Appendix A: notations related to the nonlinear terms
Denoting
$$\begin{aligned} f^n_{+,l}(s)=\sum _{j=1}^d f_{+,l,j}^n e^{-i \beta ^-_j s},\quad f^n_{-,l}(s)=\sum _{j=1}^d f_{-,l,j}^n e^{i \beta ^-_j s},\quad l=1,2,\ldots ,d, \end{aligned}$$
the expressions of \(F_{k,l}^n(s)=F_{k,l}(\varvec{y}(t_n),\dot{\varvec{y}}(t_n); s)\) (\(l=1,\ldots ,d\), \(k=\pm 1,\ldots ,\pm 5\)) are given as:
$$\begin{aligned} \begin{aligned} F^n_{0,l}(s) =&|f^n_{+,l}(s)|^2+ |f^n_{-,l}(s)|^2, \quad F^n_{1,l}(s) = 2 {\text {Re}} (-\overline{f^n_l}f^n_{-,l}(s)),\\ F^n_{2,l}(s) =&\overline{f^n_l} f_{+,l}^n(s) \\ F^n_{3,l}(s) =&f^n_{+,l}(s) \overline{f^n_{-,l}(s)}, \quad F^n_{4,l}(s) = -\overline{f^n_l}f^n_{-,l}(s),\quad F^n_{5,l}(s) = \overline{f^n_l} f^n_{-,l}(s), \\ F^n_{-k,l}(s)=&\overline{F^n_{k,l}(s)}.\end{aligned} \end{aligned}$$
(A.1)
\(g_{k,l}^n(s)=g_{k,l}(\varvec{y}(t_n),\dot{\varvec{y}}(t_n); s)\) (\(l=1,\ldots ,d\)) are defined as:
$$\begin{aligned} \begin{aligned} g^n_{0,l}(s) =&F^n_{-3,l}(s)f^n_{+,l}(s) + F^n_{0,l}(s) f^n_{-,l}(s), \\ g^n_{1,l}(s) =&F^n_{-4,l}(s) f^n_{+,l}(s)+ F^n_{1,l}(s) f^n_{-,l}(s)-F^n_{0,l}(s) f^n_l, \\ g^n_{2,l}(s) =&F^n_{-5,l}(s) f^n_{+,l}(s) + F^n_{2,l}(s) f^n_{-,l}(s),\\ g^n_{3,l}(s) =&F^n_{-2,l}(s) f^n_{-,l}(s)+ F^n_{-3,l}(s) f^n_l, \quad g^n_{4,l}(s) = F^n_{-3,l}(s) f^n_{-,l}(s) \\ g^n_{5,l}(s) =&F^n_{-4,l}(s) f^n_{+,l}(s) + F^n_{-3,l}(s) (-f^n_l),\quad g^n_{6,l}(s) = F^n_{-5,l}(s) f^n_{-,l}(s), \\ g^n_{7,l}(s) =&F^n_{0,l}(s) f^n_{+,l}(s)+ F^n_{3,l}(s) f^n_{-,l}(s),\\ g^n_{8,l}(s) =&F^n_{1,l}(s) f^n_{+,l}(s)+ F^n_{4,l}(s) f^n_{-,l}(s)+ F^n_{3,l}(s) (-f^n_l), \\ g^n_{9,l}(s) =&F^n_{2,l}(s) f^n_{+,l}(s), \quad g^n_{11,l}(s) =F^n_{3,l}(s) f^n_{+,l}(s) \\ g^n_{10,l}(s) =&F^n_{-2,l}(s) f^n_{+,l}(s) + F^n_{5,l}(s) f^n_{-,l}(s) + F^n_{0,l}(s) f^n_l,\\ g^n_{12,l}(s) =&F^n_{4,l}(s) f^n_{+,l}(s), \quad g^n_{13,l}(s) =F^n_{5,l}(s) f^n_{+,l}(s) + F^n_{3,l}(s) f^n_l. \end{aligned} \end{aligned}$$
(A.2)
The following expressions appear in the 2nd order NPI:
$$\begin{aligned} G_{+,0,k}^n(s)&=G_{+,0,k}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n); s)=e^{is/\varepsilon ^2} (q_{-1,0}(s)g^n_{1,k}(0)+q_{-1,1}(s)g^n_{2,k}(0)\nonumber \\&\quad +q_{-1,-1}(s)g^n_{3,k}(0)\nonumber \\&\quad +q_{-2,0}(s)g^n_{5,k}(0)+q_{-2,1}(s)g^n_{6,k}(0) + q_{0,1}(s)g^n_{9,k}(0) \nonumber \\&\quad + q_{0,-1}(s)g^n_{10,k}(0) +q_{1,0}(s)g^n_{12,k}(0) +q_{1,1}(s)g^n_{13,k}(0)), \nonumber \\ G_{+,1,k}^n(s)&= G_{+,1,k}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n);s)=e^{is/\varepsilon ^2} (s g^n_{7,k}(s/2) + s^2/2 g^n_{8,k}(s/2)),\nonumber \\ G_{+,2,k}^n(s)&= G_{+,2,k}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n); s)=e^{is/\varepsilon ^2}(p_{-1}(s) g_{0,k}^n(0) + q_{-1,0}(s) {\dot{g}}^n_{0,k}(0) \nonumber \\&\quad +p_{-2}(s) g_{4,k}^n(0) + q_{-2,0}(s) {\dot{g}}^n_{4,k}(0)\nonumber \\&\quad +p_{1}(s) g_{11,k}^n(0) + q_{1,0}(s) {\dot{g}}^n_{11,k}(0)), \nonumber \\ G_{+,3,k}(s)&= G_{+,3,k}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n); s) = e^{is/\varepsilon ^2}(q_{-1,0}(s) g^n_{0,k}(0) \nonumber \\&\quad + q_{-2,0}(s) g^n_{4,k}(0) + q_{1,0}(s) g^n_{11,k}(0)),\nonumber \\ G_{-,0,k}^n(s)&=G_{-,0,k}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n); s) = q_{0,1}(s)g^n_{2,k}(0)\nonumber \\&\quad +q_{0,-1}(s)g^n_{3,k}(0)+q_{-1,0}(s)g^n_{5,k}(0)\nonumber \\&\quad +q_{-1,1}(s)g^n_{6,k}(0)+q_{-1,-1}(s)g^n_{8,k}(0)+q_{1,1}(s)g^n_{9,k}(0)\nonumber \\&\quad +q_{1,-1}(s)g^n_{10,k}(0)+q_{2,0}(s)g^n_{12,k}(0)+q_{2,-1}(s)g^n_{13,k}(0),\nonumber \\ G_{-,1,k}^n(s)&= G_{-,1,k}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n);s)=s g^n_{0,k}(s/2) + s^2/2 g^n_{1,k}(s/2),\nonumber \\ G_{-,2,k}^n(s)&= G_{-,2,k}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n);s)=p_{-1}(s) g_{4,k}^n(0) + q_{-1,0}(s) {\dot{g}}^n_{4,k}(0) \nonumber \\&\quad +p_{1}(s) g_{7,k}^n(0) + q_{1,0}(s) {\dot{g}}^n_{7,k}(0)\nonumber \\&\quad +p_{2}(s) g_{11,k}^n(0) + q_{2,0}(s) {\dot{g}}^n_{11,k}(0),\nonumber \\ G_{-,3,k}(s) =&G_{-,3,k}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n);s) =q_{-1,0}(s) g^n_{4,k}(0) + q_{-2,0}(s) g^n_{7,k}(0) \nonumber \\&\qquad + q_{1,0}(s)g^n_{11,k}(0). \end{aligned}$$
(A.3)
Appendix B: proof of Lemma 3.1
Denote local truncation error by \(\varvec{\xi }^{n}:=\varvec{y}(t_{n+1}) - \varvec{S}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n)), \, \dot{\varvec{\xi }}^{n}:=\dot{\varvec{y}}(t_{n+1}) - \dot{\varvec{S}}(\varvec{y}(t_n), \dot{\varvec{y}}(t_n))\). The local error can be easily verified by following the iterations for designing the scheme 2.21.
