h(\widetilde{S}_{k+1}) + |\widetilde{S}_{k+1}|^{p-1}\Vert _{L^{\infty }(\Omega _k^c)} \le C e^{-\delta _2 t}. \end{aligned}$$

(6.40)

Similarly, for $x\in \Omega _k^c$, there exists $c>0$ such that for t large enough,

$$\begin{aligned} |x-v_kt - {\alpha }_k| \ge \frac{1}{2} \min _{j\not =k}\{|v_j-v_k|t\} - |{\alpha }_k| \ge ct, \end{aligned}$$

and thus

$$\begin{aligned} \Vert h(\widetilde{R}_k) + |\widetilde{R}_k|^{p-1}\Vert _{L^{\infty }(\Omega _k^c)} \le C e^{-\delta _2 t}. \end{aligned}$$

(6.41)

Therefore, plugging (6.40) and (6.41) into (6.39) we obtain (6.38) and finish the proof. $\square$

Lemma 6.9

There exist constants $C, \delta _2>0$ depending on $w_k^0$, $x_k^0$, $v_k$ and $\delta _0$ such that

$$\begin{aligned} |\int (|\widetilde{S}_k|^{p-3}\widetilde{S}_k^2 -| \widetilde{R}_k|^{p-3}\widetilde{R}_k^2 -| \widetilde{S}_{k+1}|^{p-3}\widetilde{S}_{k+1}^2) \overline{{\varepsilon }}^2dx|\le Ce^{-\delta _2 t}. \end{aligned}$$

(6.42)

Proof

Since

$$\begin{aligned} |\widetilde{S}_k|^{p-1} \frac{\widetilde{S}_{k}^2}{|\widetilde{S}_k^2|} = |\widetilde{R}_k|^{p-1} \frac{\widetilde{S}_{k}^2}{|\widetilde{S}_k^2|} + |\widetilde{S}_{k+1}|^{p-1} \frac{\widetilde{S}_{k}^2}{|\widetilde{S}_k^2|} + {\mathcal {O}}( | |\widetilde{S}_k|^{p-1} - |\widetilde{R}_k|^{p-1} - |\widetilde{S}_{k+1}|^{p-1} |), \end{aligned}$$

we have

$$\begin{aligned}&\big |\int (|\widetilde{S}_k|^{p-3}\widetilde{S}_k^2 -| \widetilde{R}_k|^{p-3}\widetilde{R}_k^2 -| \widetilde{S}_{k+1}|^{p-3}\widetilde{S}_{k+1}^2) \overline{{\varepsilon }}^2dx\big | \nonumber \\ \le&\int \bigg ||\widetilde{S}_k|^{p-1}\frac{\widetilde{S}_k^2}{|\widetilde{S}_k|^2} -| \widetilde{R}_k|^{p-1}\frac{\widetilde{R}_k^2}{|\widetilde{R}_k|^2} -|\widetilde{S}_{k+1}|^{p-1}\frac{\widetilde{S}_{k+1}^2}{|\widetilde{S}_{k+1}|^2}\bigg | |{\varepsilon }|^2dx \nonumber \\ \le&\int | \widetilde{R}_k|^{p-1}\bigg |\frac{\widetilde{S}_k^2}{|\widetilde{S}_k|^2} -\frac{\widetilde{R}_k^2}{|\widetilde{R}_k|^2} \bigg | |{\varepsilon }|^2dx +\int | \widetilde{S}_{k+1}|^{p-1}\bigg |\frac{\widetilde{S}_k^2}{|\widetilde{S}_k|^2} -\frac{\widetilde{S}_{k+1}^2}{|\widetilde{S}_{k+1}|^2} \bigg | |{\varepsilon }|^2dx \nonumber \\&+ {\mathcal {O}}\left( \int \big ||\widetilde{S}_k|^{p-1}-| \widetilde{S}_k|^{p-1}-| \widetilde{R}_k|^{p-1}\big | |{\varepsilon }|^2dx\right) \nonumber \\ =&\int | \widetilde{R}_k|^{p-1}\bigg |\frac{\widetilde{S}_{k}^2}{|\widetilde{S}_k|^2} -\frac{\widetilde{R}_k^2}{|\widetilde{R}_k|^2} \bigg | |{\varepsilon }|^2dx +\int | \widetilde{S}_{k+1}|^{p-1}\bigg |\frac{\widetilde{S}_k^2}{|\widetilde{S}_k|^2} -\frac{\widetilde{S}_{k+1}^2}{|\widetilde{S}_{k+1} |^2} \bigg | |{\varepsilon }|^2dx+ {\mathcal {O}}(e^{-\delta _2 t}) \nonumber \\ =:&J_1 + J_2 + {\mathcal {O}}(e^{-\delta _2 t}). \end{aligned}$$

(6.43)

where the last step is due to Lemma 6.8.

