On Uniqueness of Multi-bubble Blow-Up Solutions and Multi-solitons to $L^2$-Critical Nonlinear Schrödinger Equations

Abstract

We are concerned with the focusing $L^2$-critical nonlinear Schrödinger equations in ${{\mathbb {R}}}^d$ for dimensions $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of K pseudo-conformal blow-up solutions particularly with the low rate $(T-t)^{0+}$, as $t\rightarrow T$, $1\leqq K<{\infty }$. Moreover, we also prove the uniqueness in the energy class of multi-solitons which converge to a sum of K solitary waves with convergence rate $(1/t)^{2+}$, as $t\rightarrow {\infty }$. The uniqueness class is further enlarged to contain the multi-solitons with even lower convergence rate $(1/t)^{\frac{1}{2}+}$ in the pseudo-conformal space. Our proof is mainly based on several upgrading procedures of the convergence of remainder in the geometrical decomposition, in which the key ingredients are several monotone functionals constructed particularly in the multi-bubble case.

Appendix

In this appendix we collect the tools used in the previous sections and the proof of modulation equations in Proposition 2.5.

Lemma 6.1

([9, Theorem 1.3.7]) Let $d\geqq 1$ and $2\leqq p< \infty$. Then, there exists $C>0$ such that

$$\begin{aligned} \Vert f\Vert _{L^p}\leqq C \Vert f\Vert _{L^2}^{1 - d (\frac{1}{2}-\frac{1}{p})} \Vert \nabla f\Vert _{L^2}^{d(\frac{1}{2}-\frac{1}{p})}, \ \ \forall f\in H^1. \end{aligned}$$

(6.1)

In particular, for any $1<p<{\infty }$,

$$\begin{aligned} \Vert f\Vert _{L^p} \leqq C \Vert f\Vert _{H^1},\ \ \forall f\in L^p. \end{aligned}$$

(6.2)

Lemma 6.2

(Decoupling estimates [59, Lemma 3.1]) For every $1\leqq k\leqq K$, set

$$\begin{aligned} G_k(t,x) := {\lambda }_k^{-\frac{d}{2}} g_k(t,\frac{x-{\alpha }_k}{{\lambda }_k}) e^{i\theta _k}, \ \ with\ \ g_{k}(t,y) := g(y) e^{i(\beta _{k}(t) \cdot y - \frac{1}{4}{\gamma }_{k}(t) |y|^2)}, \end{aligned}$$

(6.3)

where $g \in C_b^2({{\mathbb {R}}}^d)$ decays exponentially fast at infinity

$$\begin{aligned} |\partial ^\nu g(y)| \leqq C e^{-\delta |y|}, \ \ |\nu |\leqq 2, \end{aligned}$$

with $C,\delta >0$, ${{\mathcal {P}}}_k:=({\lambda }_k,\alpha _k,\beta _k,\gamma _k,\theta _k) \in C([T^*,T); {\mathbb {X}})$ satisfies that for $T^*$ close to T

$$\begin{aligned} |(\alpha _k(t)-x_k)\cdot \mathbf{v_1}|\leqq \sigma , \ |x_k - {\alpha }_k(t)| \leqq 1,\ \frac{1}{2}\leqq \frac{\lambda _{k}(t)}{|\omega _{k}(T-t)|}\leqq 2, \ \ t\in [T^*,T), \end{aligned}$$

(6.4)

and $|\beta _k|+|{\gamma }_k|\leqq 1$,

$$\begin{aligned} C (T-T^*) (1+ \max _{1\leqq k\leqq K} |x_k|) \leqq 1, \end{aligned}$$

(6.5)

where C is sufficiently large but independent of T. Then, there exist $C, \delta >0$ such that for any $1\leqq k\not =l\leqq K$, $m\in {\mathbb {N}}$, for any multi-index $\nu$ with $|\nu |\leqq 2$ and for any $T^*$ close to T,

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^d} |x-{\alpha }_l|^n |\partial ^\nu G_l(t)| |x-{\alpha }_k|^m |G_k(t)| dx \leqq Ce^{-\frac{\delta }{T-t}}, \ \ t\in [T^*,T). \end{aligned}$$