By the construction of NPI (2.21), we will first estimate the first order NPI approximation \(\varvec{y}^{n,1}(s)\). Noticing function \(F(\varvec{y})\varvec{y}\) (here \(\text {diag}(\varvec{y} \varvec{y}^H) \varvec{y}\)) is Lipschitz on bounded interval \([-(M+1),(M+1)]^{d}\) with some Lipschitz constant \(L_M\), which means:
$$\begin{aligned} ||a_l|^2 a_l - |b_l|^2 b_l| \le L_M\Vert \varvec{a}-\varvec{b}\Vert _2, \quad \forall 1\le l\le d. \end{aligned}$$
For the integration kernel \(\kappa (t)\), we have \(|\kappa _l(t)| \le 2,\forall 0\le t\le \tau \). Therefore, we have
$$\begin{aligned}&|y_l(t_n+s) - y^{n,1}_l(s) |\\&\quad \le \sum _{j,k=1}^d |\overline{u_{jl}} u_{jk} \gamma _j|\int _{0}^{s} |\kappa _j(s-w)| \left| |y_k(t_n)|^2 y_k(t_n) - |y_k(t_n+w)|^2 y_k(t_n+w) \right| dw\\&\qquad +\sum _{j,k=1}^d |\overline{u_{jl}} u_{jk} \gamma _j| \bigg (\left| \int _0^s e^{-iw/\varepsilon ^2} e^{-i \beta ^-_j (s-w)} |y_k(t_n+w)|^2 y_k(t_n+w) dw \right. \\&\qquad \left. - p_{-1}(s)|y_k(t_n+w)|^2 y_k(t_n+w)\right| \\&\qquad +\left| \int _0^s e^{i \beta ^-_j (s-w)} |y_k(t_n+w)|^2 y_k(t_n+w) dw - s |y_k(t_n+w)|^2 y_k(t_n+w)\right| \bigg )\\&\quad \lesssim \int _{0}^{s} L \Vert \varvec{y}(t_n+w)-\varvec{y}(t_n)\Vert _2 dw+ \sum _{j,k=1}^d \int _0^s |e^{i \beta ^-_j (s-w)} -1| |y_k(t_n+w)|^3 dw\\&\quad \lesssim \int _{0}^{s} L w\max _{t\in [0,\tau ]}\Vert \dot{\varvec{y}}(t_n+t)\Vert _2 dw+\sum _{j,k=1}^d \int _0^s |\beta ^-_j (s - w)| |y_k(t_n+w)|^3 dw\\&\quad \lesssim s^2, \quad \forall 0\le s \le \tau , 1\le l\le d. \end{aligned}$$
Thus, we know that there exists \(\tau _0>0\) such that for \(0<s\le \tau <\tau _0\), \(\Vert \varvec{y}^{n,1}(s)\Vert _\infty \le \Vert \varvec{y}(t_n+s)\Vert _\infty +1\le M+1\).
Continue with the second NPI approximation,
$$\begin{aligned}&|y_l(t_n+s) - y^{n,2}_{l}(s) |\\&\quad \le \sum _{j,k=1}^d |\overline{u_{jl}} u_{jk} \gamma _j|\int _{0}^{s} |\kappa _j(s-w)| \left| |y^{n,1}_{k}(w)|^2 y^{n,1}_{k}(w) \right. \\&\qquad \left. -\, |y_k(t_n+w)|^2 y_k(t_n+w) \right| dw +| R_{2,l}^n(s)|\\&\qquad \le C \int _{0}^{s} L \Vert \varvec{y}(t_n+w)-\varvec{y}^{n,1}(w)\Vert _2 dw +| R_{2,l}^n(s)| \lesssim s^3+ | R_{2,l}^n(s)|, \quad \forall 1\le l\le d. \end{aligned}$$
where \(\varvec{R}_{2}^n(s) =(R_{2,1}^n(s),\ldots ,R_{2,d}^{n}(s))^T\) is the error for approximating the integrals in (2.17). By construction, we write \(\varvec{R}_{2}^n(s) = \varvec{D}_{2}^n(s) + \varvec{Q}_{2}^n(s)\), where \(\varvec{D}_{2}^n(s)\) is the error introduced by discarded term in formula (2.15) and (2.16), \(\varvec{Q}_{2}^n(s)\) is the error introduced by numerical quadrature.