Below we estimate $J_1$ and $J_2$ separately. For this purpose, let us set $d_*:= \min _{k\le j\not =l\le K}\{|v_jt + {\alpha }_j - v_lt-{\alpha }_l|\}$. Similarly, let $w_*:=\min _{k\le j\le K} w_j$, $w^*:=\max _{k\le j\le K} w_j$. For every $k\le j\le K$, set

$$\begin{aligned} \Omega _j:= \bigg \{x\in {{\mathbb {R}}}^d: |x-v_jt -{\alpha }_j| \le {\varepsilon }d_* \bigg \}, \end{aligned}$$

where ${\varepsilon }$ is a small constant to be specified below.

(i) Estimate of $J_1$. We decompose

$$\begin{aligned} J_1&= \int _{\Omega _k^c} | \widetilde{R}_k|^{p-1}\bigg |\frac{\widetilde{S}_k^2}{|\widetilde{S}_k|^2} -\frac{\widetilde{R}_k^2}{|\widetilde{R}_k|^2} \bigg | |{\varepsilon }|^2dx + \int _{\Omega _k} | \widetilde{R}_k|^{p-1}\bigg |\frac{\widetilde{S}_k^2}{|\widetilde{S}_k|^2} -\frac{\widetilde{R}_k^2}{|\widetilde{R}_k|^2} \bigg | |{\varepsilon }|^2 dx\nonumber \\&:=J_{11}+J_{12}. \end{aligned}$$

(6.44)

Note that, for $x\in \Omega _k^c$, since

$$\begin{aligned} |x-v_kt -{\alpha }_k| \ge&{\varepsilon }d_* > \frac{c}{2} t \end{aligned}$$

(6.45)

for t large enough, where $c>0$, by (1.5), there exist $C,\delta _2>0$ such that

$$\begin{aligned} J_{11}\le C\Vert R_k\Vert ^{p-1}_{L^{\infty }(\Omega ^c_k)}\Vert {\varepsilon }\Vert _{L^2}^2\le Ce^{-\delta _2 t}\Vert {\varepsilon }\Vert _{L^2}^2 . \end{aligned}$$

(6.46)

Concerning the first term $J_{12}$ in (6.44), since $Q(x)\sim e^{-\delta _0 |x|}$ (see [10]), we infer that

$$\begin{aligned} |\widetilde{R}_{k}(t,x)| \ge C e^{-\delta _0 \frac{ {\varepsilon }d_* }{w_k} }\ge C e^{-\delta _0 \frac{ {\varepsilon }d_* }{w_*} },\ \ x\in \Omega _k. \end{aligned}$$

(6.47)

On the other hand, for $x\in \Omega _k$ and any $j\not =k$,

$$\begin{aligned} |x-v_jt -{\alpha }_j| \ge |(v_k-v_j)t + {\alpha }_k-{\alpha }_j| - |x-v_kt-{\alpha }_k| \ge (1-{\varepsilon })d_*, \end{aligned}$$

which yields that

$$\begin{aligned} |\widetilde{S}_{k+1}(t,x)| \le C \sum \limits _{j=k+1}^K e^{- \delta _0 \frac{(1-{\varepsilon })d_*}{w_j}} \le C e^{-\delta _0 \frac{(1-{\varepsilon })d_*}{w^*}},\ \ x\in \Omega _k. \end{aligned}$$

(6.48)

Hence, we obtain from (6.47) and (6.48) that, for ${\varepsilon }$ small enough such that

$$\begin{aligned} {\varepsilon }<\frac{w_*}{w^*+w_*}, \end{aligned}$$

there exist $C,\delta _2>0$ such that

$$\begin{aligned} \bigg |\frac{\widetilde{S}_{k+1}(t,x)}{\widetilde{R}_k(t,x)}\bigg | \le C e^{-\delta _0 d_* (\frac{(1-{\varepsilon })}{w^*}-\frac{{\varepsilon }}{w_*})} \le C e^{-\delta _2 t},\ \ x\in \Omega _k. \end{aligned}$$

