# 散焦HIROTA方程在非零背景下的无穷守恒律Conservation Laws of the Defocusing Hirota Equation under Non-Zero Background

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The conservation laws are important indexes for the integrability of nonlinear equations. It plays an important role in the research of partial differential equation systems. Based on Lax pair, this paper studies infinite conservation laws and higher-order conserved quantities of the one-dimensional defocusing Hirota equation under non-zero backgrounds.

1. 问题背景及主要结论

$\begin{array}{l}\text{i}\left({\psi }_{t}+{\mu }_{1}\psi +\left({\nu }_{1}+{\mu }_{1}x\right){\psi }_{x}\right)+\left({\nu }_{2}+{\mu }_{2}x\right)\left({\psi }_{xx}+2{|\psi |}^{2}\psi \right)\\ \text{ }+2{\mu }_{2}\left({\psi }_{x}+\psi {\int }_{-\infty }^{x}{|\psi |}^{2}\text{d}x\right)+\text{i}\nu \left({\psi }_{xxx}+6{|\psi |}^{2}{\psi }_{x}\right)=0\end{array}$(1.1)

$\text{i}{\psi }_{t}+\left({\nu }_{2}+{\mu }_{2}x\right)\left({\psi }_{xx}+2{|\psi |}^{2}\psi \right)+2{\mu }_{2}\left({\psi }_{x}+\psi {\int }_{-\infty }^{x}{|\psi |}^{2}\text{d}x\right)=0$(1.2)

$\text{i}{\psi }_{x}+{\psi }_{xx}+2{|\psi |}^{2}\psi +\text{i}ϵ\left({\psi }_{xxx}+6{|\psi |}^{2}{\psi }_{x}\right)=0$(1.3)

$\text{i}{\psi }_{x}+\frac{1}{2}\left({\psi }_{xx}+2{|\psi |}^{2}\psi \right)+\frac{\sqrt{2}}{4}\text{i}\left({\psi }_{xxx}+6{|\psi |}^{2}{\psi }_{x}\right)=0$(1.4)

$\text{i}{\psi }_{x}+{\psi }_{xx}+{|\psi |}^{2}\psi +\text{i}\left({\psi }_{xxx}+3{|\psi |}^{2}{\psi }_{x}\right)=0$(1.5)

$\text{i}{\psi }_{x}+{\psi }_{xx}-{|\psi |}^{2}\psi +\text{i}\left({\psi }_{xxx}-3{|\psi |}^{2}{\psi }_{x}\right)=0$(1.6)

$F\left(x,t,u\right)=0$(1.7)

${\partial }_{t}\omega \left(x,t,u\right)={\partial }_{x}J\left(x,t,u\right)$(1.8)

$\frac{\text{d}}{\text{d}t}{\int }_{-\infty }^{\infty }\omega \left(x,t,u\right)\text{d}x=0$(1.9)

$\begin{array}{r}\hfill M=\left(\begin{array}{cc}{m}_{1}& {m}_{2}\\ {m}_{3}& {m}_{4}\end{array}\right)\end{array}$

$\begin{array}{r}\hfill {M}^{off}=\left(\begin{array}{cc}0& {m}_{2}\\ {m}_{3}& 0\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}^{diag}=\left(\begin{array}{cc}{m}_{1}& 0\\ 0& {m}_{4}\end{array}\right)\end{array}$

$S\left(\psi \right)={\int }_{ℝ}\left[{|{\psi }_{xx}|}^{2}+3{|\psi |}^{2}{|{\psi }_{x}|}^{2}+\frac{1}{2}{\left(\stackrel{¯}{\psi }{\psi }_{x}+\psi {\stackrel{¯}{\psi }}_{x}\right)}^{2}+{\left(1-{|\psi |}^{2}\right)}^{2}\left(1+\frac{1}{2}{|\psi |}^{2}\right)\right]\text{d}x\text{ }\text{ }$(1.10)

2. 定理的证明

$\text{i}{\psi }_{x}+\frac{1}{2}\left({\psi }_{xx}-2{|\psi |}^{2}\psi \right)+\frac{\sqrt{2}}{4}\text{i}\left({\psi }_{xxx}-6{|\psi |}^{2}{\psi }_{x}\right)=0$(2.1)