To estimate \(\varvec{D}_{2}^n(s)\), we write
$$\begin{aligned} D_{2,l}^n(s)&= \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j \int _0^{s} \kappa _j(s-w) \\&\quad \left( e^{3iw/\varepsilon ^2} \left( \sum _{m=1}^d f_{+,k,m}^n e^{-i \beta _m^- w}\right) \left( p_{-1}^2(w) |f_k^n|^2 \right) \right) dw \\&\quad +\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j \int _0^{s}\kappa _j(s-w) \\&\quad \left( -e^{3iw/\varepsilon ^2}\left( \sum _{m=1}^d f_{+,k,m}^n e^{-i \beta _m^- w}\right) \left( p_{-1}(w) w|f_k^n|^2 \right) \right) dw \\&\quad +\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j \int _0^{s} \kappa _j(s-w)\left( e^{2iw/\varepsilon ^2}\right. \\&\left. \quad \left( \sum _{m=1}^d f_{-,k,m}^n e^{i \beta _m^- w}\right) \left( p_{-1}^2(w) |f_k^n|^2 \right) \right) dw\\&\quad +\cdots , \end{aligned}$$
where other similar terms are omitted for simplicity. Noticing \(p_0(w) =w\), \(\varvec{D}_{2}^n(s)\) includes every term containing integrated product of more than one \(p_k(w),k = -1,0,1\). We estimate one of these terms, and the others can be proved similarly. Since \(\gamma _j = (1+4\varepsilon ^2 d_{jj})^{-1/2}\) and \(\kappa _j(t) = e^{i \beta _j^+ t} - e^{i \beta _j^- t}\) are bounded for \( j = 1,2,\ldots ,d\), there holds
$$\begin{aligned}&\left| \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j \int _0^{s} \kappa _j(s-w) \left( e^{3iw/\varepsilon ^2} \left( \sum _{m=1}^d f_{+,k,m}^n e^{-i \beta _m^- w}\right) \left( p_{-1}^2(w) |f_k^n|^2 \right) \right) dw\right| \\&\quad \le C \int _0^{s} \sum _{m=1}^d \left( |y_m(t_n)| + \varepsilon ^2|{\dot{y}}_m(t_n)|\right) \left( \sum _{k=1}^d w^2 |y_k^n|^6 \right) dw \lesssim \int _0^{s} w^2 dw \lesssim s^3. \end{aligned}$$
Then we can derive \(|D_{2,l}^n(s)| \lesssim s^3, l = 1,2,\ldots ,d\).
To estimate \(\varvec{Q}_{2}^n(s)\), we write
$$\begin{aligned} Q_{2,l}^n(s)&= \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \left( \int _0^s e^{i \beta ^-_j (s-w)} w g_{1,k}^n(w) dw - e^{i \beta ^-_j s/2} \frac{s^2}{2} g_{1,k}^n(\frac{s}{2})\right) \\&\quad +\cdots (\text {error of other type 1 error terms})\\&\quad +\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \left( \int _0^s e^{-\frac{i}{\varepsilon ^2}w} e^{-i \beta ^-_j (s-w)} p_1(w) g_{2,k}^n(w) dw - q_{-1,1}(s) g_{2,k}^n(0) \right) \\&\quad +\cdots (\text {error of other type 2 error terms})\\&\quad +\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \bigg (\int _0^s e^{-\frac{i}{\varepsilon ^2}w} e^{-i \beta ^-_j (s-w)} g_{0,k}^n(w) dw \\&\qquad - e^{-i \beta ^-_j s} (p_{-1}(s) g_{0,k}^n(0) + q_{-1,0}(s) (i \beta ^-_l g_{0,k}^n(0) + {\dot{g}}_{0,k}^n(0)))\bigg )\\&\quad +\cdots (\text {error of other type 3 error terms}), \end{aligned}$$
where classification of type 1, 2, 3 terms are introduced for numerical quadrature to calculate expression (2.17). For simplicity, we prove the estimates for one of each type of terms, and the proof can be easily extended to the other terms.
First we prove the boundedness of \(g_{m,k}^n(t)\), \({\dot{g}}_{m,k}^n(t)\) and \( \ddot{g}_{m,k}^n(t)\), \(m = 1,2,\ldots , 13, k = 1,2,\ldots ,d\) for \(\forall 0<t<\tau \). Notice all \(g_{m,k}^n(t)\) share a similar structure, here we only prove for \(g_{4,k}^n(t)\).
$$\begin{aligned} g_{4,k}^n(t)&= \left( \sum _{j=1}^d \overline{f_{+,k,j}^n} e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d f_{-,k,j}^n e^{i \beta _j^- t} \right) ^2 ,\\ {\dot{g}}_{4,k}^n(t)&= \left( \sum _{j=1}^d i \beta _j^- \overline{f_{+,k,j}^n} e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d f_{-,k,j}^n e^{i \beta _j^- t} \right) ^2\\&\quad + 2 \left( \sum _{j=1}^d \overline{f_{+,k,j}^n} e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d i \beta _j^- f_{-,k,j}^n e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d f_{-,k,j}^n e^{i \beta _j^- t} \right) ,\\ \ddot{g}_{4,k}^n(t)&= \left( \sum _{j=1}^d -(\beta _j^-)^2 \overline{f_{+,k,j}^n} e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d f_{-,k,j}^n e^{i \beta _j^- t} \right) ^2 \\&\quad + 4 \left( \sum _{j=1}^d i \beta _j^- \overline{f_{+,k,j}^n} e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d i \beta _j^- f_{-,k,j}^n e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d f_{-,k,j}^n e^{i \beta _j^- t} \right) \\&\quad +2 \left( \sum _{j=1}^d \overline{f_{+,k,j}^n} e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d - (\beta _j^-)^2 f_{-,k,j}^n e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d f_{-,k,j}^n e^{i \beta _j^- t} \right) \\&\quad + 2 \left( \sum _{j=1}^d \overline{f_{+,k,j}^n} e^{i \beta _j^- t} \right) \left( \sum _{j=1}^d i \beta _j^- f_{-,k,j}^n e^{i \beta _j^- t} \right) ^2. \end{aligned}$$
Since \(\beta _j^-\) are bounded \(\forall j = 1,2,\ldots ,d\), and \(|f_{+,k,j}^n|\lesssim |y_k(t_n)| + \varepsilon ^2 |{\dot{y}}_k(t_n)|\), \(|f_{-,k,j}^n|\lesssim |y_k(t_n)| + \varepsilon ^2 |{\dot{y}}_k(t_n)|\), \(|f_k^n| \le |y_k(t_n)|^3\) are also bounded by the assumption of the exact solution, we have
$$\begin{aligned} g_{k,l}^n(t) \lesssim 1, \, {\dot{g}}_{k,l}^n(t) \lesssim 1, \, \ddot{g}_{k,l}^n(t) \lesssim 1, \forall k=1,2,\ldots ,13, \, l=1,2,\ldots d. \end{aligned}$$
Then we estimate \(Q_{2,l}^n(s)\).
