Lax 对

Lax 对定义。一个非线性偏微分方程

${\displaystyle F(x,t,u,\dots )=0}$

的Lax 对 是一对线性微分算子^[1]

${\displaystyle L=L(u,\lambda )}$

${\displaystyle M=M(u,\lambda )}$

${\displaystyle [L,M]=LM-ML}$ 是交换子。

如果 ${\displaystyle F(x,t,u,\dots )=0}$可以表示为 Lax 方程：

${\displaystyle L_{t}+[L,M]=0}$ , 且 ${\displaystyle L\phi =\lambda (t)\phi }$ , 则 ${\displaystyle \lambda _{t}=0}$ , 并且 ${\displaystyle \phi }$ 满足

${\displaystyle \phi _{t}=M\phi }$

高维Lax对

1972年V.E.Zakharov,A.B.Shabat,将Lax对推广到高维^[2]

对于两个 线性方程 ${\displaystyle \phi _{x}=A\phi ,\phi _{t}=B\phi }$

其中A、B是 n x n 维矩阵; 或者更一般地，A和B可以是李代数g的元素; g可以是无限维的，参见 例如 ^[3]及其中的参考文献 。

定义 ${\displaystyle A_{t}-B_{x}+[A,B]=0}$ 为两个 线性方程 ${\displaystyle \phi _{x}=A\phi ,\phi _{t}=B\phi }$的相容条件。

实例

KdV 方程 的Lax对为

${\displaystyle L={\frac {\partial ^{2}}{\partial x^{2}}}+u}$

${\displaystyle M=-4{\frac {\partial ^{3}}{\partial x^{3}}}+6u{\frac {\partial }{\partial x}}+3{\frac {\partial u}{\partial x}}}$

非线性薛定谔方程

${\displaystyle \mathbf {A} =i\lambda {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$+${\displaystyle i{\begin{bmatrix}0&q\\r&0\end{bmatrix}}}$

${\displaystyle \mathbf {B} =2i\lambda ^{2}{\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$+${\displaystyle 2i\lambda {\begin{bmatrix}0&Q\\R&0\end{bmatrix}}}$+ ${\displaystyle {\begin{bmatrix}0&q_{x}\\-r_{x}&0\end{bmatrix}}}$-${\displaystyle i{\begin{bmatrix}rq&0\\0&-rq\end{bmatrix}}}$

sine-Gordon方程

${\displaystyle \mathbf {A} =i\lambda {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$+${\displaystyle i{\begin{bmatrix}0&q\\r&0\end{bmatrix}}}$

${\displaystyle \mathbf {B} ={\frac {1}{4i\lambda }}{\begin{bmatrix}\cos u&-i\sin u\\i\sin u&-\cos u\end{bmatrix}}}$

Sinh-Gordon方程

${\displaystyle \mathbf {A} =i\lambda {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$+${\displaystyle i{\begin{bmatrix}0&q\\r&0\end{bmatrix}}}$

${\displaystyle \mathbf {B} ={\frac {1}{4i\lambda }}{\begin{bmatrix}coshu&-isinhu\\-isinhu&-coshu\end{bmatrix}}}$

KdV 方程

${\displaystyle \mathbf {A} ={\begin{bmatrix}i\lambda &1\\u&-i\lambda \end{bmatrix}}}$

${\displaystyle \mathbf {B} ={\begin{bmatrix}4i\lambda ^{3}+2i\lambda u-u_{x}&4\lambda ^{2}+2u\\4\lambda ^{2}u+2i\lambda u_{x}+2u^{2}-u_{xx}+2u^{3}&4i\lambda ^{3}+2i\lambda u^{2}\end{bmatrix}}}$

mKdV方程

${\displaystyle \mathbf {A} ={\begin{bmatrix}-i\lambda &u\\u&i\lambda \end{bmatrix}}}$

${\displaystyle \mathbf {B} ={\begin{bmatrix}-4i\lambda ^{3}-2i\lambda u^{2}&4\lambda ^{2}u+2i\lambda u_{x}-u_{xx}+2u^{3}\\4\lambda ^{2}u-2i\lambda u_{x}-u_{xx}+2u^{2}&4i\lambda ^{3}+2i\lambda u^{2}\end{bmatrix}}}$

切触Lax对^[3]

参考文献

1. Inna p217
2. Inna p218
3. Sergyeyev A. "New integrable (3+1)-dimensional systems and contact geometry", Lett. Math. Phys. 108 (2018), no. 2, 359-376,
   arXiv:1401.2122
   doi: 10.1007/s11005-017-1013-4

• Inna Shingareva, Carlos Lizarraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, Springer Wien New York