(6.6)

Moreover, for any $h\in L^1$ or $L^2$, $1\leqq k\not = l\leqq K$, $m,n\in {\mathbb {N}}$, multi-index $\nu$ with $|\nu |\leqq 2$ and $T^*$ close to T,

$$\begin{aligned}&\int \limits _{{{\mathbb {R}}}^d} |x-{\alpha }_l|^n |\partial ^\nu G_l(t)| |x-{\alpha }_k|^m |h| \Phi _kdx \nonumber \\&\quad \leqq Ce^{-\frac{\delta }{T-t}} \min \{\Vert h\Vert _{L^1}, \Vert h\Vert _{L^2}\}, \ \ t\in [T^*,T). \end{aligned}$$

(6.7)

Coercivity of linearized operators. We recall the linearized operators from [56, 58, 59, 62]. Let Q denote the ground state that solves the elliptic equation (1.1). It is known that Q is smooth and decays at infinity exponentially fast, i.e., there exist $C, \delta >0$ such that for any multi-index $|\nu |\leqq 3$,

$$\begin{aligned} |\partial _x^\nu Q(x)|\leqq C e^{-\delta |x|}, \ \ x\in {{\mathbb {R}}}^d. \end{aligned}$$

(6.8)

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

$$\begin{aligned} L_{+}:= -\Delta + I -(1+{\frac{4}{d}})Q^{\frac{4}{d}}, \ \ L_{-}:= -\Delta +I -Q^{\frac{4}{d}}, \end{aligned}$$

(6.9)

which has the generalized null space spanned by $\{Q, xQ, |x|^2 Q, \nabla Q, \Lambda Q, \rho \}$. Here, the operator $\Lambda := \frac{d}{2}I_d + x\cdot \nabla$, and $\rho$ is the unique radial solution to the equation

$$\begin{aligned} L_{+}\rho = - |x|^2Q. \end{aligned}$$

(6.10)

One has the exponential decay of $\rho$ (see, e.g., [34, 48])

$$\begin{aligned} |\rho (x)|+|\nabla \rho (x)| \leqq Ce^{-\delta |x|}, \end{aligned}$$

(6.11)

where $C,\delta >0$, and the algebraic identities (see, e.g., [62, (B.1), (B.10), (B.15)])

$$\begin{aligned} \begin{aligned}&L_+ \nabla Q =0,\ \ L_+ \Lambda Q = -2 Q,\ \ L_+ \rho = -|x|^2 Q, \\&L_{-} Q =0,\ \ L_{-} xQ = -2 \nabla Q,\ \ L_{-} |x|^2 Q = - 4 \Lambda Q. \end{aligned}\end{aligned}$$

(6.12)

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

$$\begin{aligned} (Lf,f) :=\int f_1L_+f_1\textrm{d}x+\int f_2L_-f_2\textrm{d}x. \end{aligned}$$

(6.13)

The scalar product along the unstable directions in the null space is defined by

$$\begin{aligned} \textrm{Scal}(f)=&\langle f_1,Q\rangle ^2+\langle f_1,xQ\rangle ^2+\langle f_1,|x|^2Q\rangle ^2+\langle f_2,\nabla Q\rangle ^2\nonumber \\&+\langle f_2,\Lambda Q\rangle ^2+\langle f_2,\rho \rangle ^2. \end{aligned}$$

(6.14)

The localized coercivity of linearized operators is stated in Lemma 6.3 below.

Lemma 6.3

(Localized coercivity [59, Corollary 3.4]) Let $\phi$ be a positive smooth radial function on ${\mathbb {R}}^d$, such that $\phi (x) = 1$ for $|x|\leqq 1$, $\phi (x) = e^{-|x|}$ for $|x|\geqq 2$, $0<\phi \leqq 1$, and $\left| \frac{\nabla \phi }{\phi }\right| \leqq C$ for some $C>0$. Set $\phi _A(x) :=\phi \left( \frac{x}{A}\right)$, $A>0$. Then, for A large enough we have

$$\begin{aligned}&\int (|f|^2+|\nabla f|^2)\phi _A -(1+\frac{4}{d})Q^{\frac{4}{d}}f_1^2-Q^{\frac{4}{d}}f_2^2\textrm{d}x \nonumber \\&\quad \geqq C_1\int (|f|^2+|\nabla f|^2)\phi _A \textrm{d}x-C_2 \textrm{Scal}(f), \end{aligned}$$

(6.15)

where $C_1, C_2>0$, and $f_1, f_2$ are the real and imaginary parts of f, respectively.

