齐次蒙日-安培方程

齐次蒙日-安培方程（Homogeneous Monge-Ampère equation）是一个常见于黎曼几何的非线性偏微分方程，同时也是卡拉比-丘流形证明时曾用的工具。^[1] 广义而言，定义两个独立变量x,y，以及一个非独立变量u，蒙日-安培方程可以表述为：
${\displaystyle L[u]=A(u_{xx}u_{yy}-u_{xy}^{2})+Bu_{xx}+2Cu_{xy}+Du_{yy}+E=0,}$
这里的A,B,C,D,E为一阶变量x,y,u_x和u_y唯一的非独立函数。

解析解

根据齐次蒙日-安培方程： ${\displaystyle sys={(U[x,t])^{2}-U[x,x]*U[t,t]=0};}$
其对应的解析解为：

${\displaystyle u[1]:=C1*y^{2}+C2*x*y+C2^{2}*x^{2}/(4*C1)+C3*y+C4*x+C5}$

${\displaystyle u[2]:=(C2*y^{2}+C3*y+C3^{2}/(4*C2))/(x+C1)+C4*y+C5*x+C6}$

${\displaystyle u[3]:=(C1*x+C2*y+C3)^{k+1}/(C4*x+C5*y+C6)^{k}+C7*x+C8*y+C9}$

${\displaystyle u[4]:=(C1*x+C2*y+C3)^{k}+C4*x+C5*y+C6}$

${\displaystyle u[5]:=-(C1*x+C2*y+C3)^{k}+C4*x+C5*y+C6}$

${\displaystyle u[6]:=-(C1*x+C2*y+C3)^{k+1}/(C4*x+C5*y+C6)^{k}+C7*x+C8*y+C9}$

${\displaystyle u[7]:={\sqrt {C1*x^{2}+2*C1*x*a+C1*a^{2}+C2*x*y+C2*x*b+C2*a*y+C2*a*b+C3*y^{2}+2*C3*y*b+C3*b^{2}}}+C5*x+C6*y+C7}$

${\displaystyle u[8]:=C1*(C1*y^{2}+C2*x*y+(1/4)*C2^{2}*x^{2}/C1+C3*y+C4*x+C5)*(C2*x+C3*y+C4)+C5*x+C6*y+C7}$

${\displaystyle u[9]:=C1*((C2*y^{2}+C3*y+(1/4)*C3^{2}/C2)/(x+C1)+C4*y+C5*x+C6)*(C2*x+C3*y+C4)+C5*x+C6*y+C7}$

${\displaystyle u[10]:=C1*((C1*x+C2*y+C3)^{k+1}/(C4*x+C5*y+C6)^{k}+C7*x+C8*y+C9)*(C2*x+C3*y+C4)+C5*x+C6*y+C7}$

行波图

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot

Homogeneous Monge-Ampere equation plot