${\Phi }_{x}=\left(\text{i}\lambda {\sigma }_{3}+Q\right)\Phi$(2.2)

$\begin{array}{c}{\Phi }_{t}=\left[\sqrt{2}\left[\text{i}{\sigma }_{3}{\lambda }^{3}+Q{\lambda }^{2}+\left(-\frac{\text{i}}{2}{\sigma }_{3}{Q}_{x}+\frac{\text{i}}{2}{\sigma }_{3}{Q}^{2}\right)\lambda \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(-\frac{1}{4}{Q}_{xx}+\frac{1}{2}{Q}^{3}+\frac{1}{4}\left({Q}_{x}Q-Q{Q}_{x}\right)\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left[\text{i}{\sigma }_{3}{\lambda }^{2}+Q\lambda +\left(-\frac{\text{i}}{2}{\sigma }_{3}{Q}_{x}+\frac{\text{i}}{2}{\sigma }_{3}{Q}^{2}\right)\right]\right]\Phi \end{array}$(2.3)

$Q=\left(\begin{array}{cc}0& \psi \\ \stackrel{¯}{\psi }& 0\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{3}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$

$\begin{array}{r}\hfill \left\{\begin{array}{l}{\Phi }_{x}=\left(\text{i}\lambda {\sigma }_{3}+Q\right)\Phi \hfill \\ {\Phi }_{x}=\left(\underset{k=0}{\overset{3}{\sum }}{v}_{k}{\lambda }^{k}\right)\Phi \hfill \end{array}\end{array}$

$\Phi \left(\lambda ;x,t\right)$ 为复平面上关于 $\lambda$ 的亚纯函数。因为 ${\Phi }_{xt}={\Phi }_{tx}$，则有

$\begin{array}{r}\hfill \left\{\begin{array}{l}{\Phi }_{xt}=\left[{\left(\text{i}\lambda {\sigma }_{3}+Q\right)}_{t}+\left(\text{i}\lambda {\sigma }_{3}+Q\right)\left(\underset{k=0}{\overset{3}{\sum }}{v}_{k}{\lambda }^{k}\right)\right]\Phi \hfill \\ {\Phi }_{tx}=\left[{\left(\underset{k=0}{\overset{3}{\sum }}{v}_{k}{\lambda }^{k}\right)}_{x}+\left(\underset{k=0}{\overset{3}{\sum }}{v}_{k}{\lambda }^{k}\right)\left(\text{i}\lambda {\sigma }_{3}+Q\right)\right]\Phi \hfill \end{array}\end{array}$

${\left(\text{i}\lambda {\sigma }_{3}+Q\right)}_{t}-{\left(\underset{k=0}{\overset{3}{\sum }}{v}_{k}{\lambda }^{k}\right)}_{x}+\left[\text{i}\lambda {\sigma }_{3}+Q,\underset{k=0}{\overset{3}{\sum }}{v}_{k}{\lambda }^{k}\right]=0$

${\lambda }^{4}:\left[\text{i}{\sigma }_{3},{v}_{3}\right]=0$ (2.4)

${\lambda }^{3}:-{v}_{3,x}+\left[\text{i}{\sigma }_{3},{v}_{2}\right]+\left[Q,{v}_{3}\right]=0$ (2.5)

${\lambda }^{2}:-{v}_{2,x}+\left[\text{i}{\sigma }_{3},{v}_{1}\right]+\left[Q,{v}_{2}\right]=0$ (2.6)

${\lambda }^{1}:-{v}_{1,x}+\left[\text{i}{\sigma }_{3},{v}_{0}\right]+\left[Q,{v}_{1}\right]=0$ (2.7)

${\lambda }^{0}:{Q}_{t}-{v}_{0,x}+\left[Q,{v}_{0}\right]=0$(2.8)

$\begin{array}{c}2\text{i}{\sigma }_{3}{v}_{0}^{off}={v}_{1,x}^{off}+\left[{v}_{1}^{diag},Q\right]\\ =-\frac{\text{i}}{2}{\sigma }_{3}\alpha {Q}_{xx}+\beta {Q}_{x}+\frac{\text{i}}{2}\left({\sigma }_{3}\alpha {Q}^{3}-Q{\sigma }_{3}\alpha {Q}^{2}\right)\\ =-\frac{\text{i}}{2}{\sigma }_{3}\alpha {Q}_{xx}+\beta {Q}_{x}+\text{i}{\sigma }_{3}\alpha {Q}^{3}\end{array}$(2.9)