(6.49)

Taking into account

$$\begin{aligned} \frac{\widetilde{S}_k^2}{|\widetilde{S}_k^2|} - \frac{\widetilde{R}_k^2}{|\widetilde{R}_k^2|} =&\frac{\widetilde{S}_k^2 |\widetilde{R}_k|^2 -| \widetilde{S}_k|^2 \widetilde{R}_k^2 + 2 \widetilde{R}_k \widetilde{S}_{k+1} |\widetilde{R}_k|^2 - 2 \textrm{Re}(\widetilde{R}_k \widetilde{S}_{k+1}) \widetilde{R}_k^2 }{|\widetilde{R}_k + \widetilde{S}_{k+1}|^2 |\widetilde{R}_k|^2} \nonumber \\ \le&\frac{|\widetilde{R}_k^{-1} \widetilde{S}_{k+1} |^2 + |\widetilde{R}_k^{-1} \widetilde{S}_{k+1}|}{|1+ \widetilde{R}_k^{-1} \widetilde{S}_{k+1}|^2} \end{aligned}$$

we thus lead to

$$\begin{aligned} \bigg | \frac{\widetilde{S}_k^2}{|\widetilde{S}_k^2|} - \frac{\widetilde{R}_k^2}{|\widetilde{R}_k^2|} \bigg | \le C e^{-\delta _2 t}, \ \ x\in \Omega _k, \end{aligned}$$

(6.50)

which yields that

$$\begin{aligned} J_{12} \le Ce^{-\delta _2 t}\Vert {\varepsilon }\Vert ^2_{L^2}. \end{aligned}$$

(6.51)

Thus, plugging (6.46) and (6.51) into (6.44) we obtain

$$\begin{aligned} J_2 \le Ce^{-\delta _2 t}. \end{aligned}$$

(6.52)

(ii) Estimate of $J_2$. Set

$$\begin{aligned} \Omega = \bigcup \limits _{j=k+1}^K \Omega _j = \bigcup \limits _{j=k+1}^K \bigg \{x\in {{\mathbb {R}}}^d: |x-v_jt -{\alpha }_j| \le {\varepsilon }d_* \bigg \} \end{aligned}$$

(6.53)

and decompose

$$\begin{aligned} J_2&= \int \limits _{\Omega } | \widetilde{S}_{k+1}|^{p-1}\bigg |\frac{\widetilde{S}_k^2}{|\widetilde{S}_k|^2} -\frac{\widetilde{S}_{k+1}^2}{|\widetilde{S}_{k+1} |^2} \bigg | |{\varepsilon }|^2dx + \int \limits _{\Omega ^c} | \widetilde{S}_{k+1}|^{p-1}\bigg |\frac{\widetilde{S}_k^2}{|\widetilde{S}_k|^2} -\frac{\widetilde{S}_{k+1}^2}{|\widetilde{S}_{k+1} |^2} \bigg | |{\varepsilon }|^2dx \nonumber \\&= J_{21} + J_{22}. \end{aligned}$$

(6.54)

Note that, for every $k+1\le j\le K$, since $Q(x)\sim e^{-\delta _0|x|}$,

$$\begin{aligned} |\widetilde{R}_j(t,x)| \ge C e^{-\delta _0 \frac{{\varepsilon }d_*}{w_j}} \ge C e^{-\delta _0 \frac{{\varepsilon }d_*}{w_*}},\ \ x\in \Omega _j. \end{aligned}$$

(6.55)

Moreover, for $x\in \Omega /\Omega _j$, there exists $j'\not = j$ such that $x\in \Omega _{j'}$ and so

$$\begin{aligned} |x-v_jt-{\alpha }_j| \ge |v_{j'}t +{\alpha }_{j'} - v_jt -{\alpha }_j| - |x-v_{j'}t-{\alpha }_{j'}| \ge (1-{\varepsilon })d_*. \end{aligned}$$

This yields that

$$\begin{aligned} |\widetilde{R}_j (t,x)| \le C e^{-\delta _0\frac{(1-{\varepsilon })d_*}{w_j}} \le C e^{-\delta _0 \frac{(1-{\varepsilon })d_*}{w^*}}, \ \ x\in \Omega /\Omega _j. \end{aligned}$$