Below we prove Proposition 2.5. For this purpose, we set $f(z):= |z|^{\frac{4}{d}} z$, $d=1,2$, and

$$\begin{aligned}&f^\prime (U)\cdot R := \partial _z f(U) R + \partial _{\overline{z}} f(U) \overline{R} = (1+\frac{2}{d}) |U|^\frac{4}{d} R + \frac{2}{d} |U|^{\frac{4}{d}-2} U^2 \overline{R}, \end{aligned}$$

(6.16)

$$\begin{aligned}&f^{\prime \prime }(U)\cdot R^2 := \frac{1}{d} (1+\frac{2}{d}) |U|^{\frac{4}{d} -2} \overline{U} R^2 \nonumber \\&\quad + \frac{2}{d} (1+\frac{2}{d}) |U|^{\frac{4}{d} -2} U |R|^2 + \frac{1}{d} (\frac{2}{d}-1) |U|^{\frac{4}{d} -4} U^3 \overline{R}^2, \end{aligned}$$

(6.17)

$$\begin{aligned}&f^{\prime \prime }(U,R)\cdot R^2:=R^2 \int _0^1 t \int _0^1 \partial _{zz} f(U+st R)\textrm{d}s \textrm{d}t\nonumber \\&\quad + 2 |R|^2 \int _0^1 t \int _0^1 \partial _{z\overline{z}} f(U+st R)\textrm{d}s \textrm{d}t \nonumber \\&\qquad + \overline{R}^2 \int _0^1 t \int _0^1 \partial _{\overline{z}\overline{z}} f(U+st R)\textrm{d}s \textrm{d}t. \end{aligned}$$

(6.18)

Then, we have the expansion

$$\begin{aligned} f(U+R) = f(U) + f^\prime (U)\cdot R + f^{\prime \prime }(U,R)\cdot R^2. \end{aligned}$$

(6.19)

We also get from equation (NLS) and (2.1) that (see [59, (4.11)])

$$\begin{aligned}&i\partial _tR +\sum _{k=1}^{K} (\Delta R_{k}+(1+\frac{2}{d})|U_{k}|^{\frac{4}{d}}R_{k} +\frac{2}{d}|U_{k}|^{\frac{4}{d}-2}U_{k}^2 \overline{R_{k}} \nonumber \\&\quad + i\partial _tU_{k}+\Delta U_{k}+|U_{k}|^{\frac{4}{d}}U_{k}) + \sum \limits _{k=1}^K f^{\prime \prime }(U_k) \cdot R^2 \nonumber \\&=-H_1-H_2-H_3. \end{aligned}$$

(6.20)

Here, $H_1, H_2$ and $H_3$ contain the interactions between different blow-up profiles

$$\begin{aligned} H_1 :&= (1+\frac{2}{d})|U|^{\frac{4}{d}}R+\frac{2}{d}|U|^{\frac{4}{d}-2}U^2\overline{R} -\sum _{k=1}^{K}((1+\frac{2}{d})|U_{k}|^{\frac{4}{d}}R_{k}+\frac{2}{d}|U_{k}|^{\frac{4}{d}-2}U_{k}^2\overline{{R}_{k}}), \end{aligned}$$

(6.21)

$$\begin{aligned} H_2 :&= |U|^{\frac{4}{d}}U-\sum _{k=1}^{K} |U_{k}|^{\frac{4}{d}}U_{k}, \end{aligned}$$

(6.22)

$$\begin{aligned} H_3 :&= f^{\prime \prime }(U,R)\cdot R^2-\sum _{k=1}^{K} f''(U_k)\cdot R^2. \end{aligned}$$