${v}_{0}^{off}=-\frac{1}{4}\alpha {Q}_{xx}-\frac{\text{i}}{2}{\sigma }_{3}\beta {Q}_{x}+\frac{1}{2}{Q}^{3}\alpha$

$\begin{array}{c}{v}_{0,x}^{diag}=\frac{1}{4}\alpha {Q}_{xx}Q-Q\frac{1}{4}\alpha {Q}_{xx}+\frac{\text{i}}{2}{\sigma }_{3}\beta {Q}_{x}Q-\frac{\text{i}}{2}Q{\sigma }_{3}\beta {Q}_{x}\\ =\frac{1}{4}\alpha \left({Q}_{xx}Q-Q{Q}_{xx}\right)+\frac{\text{i}}{2}\beta \left({\sigma }_{3}\beta {Q}_{x}Q+{\sigma }_{3}Q{Q}_{x}\right)\end{array}$(2.10)

${v}_{0}^{diag}=\frac{1}{4}\alpha \left({Q}_{x}Q-Q{Q}_{x}\right)+\frac{\text{i}}{2}{\sigma }_{3}\beta {Q}^{2}$

$\begin{array}{l}{Q}_{t}-{v}_{0,x}^{off}+\left[Q,{v}_{0}^{diag}\right]=0，\\ -{v}_{0,x}^{off}=\frac{1}{4}\alpha {Q}_{xxx}+\frac{\text{i}}{2}{\sigma }_{3}\beta {Q}_{xx}-\frac{1}{2}\alpha \left({Q}_{x}{Q}^{2}+Q{Q}_{x}Q+{Q}^{2}{Q}_{x}\right)，\\ \left[Q,{v}_{0}^{diag}\right]=\frac{\alpha }{4}\left(2Q{Q}_{x}Q-{Q}^{2}{Q}_{x}-{Q}_{x}{Q}^{2}\right)-\text{i}{\sigma }_{3}\beta {Q}_{xx}\end{array}$

${Q}_{t}+\frac{\text{i}}{2}\beta \left({\sigma }_{3}{Q}_{xx}-2{\sigma }_{3}{Q}^{3}\right)+\frac{\alpha }{4}\left({Q}_{xxx}-3\left({Q}_{x}{Q}^{2}+{Q}^{2}{Q}_{x}\right)\right)=0$(2.11)

$Q=\left(\begin{array}{cc}0& \psi \\ \stackrel{¯}{\psi }& 0\end{array}\right)$ 代入式(2.11)，得到

${\psi }_{t}+\frac{\text{i}}{2}\beta \left({\psi }_{xx}-2{|\psi |}^{2}\psi \right)+\frac{\alpha }{4}\left({\psi }_{xxx}-6{|\psi |}^{2}{\psi }_{x}\right)=0$(2.12)

$\beta =-1$$\alpha =\sqrt{2}$，则可以得到：

$\text{i}{\psi }_{t}+\frac{1}{2}\left({\psi }_{xx}-2{|\psi |}^{2}\psi \right)+\text{i}\frac{\sqrt{2}}{4}\left({\psi }_{xxx}-6{|\psi |}^{2}{\psi }_{x}\right)=0$(2.13)

${\Phi }_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}\text{i}\lambda & \psi \\ \stackrel{¯}{\psi }& -\text{i}\lambda \end{array}\right)\Phi$(2.14)

${\Phi }_{t}=\left(\begin{array}{cc}{p}_{1}& {p}_{2}\\ {p}_{3}& -{p}_{1}\end{array}\right)\Phi$(2.15)

${p}_{1}=-\text{i}{\lambda }^{2}-\frac{\text{i}}{2}{|\psi |}^{2}+\sqrt{2}\text{i}{\lambda }^{3}+\frac{\sqrt{2}\text{i}\lambda }{2}{|\psi |}^{2}+\frac{1}{2}\left({\psi }_{x}\stackrel{¯}{\psi }-\psi {\stackrel{¯}{\psi }}_{x}\right)$