Type 1. We have
$$\begin{aligned}&\left| \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \left( \int _0^s e^{i \beta ^-_j (s-w)} w g_{1,k}^n(w) dw - e^{i \beta ^-_j s/2} \frac{s^2}{2} g_{1,k}^n(\frac{s}{2})\right) \right| \\&\quad \lesssim \sum _{j,k=1}^d \int _0^s ((w-s/2) |{\dot{\Theta }}_k^j(s/2)| + (w-s/2)^2 |\ddot{\Theta }_k^j(\xi _k^j(w))| )dw\\&\qquad \le \sum _{j,k=1}^d \int _0^s ( w-s/2)^2 |\ddot{\Theta }_k^j(\xi _k^j(w))| dw \\ \end{aligned}$$
where \(\Theta _k^j(t) = e^{i \beta ^-_j (s-t)} t g_{2,k}^n(t)\) and \(0< \xi _k^j(w) < w\). For derivatives, we get
$$\begin{aligned} {\dot{\Theta }}_k^j(t)&= i \beta ^-_j e^{i \beta ^-_j (s-t)} t g_{2,k}^n(t) + e^{i \beta ^-_j (s-t)} g_{2,k}^n(t) + e^{i \beta ^-_j (s-t)} t {\dot{g}}_{2,k}^n(t),\\ \ddot{\Theta }_k^j(t)&= -(\beta ^-_j)^2 e^{i \beta ^-_j (s-t)} t g_{2,k}^n(t) + e^{i \beta ^-_j (s-t)} t \ddot{g}_{2,k}^n(t)\\&\quad +2(i \beta ^-_j e^{i \beta ^-_j (s-t)} g_{2,k}^n(t)+ i \beta ^-_j e^{i \beta ^-_j (s-t)} t {\dot{g}}_{2,k}^n(t) + e^{i \beta ^-_j (s-t)} {\dot{g}}_{2,k}^n(t)), \end{aligned}$$
and \(\ddot{\Theta }_k^j(t) \lesssim 1\) for \(\forall 0 \le t \le \tau \) by the boundedness of \(g_{2,k}^n(t)\), \({\dot{g}}_{2,k}^n(t)\) and \(\ddot{g}_{2,k}^n(t)\). Thus,
$$\begin{aligned}&\left| \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \left( \int _0^s e^{i \beta ^-_j (s-w)} w g_{1,k}^n(w) dw - e^{i \beta ^-_j s/2} \frac{s^2}{2} g_{1,k}^n(\frac{s}{2})\right) \right| \\&\quad \lesssim \sum _{j,k=1}^d \int _0^s ( w-s/2)^2 dw \lesssim s^3. \end{aligned}$$
Type 2. In view of the boundedness of \(\beta ^-_j\), \(g_{2,k}^n(t)\) and \({\dot{g}}_{2,k}^n(t)\), we derive that
$$\begin{aligned}&\left| \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \left( \int _0^s e^{-\frac{i}{\varepsilon ^2}w} e^{-i \beta ^-_j (s-w)} p_1(w) g_{2,k}^n(w) dw - q_{-1,1}(s) g_{2,k}^n(0) \right) \right| \\&\quad \lesssim \sum _{j,k=1}^d \int _0^s |e^{-\frac{i}{\varepsilon ^2}w} p_1(w)| (|(e^{-i \beta ^-_j (s-w)}-1)g_{2,k}^n(w)|+|g_{2,k}^n(w)-g_{2,k}^n(0)|)dw \\&\quad \lesssim \sum _{j,k=1}^d \int _0^s w^2 \left( \left| i \beta ^-_j (s - w) e^{-i \beta ^-_j (s-\xi _{1}^j(w))}g_{2,k}^n(w)\right| + w|{\dot{g}}_{2,l}^n(\xi _{2,k}(w))|\right) dw \\&\quad \lesssim \int _0^s w^2 s\, dw \lesssim s^3, \end{aligned}$$
where \(0<\xi _{1}^j(w)<w, 0<\xi _{2,k}(w)<w\).
Type 3. We can obtain
$$\begin{aligned}&\bigg |\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \bigg (\int _0^s e^{-\frac{i}{\varepsilon ^2}w} e^{-i \beta ^-_j (s-w)} g_{0,k}^n(w) dw \\&\qquad - e^{-i \beta ^-_j s} (p_{-1}(s) g_{0,k}^n(0) + q_{-1,0}(s) (i \beta ^-_l g_{0,k}^n(0) + {\dot{g}}_{0,k}^n(0)))\bigg )\bigg |\\&\quad \lesssim \sum _{j,k=1}^d |e^{-i \beta ^-_j s}| \int _0^s |e^{-\frac{i}{\varepsilon ^2}w}| \left| e^{i \beta ^-_j w}g_{0,k}^n(w)\right. \\&\left. \qquad - \left( g_{0,k}^n(0)+ w(i \beta ^-_j g_{0,k}^n(0) + {\dot{g}}_{0,k}^n(0))\right) \right| dw \\&\quad \le \sum _{j,k=1}^d \int _0^s w^2 \left| -(\beta ^-_j)^2 e^{i \beta ^-_j \xi _k^j(w)}g_{0,k}^n(\xi _k^j(w)) + i\beta ^-_j e^{i \beta ^-_j \xi _k^j(w)} {\dot{g}}_{0,k}^n(\xi _k^j(w)) \right. \\&\left. \qquad + e^{i \beta ^-_j \xi _k^j(w)}\ddot{g}_{0,k}^n(\xi _k^j(w))\right| dw\\&\quad \lesssim \int _0^s w^2 dw \lesssim s^3, \end{aligned}$$
where \(0<\xi _k^j(w)<w\) and we have used the boundedness of \(\beta ^-_j\), \(g_{2,k}^n(t)\), \({\dot{g}}_{2,k}^n(t)\) and \(\ddot{g}_{2,k}^n(t)\).