(6.23)

Moreover, we get from equation (NLS) and (2.2) that (see [59, (4.14)])

$$\begin{aligned} i\partial _tU_{k}+\Delta U_{k}+|U_{k}|^{\frac{4}{d}}U_{k}&=\frac{e^{i\theta _{k}}}{\lambda _{k}^{2+\frac{d}{2}}} \bigg \{-(\lambda _{k}^2{\dot{\theta }}_{k}-1-|\beta _{k}|^2)Q_{k} -(\lambda _{k}^2{\dot{\beta }}_{k}+\gamma _{k}\beta _{k})\nonumber \\&\quad \cdot yQ_{k} +\frac{1}{4} (\lambda _{k}^2{\dot{\gamma }}_{k}+\gamma _{k}^2) |y|^2 Q_{k} -i(\lambda _{k}{\dot{\alpha }}_{k}-2\beta _{k})\cdot \nabla Q_{k} \nonumber \\&\quad -i(\lambda _{k}\dot{\lambda _{k}}+\gamma _{k})\Lambda Q_{k} \bigg \}\left( t,\frac{x-\alpha _{k}}{\lambda _{k}}\right) , \end{aligned}$$

(6.24)

where $Q_k$ is given by (2.15), $1\leqq k\leqq K$. The following identity also holds (see [59, (4.28)])

$$\begin{aligned} \Delta Q_k - Q_k + |Q_k|^{\frac{4}{d}} Q_k = |\beta _k - \frac{{\gamma }_k}{2}|^2 Q_k - i {\gamma }_k \Lambda Q_k + 2i \beta _k \cdot \nabla Q_k. \end{aligned}$$

(6.25)

Proof of Proposition 2.5

The proof is similar to that of [59, Proposition 4.3], it mainly relies on the almost orthogonality in Lemma 2.4 and the decoupling Lemma 6.2.

Taking the inner product of (6.20) with $\Lambda _k {U_{k}}$ and then taking the real part we get

$$\begin{aligned}&-\textrm{Im}\langle \partial _tR,\Lambda _k U_{k}\rangle +\textrm{Re}\langle \Delta R_{k}+(1+\frac{2}{d})|U_{k}|^{\frac{4}{d}}R_{k}+\frac{2}{d}|U_{k}|^{\frac{4}{d}-2}U_{k}^2\overline{{R}_{k}},\Lambda _k U_{k}\rangle \nonumber \\&+\textrm{Re}\langle i\partial _tU_{k}+\Delta U_{k}+|U_{k}|^{\frac{4}{d}}U_{k},\Lambda _k U_{k}\rangle + \textrm{Re} \langle f^{\prime \prime }(U_k) \cdot R^2, \Lambda _k U_k\rangle \nonumber \\&=-\textrm{Re} \bigg<\sum _{j\ne k} (\Delta R_{j}+(1+\frac{2}{d})|U_{j}|^{\frac{4}{d}}R_{j}+\frac{2}{d}|U_{j}|^{\frac{4}{d}-2}U_{j}^2\overline{{R}_{j}}) + H_1,\Lambda _k U_{k} \bigg> \nonumber \\&-\textrm{Re} \bigg< \sum _{j\ne k}( i\partial _tU_{j}+\Delta U_{j}+|U_{j}|^{\frac{4}{d}}U_{j}) + H_2,\Lambda _k U_{k} \bigg>\nonumber \\&-\textrm{Re} \bigg < \sum _{j\ne k}( f^{\prime \prime }(U_j)\cdot R^2 +H_3,\Lambda _k U_{k} \bigg >. \end{aligned}$$

(6.26)

For the R.H.S. of equation (6.26), we claim that

$$\begin{aligned} {\lambda }_k^2 \times (\mathrm{R.H.S.\ of}\ (6.26)) = {{\mathcal {O}}}\left( e^{-\frac{\delta }{T-t}}(1+Mod+ \Vert R\Vert _{L^2}+ \Vert R\Vert ^2_{L^2}) + D^3(t) \right) . \end{aligned}$$