${p}_{2}=-\lambda \psi +\frac{\sqrt{2}}{2}\text{i}{\psi }_{x}+\sqrt{2}{\lambda }^{2}\psi -\text{i}\lambda {\psi }_{x}+\frac{\sqrt{2}}{2}\left({|\psi |}^{2}\psi -{\psi }_{xx}\right)$

${p}_{3}=-\lambda \stackrel{¯}{\psi }-\frac{\sqrt{2}}{2}\text{i}{\stackrel{¯}{\psi }}_{x}+\sqrt{2}{\lambda }^{2}\stackrel{¯}{\psi }+\text{i}\lambda {\stackrel{¯}{\psi }}_{x}+\frac{\sqrt{2}}{2}\left({|\psi |}^{2}\stackrel{¯}{\psi }-{\stackrel{¯}{\psi }}_{xx}\right)$

$\begin{array}{r}\hfill {\Phi }_{0}={D}^{-1}C{\text{e}}^{\left[\frac{1}{\sqrt{2}}\text{i}\xi x+\left[\sqrt{2}\left({\lambda }^{2}+\frac{1}{2}\right)-\lambda \right]\text{i}\xi t\right]{\sigma }_{3}}\end{array}$

$\begin{array}{r}\hfill {\Phi }_{0,x}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}\text{i}\lambda & {\text{e}}^{-\text{i}t}\\ {\text{e}}^{\text{i}t}& -\text{i}\lambda \end{array}\right){\Phi }_{0},\end{array}$

${\left({\stackrel{˜}{\Phi }}_{0}\right)}_{x}=\frac{1}{\sqrt{2}}D\left(\begin{array}{cc}\text{i}\lambda & {\text{e}}^{-\text{i}t}\\ {\text{e}}^{\text{i}t}& -\text{i}\lambda \end{array}\right){D}^{-1}{\stackrel{˜}{\Phi }}_{0}=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}\text{i}\lambda & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\text{i}\lambda \end{array}\right){\stackrel{˜}{\Phi }}_{0}$

$\left(\begin{array}{cc}\frac{1}{\sqrt{2}}\text{i}\left(\lambda -\xi \right)& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\text{i}\left(\lambda +\xi \right)\end{array}\right)=C\left(\begin{array}{cc}\frac{1}{\sqrt{2}}\text{i}\xi & 0\\ 0& -\frac{1}{\sqrt{2}}\text{i}\xi \end{array}\right){C}^{-1}$

${\left({\stackrel{^}{\Phi }}_{0}\right)}_{x}={\left({C}^{-1}{\stackrel{˜}{\Phi }}_{0}\right)}_{x}=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}\text{i}\xi & 0\\ 0& -\frac{1}{\sqrt{2}}\text{i}\xi \end{array}\right){\stackrel{^}{\Phi }}_{0}$(2.16)

${\left({\stackrel{^}{\Phi }}_{0}\right)}_{t}=\left[\sqrt{2}\left({\lambda }^{2}+\frac{1}{2}\right)-\lambda \right]\left(\begin{array}{cc}\text{i}\xi & 0\\ 0& -\text{i}\xi \end{array}\right){\stackrel{^}{\Phi }}_{0}$(2.17)

${\stackrel{^}{\Phi }}_{0}={C}^{-1}{\stackrel{˜}{\Phi }}_{0}={\text{e}}^{\left[\frac{1}{\sqrt{2}}\text{i}\xi x+\left[\sqrt{2}\left({\lambda }^{2}+\frac{1}{2}\right)-\lambda \right]\text{i}\xi t\right]{\sigma }_{3}}$

$\begin{array}{l}{\stackrel{˜}{\Phi }}_{0}=C{\text{e}}^{\left[\frac{1}{\sqrt{2}}\text{i}\xi x+\left[\sqrt{2}\left({\lambda }^{2}+\frac{1}{2}\right)-\lambda \right]\text{i}\xi t\right]{\sigma }_{3}}\\ {\Phi }_{0}={D}^{-1}C{\text{e}}^{\left[\frac{1}{\sqrt{2}}\text{i}\xi x+\left[\sqrt{2}\left({\lambda }^{2}+\frac{1}{2}\right)-\lambda \right]\text{i}\xi t\right]{\sigma }_{3}}\end{array}$

