这是一份 Imperial帝国理工大学 PHYS96001作业代写的成功案例
where $u=-i \theta, \lambda$ is the parameter in the bulk $S$ matrix while $\eta$ and $\vartheta$ are two real parameters that characterize the solution.
The spectrum of excited boundary states was determined. It can be parametrized by a sequence of integers $\left|n_{1}, n_{2}, \ldots, n_{k}\right\rangle$, whenever the
$$
\frac{\pi}{2} \geq v_{n_{1}}>w_{n_{2}}>\cdots \geq 0
$$
condition holds, where
$$
v_{n}=\frac{\eta}{\lambda}-\frac{\pi(2 n+1)}{2 \lambda} \text { and } w_{k}=\pi-\frac{\eta}{\lambda}-\frac{\pi(2 k-1)}{2 \lambda} .
$$
The mass of such a state is
$$
m_{\left.\mid n_{1}, n_{2}, \ldots, n_{k}\right)}=M \sum_{i \text { odd }} \cos \left(v_{n_{i}}\right)+M \sum_{i \text { even }} \cos \left(w_{n_{l}}\right) .
$$
PHYS96001 COURSE NOTES ：
In the following, it will be more convenient to work with a modified Bloch function $\chi=\tilde{\chi} \exp \left[-i k x \sigma_{3}-i\left(2 k^{2}+\omega\right) t \sigma_{3}\right]$ which satisfies the equations
$$
\chi_{x}=U \chi-i k \chi \sigma_{3}, \quad \chi_{t}=V \chi-i\left(2 k^{2}+\omega\right) \chi \sigma_{3}
$$
and admits the asymptotic expansion started with the unit matrix, $\chi=I+$ $k^{-1} \chi^{(1)}+\mathcal{O}\left(k^{-2}\right)$, while the potential $Q$ is reconstructed via
$$
Q=-\left[\sigma_{3}, \chi^{(1)}\right]
$$
Suppose now that a solution of NLS homoclinic to the plane wave (2) can be obtained from Eq. (4) with the Bloch function $\chi$ being a result of dressing the Bloch function $\chi_{0}$ which satisfies Eq. (3) with $u=u_{0}$ :
$$
x=D \chi_{0}
$$
Here $D(k, x, t)$ is the dressing factor,
$$
D=I-\frac{k_{1}-\bar{k}{1}}{k-\bar{k}{1}} P, \quad D^{-1}=I+\frac{k_{1}-\bar{k}{1}}{k-k{1}} P,
$$