Optimal Error Estimate of the Extended-WKB Approximation to the High Frequency Wave-Type Equation in the Semi-classical Regime

Abstract

For the Cauchy problem of the high frequency wave-type equation with Wentzel–Kramers–Brillouin (WKB) type initial data, the extended Wentzel–Kramers–Brillouin (E-WKB) ansatz is an asymptotically valid solution. This ansatz is globally defined and formulated as an integral of coherent states over the displaced Lagrangian submanifold. This paper proves the optimal first order error estimate of the proposed E-WKB ansatz in $L^2$ norm for the wave-type equation in the semi-classical regime. The key ingredients in the proof are the moving frame technique developed in Zheng (Commun Math Sci 11:105–140, 2013) and the deep relations between the E-WKB analysis and the classical WKB analysis. Numerical results on the linear KdV equation verify the theoretical analysis.

A Proof of (13) and (14)

For (14), the action of the Weyl quantized operator H(W) has been analyzed in Theorem 4.2 of [35]. Moreover, we recall another useful formula in [35],

$$\begin{aligned} \int _{z\in \varLambda _t}(x-q){\mathcal {A}}_z\phi _{z,{\mathcal {S}}_z}^\epsilon d\nu _{z,t} =(-\, \mathrm {i}\epsilon ) \int _{z\in \varLambda _t} \mathrm {div}_{\varLambda _t} \left[ {\mathcal {A}}_z \varPi _{ \varLambda _t} \begin{pmatrix} - \mathrm {i}I\\ -I \end{pmatrix}\right] \phi _{z,{\mathcal {S}}_z}^\epsilon d\nu _{z,t}. \end{aligned}$$

(55)

Next, we prove Eq. (13). Since $\phi _{z, {\mathcal {S}}_z}^ \varepsilon$ is a smooth function defined on ${{\bar{\varLambda }}}=\cup _t \varLambda _t$,

$$\begin{aligned} \phi _{z, {\mathcal {S}}_z}^ \varepsilon = (2 \pi \epsilon )^{\frac{N}{2}} \exp (\mathrm {i} {\mathcal {S}}_z/ \epsilon ) \exp (\mathrm {i} p^\dagger (x- q/2)/ \epsilon ) \exp (-\,(x-q)^2/2 \epsilon ), \end{aligned}$$

the following result is obvious,

$$\begin{aligned} d \phi _{z, {\mathcal {S}}_z}^ \varepsilon = \left[ \frac{ \mathrm {i}}{ \epsilon } (d {\mathcal {S}}_z + d(p^\dagger (x- q/2))) - d (x-q)^2/ 2\epsilon \right] \phi _{z, {\mathcal {S}}_z}^ \varepsilon . \end{aligned}$$

The term in the square bracket is a differential 1-form on ${\bar{\varLambda }}$ and can be contracted with the vector $\dfrac{\partial }{\partial t}$. Thus, we derive

$$\begin{aligned} \begin{aligned} \partial _t \phi _{z, {\mathcal {S}}_z}^ \varepsilon =&\frac{ \mathrm {i}}{ \epsilon } \left[ \dot{{\mathcal {S}}_z} + \frac{\partial }{\partial t}(p^\dagger (x- q/2))) + \mathrm {i} \frac{\partial }{\partial t} (x-q)^2/ 2 \right] \phi _{z, {\mathcal {S}}_z}^ \varepsilon \\ =&\frac{ \mathrm {i}}{ \epsilon } \left[ \dot{{\mathcal {S}}_z} +[z,\dot{z}]/2 + (x-q)^\dagger (\dot{p}- \mathrm {i} \dot{q}) \right] \phi _{z, {\mathcal {S}}_z}^ \varepsilon . \end{aligned} \end{aligned}$$

(56)

By utilizing (12), Eq. (56) can be further simplified as

$$\begin{aligned} \partial _t \phi _{z, {\mathcal {S}}_z}^ \varepsilon = \frac{ \mathrm {i}}{ \epsilon } \left[ - H(z) + (x-q)^\dagger (\dot{p} - \mathrm {i} \dot{q})\right] \phi _{z, {\mathcal {S}}_z}^ \varepsilon . \end{aligned}$$

(57)

In order to compute Eq. (13), one first rewrites it into an equivalent form

$$\begin{aligned} - \mathrm {i}\epsilon \partial _t\int _{z\in \varLambda _t}{\mathcal {A}}_{z}\phi _{z, {\mathcal {S}}_{z}}^\epsilon d\nu _{z,t} = -\, \mathrm {i}\epsilon \int _{\varLambda _t} {\mathcal {L}}_t\left[ {\mathcal {A}}_{z}\phi _{z, {\mathcal {S}}_{z}}^\epsilon d\nu _{z,t} \right] , \end{aligned}$$