Therefore, combing all the three types of estimates, we have \(|Q_{2,l}^n(s)| \lesssim s^3, l =1,2,\ldots ,d\) and
$$\begin{aligned} | y_l(t_n +s) - y_l^{n,2}(s) |&\le |D_{2,l}^n(s)| + |Q_{2,l}^n(s)| +O(s^3)\lesssim s^3, \quad \forall 1\le l\le d, \end{aligned}$$
which leads to the estimates for local truncation error \(\varvec{\xi }^{n}\) follows:
$$\begin{aligned} |\xi ^{n+1}_l|&=|y_l(t_n +\tau ) - S_l(\varvec{y}(t_n),\dot{\varvec{y}}(t_n))|\\&= |y_l(t_n +\tau ) - y_l^{n,2}(\tau )| \lesssim \tau ^3, \quad \forall 1\le l\le d. \end{aligned}$$
For the truncation error \(\dot{\varvec{\xi }}^{n+1}\), we have similar estimates. Given first NPI approximation \(y^{n,1}(s)\), in view of \(|{\dot{\kappa }}_j(t)| = |i \beta _j^+ e^{\beta _j^+ t}- i \beta _j^- e^{\beta _j^- t}| \le |\beta _j^+| + |\beta _j^-| \lesssim \varepsilon ^{-2}\), we get
$$\begin{aligned}&|{\dot{y}}_l(t_n+s) - {\dot{y}}^{n,2}_{l}(s) |\\&\quad \le \sum _{j,k=1}^d |\overline{u_{jl}} u_{jk} \gamma _j|\int _{0}^{s} |{\dot{\kappa }}_j(s-w)| \left| |y^{n,1}_{k}(w)|^2 y^{n,1}_{k}(w)\right. \\&\left. \qquad - |y_k(t_n+w)|^2 y_k(t_n+w) \right| dw+|{\dot{R}}_{2,l}^n(s)|\\&\quad \le C \varepsilon ^{-2} \int _{0}^{s} L \Vert \varvec{y}(t_n+w)-\varvec{y}^{n,1}(s)\Vert _2 dw\\&\quad +|{\dot{R}}_{2,l}^n(s)|\le |{\dot{R}}_{2,l}^n(s)|+C\varepsilon ^{-2} s^3, \quad \forall 1\le l\le d, \end{aligned}$$
where the remainder function \(\dot{\varvec{R}}_{2}^n(s)=({\dot{R}}_{2,1}^n(s),\ldots ,{\dot{R}}_{2,d}^n(s))^T\) can be divided into two terms as \(\dot{\varvec{R}}_{2}^n(s) = \dot{\varvec{D}}_{2}^n(s) + \dot{\varvec{Q}}_{2}^n(s)\). \(\dot{\varvec{Q}}_{2}^n(s)\) includes the quadrature error of the integral approximation in (2.8) and \(\dot{\varvec{D}}_{2}^n(s)\) includes every discarded terms as
$$\begin{aligned} D_{2,l}^n(s)&= \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j \int _0^{s} {\dot{\kappa }}_j(s-w) \\&\quad \left( e^{3iw/\varepsilon ^2} \left( \sum _{m=1}^d f_{+,k,m}^n e^{-i \beta _m^- w}\right) \left( p_{-1}^2(w) |f_k^n|^2 \right) \right) dw \\&\quad +\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j \int _0^{s}{\dot{\kappa }}_j(s-w)\\&\quad \left( -e^{3iw/\varepsilon ^2}\left( \sum _{m=1}^d f_{+,k,m}^n e^{-i \beta _m^- w}\right) \left( p_{-1}(w) w|f_k^n|^2 \right) \right) dw \\&\quad +\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j \int _0^{s} {\dot{\kappa }}_j(s-w)\\&\quad \left( e^{2iw/\varepsilon ^2}\left( \sum _{m=1}^d f_{-,k,m}^n e^{i \beta _m^- w}\right) \left( p_{-1}^2(w) |f_k^n|^2 \right) \right) dw \\&\quad +\cdots , \end{aligned}$$
We estimate the first term, and other terms can be done analogously. By direct computation, we have
$$\begin{aligned}&\left| \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j \int _0^{s} {\dot{\kappa }}_j(s-w) \left( e^{3iw/\varepsilon ^2} (\sum _{m=1}^d f_{+,k,m}^n e^{-i \beta _m^- w}) \left( p_{-1}^2(w) |f_k^n|^2 \right) \right) dw\right| \\&\quad \le C \varepsilon ^{-2} \int _0^{s} \sum _{m=1}^d \left( |y_m(t_n)| + \varepsilon ^2|{\dot{y}}_m(t_n)|\right) \left( \sum _{k=1}^d w^2 |y_k^n|^6 \right) dw \lesssim \varepsilon ^{-2} \int _0^{s} w^2 dw \lesssim \varepsilon ^{-2} s^3. \end{aligned}$$
Thus, \({\dot{D}}_{2,l}^n(s) \lesssim \varepsilon ^{-2} s^3,\, l =1,2,\ldots , d\). For \(\dot{\varvec{Q}}_2^n(s)\), we know
$$\begin{aligned} {\dot{Q}}_{2,l}^n(s)&= \sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \left( i \beta _j^- \int _0^s e^{i \beta ^-_j (s-w)} w g_{1,k}^n(w) dw - i \beta _j^- e^{i \beta ^-_j s/2} \frac{s^2}{2} g_{1,k}^n(\frac{s}{2})\right) \\&\quad +\cdots (\text {error of other type 1 error terms})\\&\quad +\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \left( i \beta _j^+ \int _0^s e^{-\frac{i}{\varepsilon ^2}w} e^{-i \beta ^-_j (s-w)} p_1(w) g_{2,k}^n(w) dw\right. \\&\left. \quad - i \beta _j^+ q_{-1,1}(s) g_{2,k}^n(0) \right) \\&\quad +\cdots (\text {error of other type 2 error terms})\\&\quad +\sum _{j,k=1}^d i\overline{u_{jl}} u_{jk} \gamma _j e^{\frac{i}{\varepsilon ^2}s} \bigg (i \beta _j^+ \int _0^s e^{-\frac{i}{\varepsilon ^2}w} e^{-i \beta ^-_j (s-w)} g_{0,k}^n(w) dw \\&\quad - i \beta _j^+ e^{-i \beta ^-_j s} (p_{-1}(s) g_{0,k}^n(0) + q_{-1,0}(s) (i \beta ^-_l g_{0,k}^n(0) + {\dot{g}}_{0,k}^n(0)))\bigg )\\&\quad +\cdots (\text {error of other type 3 error terms}). \end{aligned}$$
As \(\beta ^+_j =-\beta ^-_j +\varepsilon ^{-2} \lesssim \varepsilon ^{-2}\) and \(\beta _l^- \lesssim 1 \lesssim \varepsilon ^{-2} \), it is easy to prove \({\dot{Q}}_{2,l}^n(s) \lesssim \varepsilon ^{-2} s^3,\, l =1,2,\ldots , d\) by the same method for estimating \(Q_{2,l}^n(s)\).
Now the local truncation error \(\dot{\varvec{\xi }}^{n}\) has the following estimates:
$$\begin{aligned} |{\dot{\xi }}^{n}_l|&=|{\dot{y}}_l(t_n +\tau ) - {\dot{S}}_l(\varvec{y}(t_n),\dot{\varvec{y}}(t_n))|= |{\dot{y}}_l(t_n +\tau ) - {\dot{y}}_l^{n,2}(\tau )| \\&\quad \le |{\dot{D}}_{2,l}^n(\tau )| + |{\dot{Q}}_{2,l}^n(\tau )| +C\tau ^3/\varepsilon ^2 \lesssim \varepsilon ^{-2} \tau ^3, \quad \forall 1\le l\le d. \end{aligned}$$
The proof of Lemma 3.1 is complete.