(6.27)

To this end, we have from [59, (4.19), (4.20)] that the first two terms on the R.H.S. of (6.26) are bounded by

$$\begin{aligned} C (T-t)^{-2} e^{-\frac{\delta }{T-t}} (1+ Mod+ \Vert R\Vert _{L^2}), \end{aligned}$$

(6.28)

where $C,\delta >0$. Moreover, by Lemma 6.2,

$$\begin{aligned} \bigg |\textrm{Re} \langle \sum _{j\ne k}( f^{\prime \prime }(U_j)\cdot R^2 ,\Lambda _k U_{k} \rangle \bigg | \leqq C (T-t)^{-2} e^{-\frac{\delta }{T-t}} \Vert R\Vert ^2_{L^2}. \end{aligned}$$

(6.29)

Using (6.18) we also have that

$$\begin{aligned}&|\textrm{Re} \langle H_3, \Lambda U_k\rangle | \nonumber \\&\quad \leqq C \sum \limits _{{{\tilde{z}}}, {{\tilde{z}}}^* \in \{z, \overline{z}\}} \bigg (\bigg<|R|^2 \int _0^1 t \int _0^1 |\partial _{{{\tilde{z}}}{{\tilde{z}}}^*} f(U) - \sum \limits _{k=1}^K \partial _{{{\tilde{z}}}{{\tilde{z}}}^*}f(U_k)|\textrm{d}s \textrm{d}t, |\Lambda _k U_k| \bigg> \nonumber \\&\qquad + \bigg < |R|^2 \int _0^1 t \int _0^1 |\partial _{\tilde{z}{{\tilde{z}}}^*} f(U+stR) - \partial _{{{\tilde{z}}}{{\tilde{z}}}^*}f(U)| \textrm{d}s \textrm{d}t, |\Lambda _k U_k|\bigg >\bigg ). \end{aligned}$$

(6.30)

Using Lemma 6.2 again we see that the first inner product above only contributes $e^{-\frac{\delta }{T-t}}\Vert R\Vert _{L^2}^2$, while the second one can be bounded by

$$\begin{aligned} \int |R|^3 (|U|^{\frac{4}{d} -2} + |R|^{\frac{4}{d} -2}) |\Lambda _k U_k| dx&\leqq C (T-t)^{-2} (\Vert {\varepsilon }_k\Vert _{L^3}^3 +\Vert {\varepsilon }_k\Vert _{L^{1+\frac{4}{d}}}^{1+\frac{4}{d}}) \nonumber \\&\leqq C (T-t)^{-2} D^3, \end{aligned}$$

(6.31)

where the renormalized remainder ${\varepsilon }_k$ is given by (2.57) and we also used (6.2) in the last step.

Thus, combining estimates (6.28–6.31) and using (2.9) we obtain (6.27), as claimed.

Regarding the L.H.S. of (6.26), using the almost orthogonality (2.51), equation (6.24) and (6.25) we have that (see [59, (4.26), (4.27), (4.31)])

$$\begin{aligned} \textrm{Im}\langle \partial _t R,\Lambda _k U_{k}\rangle&= \textrm{Im}\langle \Lambda _k R_{k},\partial _t U_{k}\rangle + {{\mathcal {O}}}(e^{-\frac{\delta }{T-t}}(1+Mod) \Vert R\Vert _{L^2}), \end{aligned}$$

(6.32)

$$\begin{aligned} {\lambda }_k^2 \textrm{Im}\langle \Lambda _k R_{k},\partial _t U_{k}\rangle&= \textrm{Re}\langle \varepsilon _{k},\Lambda Q_{k}\rangle + {\gamma }_k\textrm{Im}\langle \Lambda \varepsilon _{k},\Lambda Q_{k}\rangle -2 {\beta }_k\textrm{Im}\langle \Lambda \varepsilon _{k},\nabla Q_{k}\rangle \nonumber \\&+ {{\mathcal {O}}}((Mod+P^2)\Vert R\Vert _{L^2}), \end{aligned}$$