${\left(\begin{array}{c}{\phi }_{1}\\ {\phi }_{2}\end{array}\right)}_{x}=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}\text{i}\lambda & \frac{1}{\sqrt{2}}\psi \\ \frac{1}{\sqrt{2}}\stackrel{¯}{\psi }& -\frac{1}{\sqrt{2}}\text{i}\lambda \end{array}\right)\left(\begin{array}{c}{\phi }_{1}\\ {\phi }_{2}\end{array}\right)$(2.18)

$\begin{array}{r}\hfill \begin{array}{r}\hfill \frac{{\phi }_{1,x}}{{\phi }_{1}}={\left(\mathrm{ln}{\phi }_{1}\right)}_{x}=\frac{1}{\sqrt{2}}\text{i}\lambda +\frac{1}{\sqrt{2}}\frac{{\phi }_{2}}{{\phi }_{1}}.\end{array}\end{array}$

$\begin{array}{c}{\left(\frac{{\phi }_{2}}{{\phi }_{1}}\right)}_{x}=\frac{{\phi }_{2,x}}{{\phi }_{1}}-\frac{{\phi }_{2}{\phi }_{1,x}}{{\phi }_{1}^{2}}\\ =\frac{\frac{1}{\sqrt{2}}\stackrel{¯}{\psi }{\phi }_{1}-\frac{1}{\sqrt{2}}\text{i}\lambda {\phi }_{2}}{{\phi }_{1}}-\frac{\frac{1}{\sqrt{2}}\text{i}\lambda {\phi }_{1}{\phi }_{2}+\frac{1}{\sqrt{2}}\psi {\phi }_{2}^{2}}{{\phi }_{1}^{2}}\\ =\frac{1}{\sqrt{2}}\stackrel{¯}{\psi }-\sqrt{2}\text{i}\lambda \frac{{\phi }_{2}}{{\phi }_{1}}-\frac{1}{\sqrt{2}}\psi {\left(\frac{{\phi }_{2}}{{\phi }_{1}}\right)}^{2}\end{array}$(2.19)

${\varphi }_{x}=\frac{1}{\sqrt{2}}\stackrel{¯}{\psi }-\sqrt{2}\text{i}\lambda \varphi -\frac{1}{\sqrt{2}}\psi {\varphi }^{2}$

$\begin{array}{c}\underset{k=1}{\overset{+\infty }{\sum }}{\omega }_{k,x}{\lambda }^{-k}=\frac{1}{\sqrt{2}}\stackrel{¯}{\psi }-\sqrt{2}\text{i}\lambda \underset{k=1}{\overset{+\infty }{\sum }}{\omega }_{k}{\lambda }^{-k}-\frac{1}{\sqrt{2}}\psi {\left(\underset{k=1}{\overset{+\infty }{\sum }}{\omega }_{k}{\lambda }^{-k}\right)}^{2}\\ =\frac{1}{\sqrt{2}}\stackrel{¯}{\psi }-\sqrt{2}\text{i}\underset{k=0}{\overset{+\infty }{\sum }}{\omega }_{k+1}{\lambda }^{-k}-\frac{1}{\sqrt{2}}\psi \underset{k=2}{\overset{+\infty }{\sum }}\left(\underset{j=1}{\overset{k}{\sum }}{\omega }_{j}{\omega }_{k-j}\right){\lambda }^{-k}\end{array}$

$\begin{array}{l}{\lambda }^{0}:\text{i}\sqrt{2}{\omega }_{1}-\frac{1}{\sqrt{2}}\stackrel{¯}{\psi }=0\\ {\lambda }^{-1}:{\omega }_{1,x}+\sqrt{2}\text{i}{\omega }_{2}=0\\ {\lambda }^{-2}:{\omega }_{2,x}+\sqrt{2}\text{i}{\omega }_{3}+\frac{1}{\sqrt{2}}\psi {\omega }_{1}^{2}=0\\ {\lambda }^{-3}:{\omega }_{3,x}+\sqrt{2}\text{i}{\omega }_{4}+\frac{1}{\sqrt{2}}\psi 2{\omega }_{1}{\omega }_{2}=0\\ {\lambda }^{-k}:{\omega }_{k+1}=\frac{1}{\sqrt{2}}\text{i}\left[{\omega }_{k,x}+\frac{1}{\sqrt{2}}\psi {\sum }_{j=1}^{k}{\omega }_{j}{\omega }_{k-j}\right],\left(k\ge 4\right)\end{array}$