Hence, for ${\varepsilon }$ very small such that

$$\begin{aligned} {\varepsilon }<\frac{w_*}{w_*+w^*}, \end{aligned}$$

we obtain that

$$\begin{aligned} |\widetilde{S}_{k+1}|\ge C e^{-\delta _0 \frac{{\varepsilon }d_*}{w_*}} - C' e^{-\delta _0 \frac{(1-{\varepsilon })d_*}{w^*}} \ge \frac{1}{2} C e^{-\delta _0 \frac{{\varepsilon }d_*}{w_*}}, \ \ x\in \Omega ,\ k+1\le j\le K, \end{aligned}$$

(6.56)

which yields that there exist $C,\delta >0$ such that

$$\begin{aligned} |\widetilde{S}_{k+1}| \ge C e^{-\delta _0 \frac{{\varepsilon }d_*}{w_*}}, \ \ x\in \Omega . \end{aligned}$$

(6.57)

Moreover, for any $x\in \Omega$, there exists $k+1\le j\le K$ such that $x\in \Omega _j$ and so

$$\begin{aligned} |x-v_kt -{\alpha }_k| \ge |(v_j-v_k)t -({\alpha }_j-{\alpha }_k)| - |x-v_jt -{\alpha }_j| \ge (1-{\varepsilon }) d_*, \end{aligned}$$

(6.58)

which yields that

$$\begin{aligned} |\widetilde{R}_k(t,x)| \le e^{-\delta _0\frac{(1-{\varepsilon })d_*}{w_k}}\le e^{-\delta _0\frac{(1-{\varepsilon })d_*}{w^*}}, \ \ x\in \Omega . \end{aligned}$$

(6.59)

Thus, we infer that for ${\varepsilon }$ possibly smaller such that

$$\begin{aligned} {\varepsilon }< \frac{w_*}{w_*+w^*}, \end{aligned}$$

then for $x\in \Omega$,

$$\begin{aligned} \bigg |\frac{\widetilde{R}_k(t,x)}{\widetilde{S}_{k+1}(t,x)}\bigg | \le C e^{-\delta _0d_*(\frac{(1-{\varepsilon })}{w^*} - \frac{{\varepsilon }}{w_*} )} \le C e^{-\delta _2 t},\ \ x\in \Omega . \end{aligned}$$

(6.60)

Then, similar to (6.50), we have

$$\begin{aligned} \bigg | \frac{\widetilde{S}_k^2}{|\widetilde{S}_k^2|} - \frac{\widetilde{S}_{k+1}^2}{|\widetilde{S}_{k+1}^2|} \bigg | \le C \frac{|\widetilde{S}_{k+1}^{-1}\widetilde{R}_k |^2 + |\widetilde{S}_{k+1}^{-1}\widetilde{R}_k |}{|1+\widetilde{S}_{k+1}^{-1} \widetilde{R}_k |^2} \le C e^{-\delta _2 t}, \ \ x\in \Omega , \end{aligned}$$

(6.61)

which yields that

$$\begin{aligned} J_{21} \le C e^{-\delta _2 t} \Vert {\varepsilon }\Vert ^2_{L^2}. \end{aligned}$$

(6.62)

Concerning $J_{22}$, we see that for $x\in \Omega ^c$, for $k+1\le j\le K$,

$$\begin{aligned} |x-v_jt -{\alpha }_j| \ge {\varepsilon }d_*, \end{aligned}$$

and so

$$\begin{aligned} |\widetilde{R}_j(t,x)| \le C e^{-\delta _0 \frac{{\varepsilon }d_*}{w_j}},\ \ x\in \Omega ^c. \end{aligned}$$

(6.63)

This yields that there exist $C,\delta >0$ such that

$$\begin{aligned} |\widetilde{S}_{k+1}| \le C \sum \limits _{j=k+1}^K |\widetilde{R}_j| \le C e^{-\delta _2 t},\ \ x\in \Omega ^c, \end{aligned}$$

(6.64)

and thus

$$\begin{aligned} J_{22} \le C e^{-\delta _2 t} \Vert {\varepsilon }\Vert ^2_{L^2}. \end{aligned}$$

(6.65)

Thus, we obtain from (6.54), (6.62) and (6.65) that

$$\begin{aligned} J_2 \le Ce^{-\delta _2 t}. \end{aligned}$$

(6.66)

Therefore, plugging (6.52) and (6.66) into (6.43) we prove (6.42) and thus finish the proof. $\square$

Now, estimate (4.45) follows from Lemmas 6.6, 6.7, 6.8 and 6.9.