Appendix C: proof of Lemma 3.2
From (2.5) and (3.3), it is easy to find that
$$\begin{aligned} U\varvec{S}_{L}(\varvec{y},\dot{\varvec{y}})=&\beta ^{-1}(e^{i\beta ^-\tau }\beta ^+-e^{i\beta ^+\tau }\beta ^-) U\varvec{y}+i\beta ^{-1}(e^{i\beta ^-\tau }-e^{i\beta ^+\tau })\Lambda ^{-1} \Lambda U\dot{\varvec{y}}, \end{aligned}$$
(C.1)
$$\begin{aligned} \Lambda U\dot{\varvec{S}}_{L}(\varvec{y},\dot{\varvec{y}})=&\beta ^{-1}\Lambda ^{-1}(-ie^{i\beta ^-\tau }+ie^{i\beta ^+\tau })U\varvec{y}-\beta ^{-1}(e^{i\beta ^-\tau }\beta ^--e^{i\beta ^+\tau }\beta ^+)\Lambda U\dot{\varvec{y}}. \end{aligned}$$
(C.2)
By direct computation, it is easy to verify that Q is unitary. As \(\varvec{S}_L\) and \(\dot{\varvec{S}}_L\) are linear in \(\varvec{y}\) and \(\dot{\varvec{y}}\), we conclude that
$$\begin{aligned} \begin{bmatrix} U\varvec{S}(\varvec{y}^0,\dot{\varvec{y}}^0)-U\varvec{S}(\varvec{y}^1,\dot{\varvec{y}}^1)\\ \Lambda U\dot{\varvec{S}}(\varvec{y}^0,\dot{\varvec{y}}^0)-\Lambda U\varvec{S}(\varvec{y}^1,\dot{\varvec{y}}^1) \end{bmatrix} =Q\begin{bmatrix} U(\varvec{y}^0-\varvec{y}^1)\\ \Lambda U(\dot{\varvec{y}}^0-\dot{\varvec{y}}^1) \end{bmatrix}. \end{aligned}$$
(C.3)
The properties of Q and \(Q^k\) can be verified by direct computation and the fact that \(\beta _l^+=O(1/\varepsilon ^2), \beta _l^-=O(1)\).
For the nonlinear parts, we obtain the component-wise forms of \(U \varvec{S}_{NL}\) and \(\Lambda U \dot{\varvec{S}}_{NL}\) as (\(j=1,\ldots ,d\))
$$\begin{aligned} \begin{aligned} (US_{NL})_j(\varvec{y}, \dot{\varvec{y}})&:= \sum _{k=1}^d i u_{jk} \gamma _j(G_{+,0,k}(\varvec{y},\dot{\varvec{y}};\tau ) + e^{-i \beta ^-_j \tau /2} G_{+,1,k}(\varvec{y},\dot{\varvec{y}};\tau ) \\&\quad + e^{-i \beta ^-_j \tau } G_{+,2,k}(\varvec{y},\dot{\varvec{y}};\tau ) + i \beta ^-_j e^{-i \beta ^-_j \tau } G_{+,3,k}(\varvec{y},\dot{\varvec{y}};\tau ) \\&\quad + G_{-,0,k}(\varvec{y},\dot{\varvec{y}};\tau ) \\&\quad + e^{i \beta ^-_j \tau /2} G_{-,1,k}(\varvec{y},\dot{\varvec{y}};\tau ) + e^{i \beta ^-_j \tau } G_{-,2,k}(\varvec{y},\dot{\varvec{y}};\tau )\\&\quad - i \beta ^-_j e^{i \beta ^-_j \tau } G_{-,3,k}(\varvec{y},\dot{\varvec{y}};\tau )),\\ (\Lambda U \dot{\varvec{S}}_{NL})_j(\varvec{y}, \dot{\varvec{y}})&:= -\sum _{k=1}^d u_{jk} \gamma _j(-\beta _j^-\beta _j^+)^{-1/2}((\beta ^+_j (G_{+,0,k}(\varvec{y},\dot{\varvec{y}};\tau ) \\&\quad + e^{-i \beta ^-_j \tau /2} G_{+,1,k}(\varvec{y},\dot{\varvec{y}};\tau )\\&\quad + e^{-i \beta ^-_j \tau } G_{+,2,k}(\varvec{y},\dot{\varvec{y}};\tau ) + i \beta ^-_j e^{-i \beta ^-_j \tau } G_{+,3,k}(\varvec{y},\dot{\varvec{y}};\tau ))\\&\quad +\beta _j^- (G_{-,0,k}(\varvec{y},\dot{\varvec{y}};\tau ) \\&\quad + e^{i \beta ^-_j \tau /2} G_{-,1,k}(\varvec{y},\dot{\varvec{y}};\tau ) + e^{i \beta ^-_j \tau } G_{-,2,k}(\varvec{y},\dot{\varvec{y}};\tau ) \\&\quad - i \beta ^-_j e^{i \beta ^-_j \tau } G_{-,3,k}(\varvec{y},\dot{\varvec{y}};\tau ))). \end{aligned} \end{aligned}$$
(C.4)
Now, we estimate \((U\varvec{S}_{NL})_l(\varvec{y}^0,\dot{\varvec{y}}^0)-(U\varvec{S}_{NL})_l(\varvec{y}^1,\dot{\varvec{y}}^1)\) (\(l=1,\ldots ,d\)). Since \(|u_{jk}| \le 1\), \(|\gamma _l|=\frac{1}{\varepsilon ^2\beta _l^+} \le 1\), \(|\beta _l^-| \le \rho (A)\) for \(\forall j,k,l=1,2,\ldots ,d\), we only need to consider \(G_{+,j,k}(\varvec{y}^0,\dot{\varvec{y}}^0;\tau )-G_{+,j,k}(\varvec{y}^1,\dot{\varvec{y}}^1;\tau ), j=0,1,2,3\), and the estimates fall into the following three types in view of the definitions of \(G_{\pm ,j,k}\) in (A.3).