$\begin{array}{l}{\omega }_{1}=-\frac{\text{i}}{2}\stackrel{¯}{\psi }\\ {\omega }_{2}=\frac{\sqrt{2}}{4}{\stackrel{¯}{\psi }}_{x}\\ {\omega }_{3}=\frac{\text{i}}{8}\left(2{\stackrel{¯}{\psi }}_{xx}-{\stackrel{¯}{\psi }}^{2}\psi \right)\\ {\omega }_{4}=\frac{1}{\sqrt{2}}\left(\frac{1}{4}\psi \stackrel{¯}{\psi }{\stackrel{¯}{\psi }}_{x}+\frac{1}{8}{\left(\psi {\stackrel{¯}{\psi }}^{2}-2{\stackrel{¯}{\psi }}_{xx}\right)}_{x}\right)\end{array}$

$\begin{array}{c}{\omega }_{5}=\text{i}\frac{1}{\sqrt{2}}\left[{\omega }_{4,x}+\frac{1}{\sqrt{2}}\psi \left({\omega }_{1}{\omega }_{3}+{\omega }_{2}{\omega }_{2}+{\omega }_{3}{\omega }_{1}\right)\right]\\ =\frac{\text{i}}{2}\left[\frac{1}{4}{\left(\psi \stackrel{¯}{\psi }{\stackrel{¯}{\psi }}_{x}\right)}_{x}+\frac{1}{8}\left[{\left(\psi {\stackrel{¯}{\psi }}^{2}\right)}_{xx}-2{\stackrel{¯}{\psi }}_{xxxx}\right]+\psi \left[\frac{1}{8}\stackrel{¯}{\psi }\left(2{\stackrel{¯}{\psi }}_{xx}-\psi {\stackrel{¯}{\psi }}^{2}\right)+\frac{1}{8}{\stackrel{¯}{\psi }}_{x}^{2}\right]\right]\\ =\frac{\text{i}}{8}\left[{\left(\stackrel{¯}{\psi }\psi {\stackrel{¯}{\psi }}_{x}\right)}_{x}+\frac{1}{2}{\left(\psi {\stackrel{¯}{\psi }}^{2}\right)}_{xx}-{\stackrel{¯}{\psi }}_{xxxx}+\psi \left[\stackrel{¯}{\psi }\left({\stackrel{¯}{\psi }}_{xx}-\frac{1}{2}\psi {\stackrel{¯}{\psi }}^{2}\right)+\frac{1}{2}{\stackrel{¯}{\psi }}_{x}^{2}\right]\right]\end{array}$

$\frac{{\phi }_{1,x}}{{\phi }_{1}}=\frac{{\Theta }_{x}+\frac{1}{\sqrt{2}}\text{i}\xi \Theta }{\Theta }=\frac{1}{\sqrt{2}}\underset{k=1}{\overset{+\infty }{\sum }}\left(\psi {\omega }_{k}\right){\lambda }^{-k}+\frac{\text{i}\lambda }{\sqrt{2}}$

$\frac{{\Theta }_{x}}{\Theta }=\frac{1}{\sqrt{2}}{\sum }_{k=1}^{+\infty }\left(\psi {\omega }_{k}\right){\lambda }^{-k}+\frac{\text{i}\lambda }{\sqrt{2}}-\frac{\text{i}\xi }{\sqrt{2}}$(2.20)

$\begin{array}{c}\frac{{\Theta }_{x}}{\Theta }=\frac{1}{\sqrt{2}}{\sum }_{k=1}^{+\infty }\left(\psi {\omega }_{k}\right){\lambda }^{-k}+\frac{\text{i}\lambda }{\sqrt{2}}-\frac{1}{\sqrt{2}}\text{i}\xi \\ =\frac{1}{\sqrt{2}}{\sum }_{k=1}^{+\infty }\left(\psi {\omega }_{k}\right){\lambda }^{-k}+\frac{\text{i}\lambda }{\sqrt{2}}-\frac{1}{\sqrt{2}}\text{i}\lambda \left({\sum }_{k=0}^{+\infty }\frac{{\prod }_{j=0}^{k-1}\left(j-\frac{1}{2}\right)}{k!}{\lambda }^{-2k}\right)\\ =\frac{1}{\sqrt{2}}{\sum }_{k=1}^{+\infty }\left(\psi {\omega }_{k}\right){\lambda }^{-k}-\frac{1}{\sqrt{2}}\text{i}\left({\sum }_{k=1}^{+\infty }\frac{{\prod }_{j=0}^{k-1}\left(j-\frac{1}{2}\right)}{k!}{\lambda }^{-2k+1}\right)\end{array}$(2.21)