(6.33)

and

$$\begin{aligned} {\lambda }_k^2 \textrm{Re}\langle i\partial _tU_{k}+\Delta U_{k}+|U_{k}|^{\frac{4}{d}}U_{k},\Lambda _k U_{k}\rangle = - \frac{1}{4}\Vert yQ\Vert _2^2({\lambda }_k^2\dot{{\gamma }_k}+{\gamma }_k^2) + {{\mathcal {O}}}(|\beta _k|Mod). \end{aligned}$$

(6.34)

Moreover, by (6.17) and the change of variables,

$$\begin{aligned} {\lambda }_k^2 \textrm{Re}\langle f''(U_k)\cdot R^2,\Lambda _k U_{k}\rangle&= \textrm{Re}\int (1+\frac{2}{d})|Q_k|^{\frac{4}{d}}|{\varepsilon }_k|^2 + \frac{2}{d} |Q_k|^{\frac{4}{d}-2} Q_k^2 \overline{{\varepsilon }_k}^2 \textrm{d}y \nonumber \\&+\textrm{Re}\int (y \cdot \nabla \overline{Q_k}) (f^{\prime \prime }(Q_k)\cdot {\varepsilon }^2_k) \textrm{d}y + {{\mathcal {O}}}(e^{-\frac{\delta }{T-t}}\Vert R\Vert _{L^2}^2). \end{aligned}$$

(6.35)

Thus, combining (6.32–6.35) altogether and rearranging the terms according to the orders of renormalized variable ${\varepsilon }_k$ we get that

$$\begin{aligned}&{\lambda }_k^2 \times (\mathrm{L.H.S.\ of}\ (6.26)) \nonumber \\&=- \frac{1}{4}\Vert yQ\Vert _2^2({\lambda }_k^2\dot{{\gamma }_k}+{\gamma }_k^2) \nonumber \\&+ \textrm{Re}\langle \Delta \varepsilon _{k}-\varepsilon _k+(1+\frac{2}{d})|Q_{k}|^{\frac{4}{d}}\varepsilon _{k}+\frac{2}{d}|Q_{k}|^{\frac{4}{d}-2}Q_{k}^2\overline{{\varepsilon }_{k}},\Lambda Q_{k}\rangle \end{aligned}$$

(6.36)

$$\begin{aligned}&- {\gamma }_k\textrm{Im}\langle \Lambda \varepsilon _{k},\Lambda Q_{k}\rangle +2{\beta }_k\textrm{Im}\langle \Lambda \varepsilon _{k},\nabla Q_{k}\rangle \end{aligned}$$

(6.37)

$$\begin{aligned}&+ \textrm{Re}\int (1+\frac{2}{d})|Q_k|^{\frac{4}{d}}|\varepsilon _k|^2 + \frac{2}{d} |Q_k|^{\frac{4}{d}-2}Q_k^2 \overline{\varepsilon _k}^2 \textrm{d}y \nonumber \\&+ \textrm{Re}\int (y \cdot \nabla \overline{Q_k})(f^{\prime \prime }(Q_k)\cdot {\varepsilon }_k^2) \textrm{d}y \end{aligned}$$

(6.38)

$$\begin{aligned}&+ {{\mathcal {O}}}\left( (Mod+P^2)\Vert R\Vert _{L^2} + P \ Mod + e^{-\frac{\delta }{T-t}}(1+Mod)\right) . \end{aligned}$$

(6.39)

For the linear part on the R.H.S. above, using the linearized operators $L_{\pm }$ in (6.9) we have (see [59, (4.37)])

$$\begin{aligned} (6.36)+ (6.37) = {M_k } - \int |R |^2\Phi _k\textrm{d}x + {{\mathcal {O}}}( (P^2+e^{-\frac{\delta }{T-t}}) \Vert R\Vert _{L^2}) . \end{aligned}$$

(6.40)