Type 1. Terms containing \(g_{k,l}(\cdot ;0)\), for example \(e^{is/\varepsilon ^2}q_{-1,0}(\tau )(g_{1,l}^0(0)- g_{1,l}^{1}(0))\), where \(g_{1,l}^{0}(s)=g_{1,l}(\varvec{y}^0,\dot{\varvec{y}}^0;s), g_{1,l}^{1}(s)=g_{1,l}(\varvec{y}^1,\dot{\varvec{y}}^1;s)\). Then we can estimate the difference by:
$$\begin{aligned}&e^{is/\varepsilon ^2}q_{-1,0}(\tau )(g_{0,l}^0(0)- g_{0,l}^{1}(0)) \\&\quad = -e^{is/\varepsilon ^2}q_{-1,0}(\tau ) \left( \left( \sum _{j=1}^d \overline{f^0_{+,l,j}}\right) \left( \sum _{j=1}^d f^0_{-,l,j}\right) \left( \sum _{j=1}^d f^0_{+,l,j}\right) \right. \\&\qquad - \left( \sum _{j=1}^d \overline{f^{1}_{+,l,j}}\right) \left( \sum _{j=1}^d f^{1}_{-,l,j}\right) \left( \sum _{j=1}^d f^{1}_{+,l,j}\right) \\&\qquad + \left( \sum _{j=1}^d f^0_{+,l,j}\right) \left( \sum _{j=1}^d \overline{f^0_{+,l,j}}\right) \left( \sum _{j=1}^d f^0_{-,l,j}\right) \\&\left. \qquad - \left( \sum _{j=1}^d f^{1}_{+,l,j}\right) \left( \sum _{j=1}^d \overline{f^{1}_{+,l,j}}\right) \left( \sum _{j=1}^d f^{1}_{-,l,j}\right) +\cdots \right) , \end{aligned}$$
where \(f_{\pm ,l,j}^{m}=f_{\pm ,l,j}(\varvec{y}^m,\dot{\varvec{y}}^m)\) (\(m=0,1\)) are given in (2.13), and some terms with similar structure are omitted for brevity. \(\forall j,k,l,m = 1,2,\ldots ,d\), there holds
$$\begin{aligned}&|\overline{f_{+,l,j}^0} f_{-,l,k}^0 f_{+,l,m}^0-\overline{f_{+,l,j}^{1}} f_{-,l,k}^{1} f_{+,l,m}^{1}|\\&\quad \le |\overline{f_{+,l,j}^0} - \overline{f_{+,l,j}^{1}}| |f_{-,l,k}^0 f_{+,l,m}^0| \\&\qquad + |\overline{f_{+,l,j}^{1}}| \big (|f_{-,l,k}^0-f_{-,l,k}^{1}||f_{+,l,m}^0|-|f_{-,l,k}^{1}||f_{+,l,m}^0-f_{+,l,m}^{1}|\big ). \end{aligned}$$
Under the hypothesis of the lemma where \(\Vert \varvec{y}^0\Vert _{\infty } +\varepsilon ^2 \Vert \dot{\varvec{y}}^0\Vert _{\infty } \le M+1\) (\(m=0,1\)), from (2.11), we know for
$$\begin{aligned} |h_{-,j}^{k}(\varvec{y}^0,\dot{\varvec{y}}^0)|&\le \frac{\beta ^+_j}{\beta _j}|y^0_k| + \frac{1}{\beta _j} |{\dot{y}}^0_k| \le (|y^0_k| + \varepsilon ^2 |{\dot{y}}^0_k|) \le M+1. \end{aligned}$$
Analogously \(|h_{\pm ,j}^{k} (\varvec{y}^m,\dot{\varvec{y}}^m))| \le M+1\) (\(m=0,1, j,k=1,\ldots ,d\)), and by the unitary property of U and (2.13), \(|f_{\pm ,l,j}^m| \lesssim 1\) for \(m=1,2,\) and \(l,j =1,2,\ldots ,d\). In addition, we have
$$\begin{aligned} |h_{-,j}^{k}(\varvec{y}^0,\dot{\varvec{y}}^0)-h_{-,j}^{k}(\varvec{y}^1,\dot{\varvec{y}}^1)|&\le \frac{\beta ^+_j}{\beta _j}| y_k^0-y^1_k| + \frac{1}{\beta _j} |{\dot{y}}^0_k-{\dot{y}}_k^1|\nonumber \\&\le \Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2. \end{aligned}$$
(C.5)
Recalling \(|q_{-1,0}(\tau )| \lesssim \tau ^2\), we derive from (C.5) that
$$\begin{aligned}&|e^{is/\varepsilon ^2}q_{-1,0}(\tau )(g_{0,l}^0(0)- g_{0,l}^{1}(0))| \\&\quad \le |q_{-1,0}(\tau )| \left( |\left( \sum _{j=1}^d \overline{f^0_{+,l,j}}\right) \left( \sum _{j=1}^d f^0_{-,l,j}\right) \left( \sum _{j=1}^d f^0_{+,l,j}\right) \nonumber \right. \\&\left. \qquad - \left( \sum _{j=1}^d \overline{f^{1}_{+,l,j}}\right) \left( \sum _{j=1}^d f^{1}_{-,l,j}\right) \left( \sum _{j=1}^d f^{1}_{+,l,j}\right) | +\ldots \right) ,\\&\quad \lesssim \tau ( \Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2). \end{aligned}$$
Type 2. Terms containing \(g_{k,l}(\tau /2)\), for example \(e^{is/\varepsilon ^2}\tau (g_{7,l}^n(\tau /2)- g_{7,l}^{[n]}(\tau /2))\). By definition, we have
$$\begin{aligned}&e^{is/\varepsilon ^2}\tau (g_{7,l}^0(\tau /2)- g_{7,l}^{1}(\tau /2))\\&\quad = e^{is/\varepsilon ^2} \tau \left( \left( \sum _{j=1}^d f^0_{+,l,j} e^{-i \beta _j^- \tau /2}\right) \left( \sum _{j=1}^d \overline{f^0_{+,l,j}} e^{i \beta _j^- \tau /2}\right) \left( \sum _{j=1}^d f^0_{+,l,j} e^{-i \beta _j^- \tau /2}\right) \right. \\&\qquad - \left( \sum _{j=1}^d f^{1}_{+,l,j} e^{-i \beta _j^- \tau /2}\right) \left( \sum _{j=1}^d \overline{f^{1}_{+,l,j}} e^{i \beta _j^- \tau /2}\right) \left( \sum _{j=1}^d f^{1}_{+,l,j} e^{-i \beta _j^- \tau /2}\right) \\&\qquad + \left( \sum _{j=1}^d f^0_{-,l,j} e^{i \beta _j^- \tau /2}\right) \left( \sum _{j=1}^d \overline{f^0_{-,l,j}} e^{-i \beta _j^- \tau /2}\right) \left( \sum _{j=1}^d f^0_{+,l,j} e^{-i \beta _j^- \tau /2}\right) \\&\left. \qquad -\left( \sum _{j=1}^d f^{1}_{-,l,j} e^{i \beta _j^- \tau /2}\right) \left( \sum _{j=1}^d \overline{f^{1}_{-,l,j}} e^{-i \beta _j^- \tau /2}\right) \left( \sum _{j=1}^d f^{1}_{+,l,j} e^{-i \beta _j^- \tau /2}\right) +\cdots \right) . \end{aligned}$$