${\left(\begin{array}{c}{\phi }_{1}\\ {\phi }_{2}\end{array}\right)}_{t}=\left(\begin{array}{cc}{p}_{1}& {p}_{2}\\ {p}_{3}& -{p}_{1}\end{array}\right)\left(\begin{array}{c}{\phi }_{1}\\ {\phi }_{2}\end{array}\right)$(2.22)

$\frac{{\phi }_{1,t}}{{\phi }_{1}}={\left(\mathrm{ln}{\phi }_{1}\right)}_{t}={p}_{1}+{p}_{2}\varphi =\underset{k=-3}{\overset{+\infty }{\sum }}{T}_{k}{\lambda }^{-k}$

$\begin{array}{l}{T}_{-3}=\sqrt{2}\text{i},\\ {T}_{-2}=-\text{i},\\ {T}_{-1}=\frac{\sqrt{2}\text{i}}{2}{|\psi |}^{2}+\sqrt{2}\psi {\omega }_{1},\\ {T}_{0}=-\frac{\text{i}}{2}{|\psi |}^{2}+\frac{1}{2}\left({\psi }_{x}\stackrel{¯}{\psi }-{\psi }_{x}{\stackrel{¯}{\psi }}_{x}\right)-\psi {\omega }_{1}-\text{i}{\psi }_{x}{\omega }_{1}+\sqrt{2}\psi {\omega }_{2},\\ {T}_{k}=\frac{\sqrt{2}\text{i}}{2}{\psi }_{x}{\omega }_{k}+\frac{\sqrt{2}}{2}\left({|\psi |}^{2}\psi -{\psi }_{xx}\right){\omega }_{1}-\left(\psi +\text{i}{\psi }_{x}\right){\omega }_{k+1}+\sqrt{2}\psi {\omega }_{k+2},\left(k\ge 1\right)\end{array}$

$\begin{array}{l}{\left(\frac{{\phi }_{1,x}}{{\phi }_{1}}\right)}_{t}={\left(\mathrm{ln}{\phi }_{1}\right)}_{xt}=\frac{1}{\sqrt{2}}\underset{k=1}{\overset{+\infty }{\sum }}{\left(\psi {\omega }_{k}\right)}_{t}{\lambda }^{-k}+{\left(\frac{\text{i}\lambda }{\sqrt{2}}\right)}_{t}\\ {\left(\frac{{\phi }_{1,t}}{{\phi }_{1}}\right)}_{x}={\left(\mathrm{ln}{\phi }_{1}\right)}_{tx}=\underset{k=-3}{\overset{+\infty }{\sum }}{\left({T}_{k}\right)}_{x}{\lambda }^{-k}\end{array}$

${\phi }_{1}=\Theta \left(\lambda ;x,t\right){\text{e}}^{\left[\frac{1}{\sqrt{2}}\text{i}\xi x+\left[\sqrt{2}\left({\lambda }^{2}+\frac{1}{2}\right)-\lambda \right]\text{i}\xi t\right]}$，则有

${\left(\frac{{\Theta }_{x}}{\Theta }\right)}_{t}={\left(\frac{{\Theta }_{t}}{\Theta }\right)}_{x}={\left(\frac{1}{\sqrt{2}}\underset{k=1}{\overset{+\infty }{\sum }}\left(\psi {\omega }_{k}\right){\lambda }^{-k}-\frac{1}{\sqrt{2}}\text{i}\left(\underset{k=1}{\overset{+\infty }{\sum }}\frac{{\prod }_{j=0}^{k-1}\left(j-\frac{1}{2}\right)}{k!}{\lambda }^{-2k+1}\right)\right)}_{t}=\underset{k=-3}{\overset{+\infty }{\sum }}\left({T}_{k}\right){\lambda }^{-k}$