Hence, we obtain that

$$\begin{aligned}&{\lambda }_k^2 \times (\mathrm{L.H.S.\ of}\ (6.26)) = - \frac{1}{4} \Vert yQ\Vert _{L^2}^2 ( {\lambda }_k^2 {\dot{{\gamma }}}_k + {\gamma }_k^2) + M_k - \int |R|^2 \Phi _k \textrm{d}x \nonumber \\&\quad + \textrm{Re}\int (1+\frac{2}{d})|Q_k|^{\frac{4}{d}}|\varepsilon _k|^2 + \frac{2}{d} |Q_k|^{\frac{4}{d}-2}Q_k^2 \overline{\varepsilon _k}^2 \textrm{d}y \nonumber \\&\quad + \textrm{Re}\int (y \cdot \nabla \overline{Q_k}) (f^{\prime \prime }(Q_k)\cdot {\varepsilon }_k^2) dy \nonumber \\&\quad + {{\mathcal {O}}}\left( (P+\Vert R\Vert _{L^2}+e^{-\frac{\delta }{T-t}})Mod + P^2\Vert R\Vert _{L^2} + e^{-\frac{\delta }{T-t}}\right) , \end{aligned}$$

(6.41)

which, along with (2.8) and (6.27), yields (2.56).

Moreover, we can also bound the quadratic terms of ${\varepsilon }_k$ in (6.38) by

$$\begin{aligned} (6.38) \leqq C \Vert {\varepsilon }_k\Vert _{L^2}^2 \leqq C\Vert R\Vert _{L^2}^2. \end{aligned}$$

(6.42)

This, along with (6.41), yields that

$$\begin{aligned}&{\lambda }_k^2 \times (\mathrm{L.H.S.\ of}\ (6.26)) =- \frac{1}{4}\Vert yQ\Vert _2^2({\lambda }_k^2\dot{{\gamma }_k}+{\gamma }_k^2)+ M_k \nonumber \\&\quad + {{\mathcal {O}}}\left( (P+\Vert R\Vert _{L^2}+e^{-\frac{\delta }{T-t}})Mod + P^2 \Vert R\Vert _{L^2} + \Vert R\Vert _{L^2}^2 + e^{-\frac{\delta }{T-t}}\right) . \end{aligned}$$

(6.43)

Thus, combining (6.27) and (6.43) together we obtain that, for every $1\leqq k\leqq K$,

$$\begin{aligned}&|{\lambda }_k^2\dot{{\gamma }_k}+{\gamma }_k^2|\nonumber \\&\quad \leqq C \bigg ( (P + \Vert R\Vert _{L^2}+e^{-\frac{\delta }{T-t}} )Mod +|M_k| + P^2\Vert R\Vert _{L^2}+\Vert R\Vert _{L^2}^2 + D^3 + e^{-\frac{\delta }{T-t}} \bigg ), \end{aligned}$$

(6.44)

which, along with (2.54) and (2.61), yields that

$$\begin{aligned}&\sum _{k=1}^{K}|{\lambda }_k^2\dot{{\gamma }_k}+{\gamma }_k^2| \nonumber \\&\quad \leqq C \bigg ( (P+ \Vert R\Vert _{L^2}+e^{-\frac{\delta }{T-t}})Mod +\sum _{k=1}^{K} |M_k| +P^2 D +D^2 + e^{-\frac{\delta }{T-t}} \bigg ). \end{aligned}$$

(6.45)

Similar arguments apply also to the remaining four modulation equations. Actually, taking the inner products of equation (6.20) with $i(x-{\alpha }_k) U_k$, $i|x-{\alpha }_k|^2 {U_k}$, $\nabla {U_k}$, $\varrho _{k}$, respectively, then taking the real parts and using analogous arguments as above, we obtain

$$\begin{aligned} Mod&\leqq C\bigg ( (P+\Vert R\Vert _{L^2}+e^{-\frac{\delta }{T-t}})Mod +\sum _{k=1}^{K} |M_k| + P^2 D + D^2 + e^{-\frac{\delta }{T-t}} \bigg ). \end{aligned}$$

(6.46)

Therefore, using (2.6) and (2.7) we may take t close to T such that $P(t)+ \Vert R(t)\Vert _{L^2} + e^{-\frac{\delta }{T-t}}$ is sufficiently small to obtain (2.53). The proof of Proposition 2.5 is complete. $\square$