Then \(\forall j,k,l,m = 1,2,\ldots ,d\), there holds
$$\begin{aligned}&|f_{+,l,j}^0 e^{-i \beta _j^- \tau /2} \overline{f_{+,l,k}^0} e^{i \beta _k^- \tau /2} f_{+,l,m}^0 e^{-i \beta _m^- \tau /2}\\&\qquad -f_{+,l,j}^{1} e^{-i \beta _j^- \tau /2} \overline{f_{+,l,k}^{1}} e^{i \beta _k^- \tau /2} f_{+,l,m}^{1} e^{-i \beta _m^- \tau /2}| \\&\quad \le |f_{+,l,j}^0 \overline{f_{+,l,k}^0} f_{+,l,m}^0-f_{+,l,j}^{1} \overline{f_{+,l,k}^{1}} f_{+,l,m}^{1}| \lesssim \Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2. \end{aligned}$$
where the last inequality can be derived by the same arguments for type 1 terms. Thus, we can get
$$\begin{aligned} |e^{is/\varepsilon ^2}\tau (g_{7,l}^n(\tau /2)- g_{7,l}^{[n]}(\tau /2))| \lesssim \tau (\Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2). \end{aligned}$$
Type 3. Terms containing \({\dot{g}}_{k,l}(0)\), for example \(e^{is/\varepsilon ^2}q_{-1,0}(\tau )({\dot{g}}_{0,l}^0(0)- {\dot{g}}_{0,l}^{1}(0))\). For \(l=1,\ldots ,d\), we know
$$\begin{aligned}&e^{is/\varepsilon ^2}q_{-1,0}(\tau )({\dot{g}}_{0,l}^0(0)- {\dot{g}}_{0,l}^{1}(0))\\&\quad = e^{is/\varepsilon ^2}q_{-1,0}(\tau ) \left( \left( \sum _{j=1}^d -i\beta _j^- f^0_{+,l,j} \right) \left( \sum _{j=1}^d \overline{f^0_{+,l,j}} \right) \left( \sum _{j=1}^d f^0_{+,l,j} \right) \right. \\&\qquad - \left( \sum _{j=1}^d -i\beta _j^- f^{1}_{+,l,j} \right) \left( \sum _{j=1}^d \overline{f^{1}_{+,l,j}} \right) \left( \sum _{j=1}^d f^{1}_{+,l,j}\right) \\&\qquad + \left( \sum _{j=1}^d i\beta _j^- f^0_{-,l,j} \right) \left( \sum _{j=1}^d \overline{f^0_{-,l,j}} \right) \left( \sum _{j=1}^d f^0_{+,l,j} \right) \\&\left. \qquad - \left( \sum _{j=1}^d i\beta _j^- f^{1}_{-,l,j} \right) \left( \sum _{j=1}^d \overline{f^{1}_{-,l,j}} \right) \left( \sum _{j=1}^d f^{1}_{+,l,j}\right) +\cdots \right) . \end{aligned}$$
Noticing the boundedness of \(\beta _j^-\) (independent of \(\varepsilon \)), by the same arguments for type 1 terms, we have for \( j,k,l,m = 1,2,\ldots ,d\),
$$\begin{aligned}&|-i \beta _j^- f_{+,l,j}^0 \overline{f_{+,l,k}^0} f_{+,l,m}^0 + i \beta _j^- f_{+,l,j}^{1} \overline{f_{+,l,k}^{1}} f_{+,l,m}^{1} | \le \Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2. \end{aligned}$$
Therefore, in view of the fact \(|q_{-1,0}(\tau )|\lesssim \tau ^2\), we obtain
$$\begin{aligned} |e^{is/\varepsilon ^2}&q_{-1,0}(\tau )({\dot{g}}_{0,l}^0(0)- {\dot{g}}_{0,l}^{1}(0))|&\lesssim \tau ^2 (\Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2)\\&\lesssim \tau (\Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2). \end{aligned}$$
Combining all the three cases above, we can prove the estimates for the nonlinear part \(U\varvec{S}_{NL}\) as
$$\begin{aligned}&\Vert U\varvec{S}_{NL}(\varvec{y}^0,\dot{\varvec{y}}^0)- U\varvec{S}_{NL}(\varvec{y}^1,\dot{\varvec{y}}^1)\Vert _2 \\&\quad \lesssim \tau (\Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2). \end{aligned}$$
For the nonlinear part of \(\Lambda U\dot{\varvec{S}}\) in (C.4), noticing all terms appearing in \(\Lambda U\dot{\varvec{S}}\) differ from corresponding terms in \(U\varvec{S}\) by a factor \(i\frac{\beta _j^+}{\sqrt{-\beta _l^+ \beta _l^-}}\) or \(i\frac{\beta _j^-}{\sqrt{-\beta _l^+ \beta _l^-}}\), \(l,j=1,2,\ldots ,d\), recalling the coefficients \(\beta _j^\pm \) (2.3),
$$\begin{aligned} |\beta _j^+|/ \sqrt{-\beta _l^+ \beta _l^-} \lesssim \frac{1}{\varepsilon },\quad |\beta _j^-|/ \sqrt{-\beta _l^+ \beta _l^-} \lesssim 1,\quad j=1,\ldots ,d, \end{aligned}$$
we have
$$\begin{aligned} \Vert U\dot{\varvec{S}}_{NL}(\varvec{y}^0,\dot{\varvec{y}}^0)- U\dot{\varvec{S}}_{NL}(\varvec{y}^1,\dot{\varvec{y}}^1)\Vert _2 \lesssim \frac{\tau }{\varepsilon } \left( \Vert \varvec{y}^0-\varvec{y}^1\Vert _2+\varepsilon ^2\Vert \dot{\varvec{y}}^0-\dot{\varvec{y}}^1\Vert _2\right) . \end{aligned}$$
Setting \(\varvec{\eta }= U\varvec{S}_{NL}(\varvec{y}^0,\dot{\varvec{y}}^0)- U\varvec{S}_{NL}(\varvec{y}^1,\dot{\varvec{y}}^1) \) and \( \dot{\varvec{\eta }}=U\dot{\varvec{S}}_{NL}(\varvec{y}^0,\dot{\varvec{y}}^0)- U\dot{\varvec{S}}_{NL}(\varvec{y}^1,\dot{\varvec{y}}^1)\), combining (C.3), we draw the conclusions in Lemma 3.2.