$k=2n+1,n\in N$，即k为奇数时，

${\left(\frac{1}{\sqrt{2}}\psi {\omega }_{k}-\frac{{\prod }_{j=0}^{k-1}\left(j-\frac{1}{2}\right)}{k!}\right)}_{t}={\left({T}_{k}\right)}_{x}$(2.23)

$k=2n,n\in N$，即k为偶数时，

${\left(\frac{1}{\sqrt{2}}\psi {\omega }_{k}\right)}_{t}={\left({T}_{k}\right)}_{x}$(2.24)

${\left(\frac{1}{\sqrt{2}}\psi {\omega }_{1}+\frac{\sqrt{2}}{4}\text{i}\right)}_{t}={\left(\frac{1}{\sqrt{2}}\left(-{|\psi |}^{2}+1\right)\right)}_{t}={\left({T}_{1}\right)}_{x}$(2.25)

$\begin{array}{l}{\left(\frac{1}{\sqrt{2}}\psi {\omega }_{5}+\frac{\sqrt{2}}{32}\text{i}\right)}_{t}\\ ={\left\{\frac{1}{\sqrt{2}}\psi \left\{\frac{\text{i}}{8}\left[{\left(\stackrel{¯}{\psi }\psi {\stackrel{¯}{\psi }}_{x}\right)}_{x}+\frac{1}{2}{\left(\psi {\stackrel{¯}{\psi }}^{2}\right)}_{xx}-{\stackrel{¯}{\psi }}_{xxxx}+\psi \left[\stackrel{¯}{\psi }\left({\stackrel{¯}{\psi }}_{xx}-\frac{1}{2}\psi {\stackrel{¯}{\psi }}^{2}\right)+\frac{1}{2}{\stackrel{¯}{\psi }}_{x}^{2}\right]\right]\right\}+\frac{\sqrt{2}}{32}\text{i}\right\}}_{t}\\ ={\left({T}_{5}\right)}_{x}\end{array}$(2.26)

${\int }_{ℝ}{|{\psi }_{xx}|}^{2}+\frac{1}{2}\left({\psi }_{x}^{2}{\stackrel{¯}{\psi }}^{2}+2\stackrel{¯}{\psi }{\stackrel{¯}{\psi }}_{x}\psi {\psi }_{x}+{\psi }^{2}{\stackrel{¯}{\psi }}_{x}^{2}\right)+3{|\psi |}^{2}{|{\psi }_{x}|}^{2}+\frac{1}{2}{|\psi |}^{6}-\frac{1}{2}\text{d}x$(2.27)

${\int }_{ℝ}\frac{1}{\sqrt{2}}\left(-{|\psi |}^{2}+1\right)\text{d}x$(2.28)

$\begin{array}{l}{\int }_{ℝ}{|{\psi }_{xx}|}^{2}+\frac{1}{2}\left({\psi }_{x}^{2}{\stackrel{¯}{\psi }}^{2}+2\stackrel{¯}{\psi }{\stackrel{¯}{\psi }}_{x}\psi {\psi }_{x}+{\psi }^{2}{\stackrel{¯}{\psi }}_{x}^{2}\right)+3{|\psi |}^{2}{|{\psi }_{x}|}^{2}+\frac{1}{2}{|\psi |}^{6}-\frac{1}{2}\text{d}x+\frac{3\sqrt{2}}{2}{\int }_{ℝ}\frac{1}{\sqrt{2}}\left(-{|\psi |}^{2}+1\right)\text{d}x\\ ={\int }_{ℝ}\left[{|{\psi }_{xx}|}^{2}+3{|\psi |}^{2}{|{\psi }_{x}|}^{2}+\frac{1}{2}{\left(\stackrel{¯}{\psi }{\psi }_{x}+\psi \stackrel{¯}{{\psi }_{x}}\right)}^{2}+{\left(1-{|\psi |}^{2}\right)}^{2}\left(1+\frac{1}{2}{|\psi |}^{2}\right)\right]\text{d}x\\ =S\left(\psi \right)\end{array}$(2.29)

$\text{i}\left({\psi }_{x}+{\psi }_{t}\right)+{\psi }_{xx}-{|\psi |}^{2}\psi +\text{i}\left({\psi }_{xxx}-3{|\psi |}^{2}{\psi }_{x}\right)=0$

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