(58)

where ${\mathcal {L}}_t$ denotes the Lie derivative corresponding to the vector field $\dfrac{\partial }{\partial t}$. Actually, by invoking Eq. (57), we have

$$\begin{aligned} {\mathcal {L}}_t\left[ {\mathcal {A}}_{z} \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t} \right]&= ~\dot{ {\mathcal {A}}_z} \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t} + \frac{ \mathrm {i}}{ \epsilon } \left[ - H(z) + (x-q)^\dagger (\dot{p} - \mathrm {i} \dot{q})\right] {\mathcal {A}}_z \phi _{z, {\mathcal {S}}_z}^ \varepsilon d \nu _{z,t}\\&\quad + \mathrm{tr}[ \varPi _{ \varLambda _t} J \nabla ^2 H] {\mathcal {A}}_z\phi _{z, {\mathcal {S}}_z}^ \varepsilon d \nu _{z,t}. \end{aligned}$$

Thus, by employing (55), Eq. (58) becomes

$$\begin{aligned} - \mathrm {i}\epsilon \partial _t\int _{z\in \varLambda _t}{\mathcal {A}}_{z}\phi _{z, {\mathcal {S}}_{z}}^\epsilon d\nu _{z,t}&= \int _{z\in \varLambda _t}\left\{ -H(z){\mathcal {A}}_{z} +(-\, \mathrm {i}\epsilon )\left[ \dot{{\mathcal {A}}}_z +\mathrm {tr}[\varPi _{ \varLambda _t}J\nabla ^2 H(z)]{\mathcal {A}}_z\right] \right\} \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t}\\&\quad +\int _{z\in \varLambda _t}(x-q)^\dagger ({\dot{p}}- \mathrm {i}{\dot{q}}){\mathcal {A}}_z\phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t}\\&= \int _{z\in \varLambda _t}\left\{ -H(z){\mathcal {A}}_{z}+(-\, \mathrm {i}\epsilon )\left[ \dot{{\mathcal {A}}}_z +\mathrm {tr}[\varPi _{\varLambda _t}J\nabla ^2 H(z)]{\mathcal {A}}_z\right] \right\} \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t}\\&\quad +(-\, \mathrm {i} \epsilon )\int _{z\in \varLambda _t} \mathrm{div}_{ \varLambda _t}\left[ \varPi _{ \varLambda _t} \begin{pmatrix} - \mathrm {i}I\\ -I\end{pmatrix} ({\dot{p}}- \mathrm {i}{\dot{q}}){\mathcal {A}}_z \right] \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t}\\&= \int _{z\in \varLambda _t}\left\{ -H(z){\mathcal {A}}_{z}+(-\, \mathrm {i}\epsilon )\left[ \dot{{\mathcal {A}}}_z +\mathrm {tr}[\varPi _{\varLambda _t}J\nabla ^2 H(z)]{\mathcal {A}}_z\right] \right\} \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t}\\&\quad +(-\, \mathrm {i} \epsilon )\int _{z\in \varLambda _t} \mathrm{div}_{ \varLambda _t}\left[ \varPi _{ \varLambda _t} \begin{pmatrix} - \mathrm {i}I\\ -I\end{pmatrix} \begin{pmatrix} - I&\quad - \mathrm {i}I \end{pmatrix} \begin{pmatrix} -\dot{p}\\ \dot{q}\end{pmatrix}{\mathcal {A}}_z \right] \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t}\\&=\int _{z\in \varLambda _t}\left\{ - H(z){\mathcal {A}}_{z} +(-\, \mathrm {i}\epsilon )\left[ \dot{{\mathcal {A}}}_z +\mathrm {tr}[\varPi _{\varLambda _t}J\nabla ^2 H(z)] {\mathcal {A}}_z\right] \right\} \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t}\\&\quad +(-\, \mathrm {i}\epsilon )\int _{z\in \varLambda _t}\mathrm {div}_{\varLambda _t}\left[ \varPi _{\varLambda _t}(\mathrm {i}I-J)\nabla H(z){\mathcal {A}}_z\right] \phi _{z, {\mathcal {S}}_z}^ \varepsilon d\nu _{z,t}. \end{aligned}$$

Thus, Eq. (13) is proved.