Lax 对

Lax 对定义。一个非线性偏微分方程 F ( x , t , u , … ) = 0 {\displaystyle F(x,t,u,\dots )=0} 的Lax 对 是一对线性微分算子 L = L ( u , λ ) {\displaystyle L=L(u,\lambda )} M = M ( u , λ ) {\displaystyle M=M(u,\lambda )} [ L , M ] = L M − M L {\displaystyle [L,M]=LM-ML} 是交换子。 如果 F ( x , t , u , … ) = 0 {\displaystyle F(x,t,u,\dots )=0} 可以表示为 Lax 方程： L t + [ L , M ] = 0 {\displaystyle L_{t}+[L,M]=0} , 且 L ϕ = λ ( t ) ϕ {\displaystyle L\phi =\lambda (t)\phi } , 则 λ t = 0 {\displaystyle \lambda _{t}=0} , 并且 ϕ {\displaystyle \phi } 满足 ϕ t = M ϕ {\displaystyle \phi _{t}=M\phi } 1972年V.E.Zakharov,A.B.Shabat,将Lax对推广到高维 对于两个 线性方程 ϕ x = A ϕ , ϕ t = B ϕ {\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 及其中的参考文献 。 定义 A t − B x + [ A , B ] = 0 {\displaystyle A_{t}-B_{x}+[A,B]=0} 为两个 线性方程 ϕ x = A ϕ , ϕ t = B ϕ {\displaystyle \phi _{x}=A\phi ,\phi _{t}=B\phi } 的相容条件。 KdV 方程 的Lax对为 L = ∂ 2 ∂ x 2 + u {\displaystyle L={\frac {\partial ^{2}}{\partial x^{2}}}+u} M = − 4 ∂ 3 ∂ x 3 + 6 u ∂ ∂ x + 3 ∂ u ∂ x {\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 A = i λ [ 1 0 0 − 1 ] {\displaystyle \mathbf {A} =i\lambda {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}} + i [ 0 q r 0 ] {\displaystyle i{\begin{bmatrix}0&q\\r&0\end{bmatrix}}} B = 2 i λ 2 [ 1 0 0 − 1 ] {\displaystyle \mathbf {B} =2i\lambda ^{2}{\begin{bmatrix}1&0\\0&-1\end{bmatrix}}} + 2 i λ [ 0 Q R 0 ] {\displaystyle 2i\lambda {\begin{bmatrix}0&Q\\R&0\end{bmatrix}}} + [ 0 q x − r x 0 ] {\displaystyle {\begin{bmatrix}0&q_{x}\\-r_{x}&0\end{bmatrix}}} - i [ r q 0 0 − r q ] {\displaystyle i{\begin{bmatrix}rq&0\\0&-rq\end{bmatrix}}} sine-Gordon方程 A = i λ [ 1 0 0 − 1 ] {\displaystyle \mathbf {A} =i\lambda {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}} + i [ 0 q r 0 ] {\displaystyle i{\begin{bmatrix}0&q\\r&0\end{bmatrix}}} B = 1 4 i λ [ cos ⁡ u − i sin ⁡ u i sin ⁡ u − cos ⁡ u ] {\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 A = i λ [ 1 0 0 − 1 ] {\displaystyle \mathbf {A} =i\lambda {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}} + i [ 0 q r 0 ] {\displaystyle i{\begin{bmatrix}0&q\\r&0\end{bmatrix}}} B = 1 4 i λ [ c o s h u − i s i n h u − i s i n h u − c o s h u ] {\displaystyle \mathbf {B} ={\frac {1}{4i\lambda }}{\begin{bmatrix}coshu&-isinhu\\-isinhu&-coshu\end{bmatrix}}} KdV 方程 A = [ i λ 1 u − i λ ] {\displaystyle \mathbf {A} ={\begin{bmatrix}i\lambda &1\\u&-i\lambda \end{bmatrix}}} B = [ 4 i λ 3 + 2 i λ u − u x 4 λ 2 + 2 u 4 λ 2 u + 2 i λ u x + 2 u 2 − u x x + 2 u 3 4 i λ 3 + 2 i λ u 2 ] {\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 A = [ − i λ u u i λ ] {\displaystyle \mathbf {A} ={\begin{bmatrix}-i\lambda &u\\u&i\lambda \end{bmatrix}}} B = [ − 4 i λ 3 − 2 i λ u 2 4 λ 2 u + 2 i λ u x − u x x + 2 u 3 4 λ 2 u − 2 i λ u x − u x x + 2 u 2 4 i λ 3 + 2 i λ u 2 ] {\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 Inna p217 Inna p218 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