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，同時也是卡拉比-丘流形證明時曾用的工具。 廣義而言，定義兩個獨立變量x,y，以及一個非獨立變量u，蒙日-安培方程可以表述為： L [ u ] = A ( u x x u y y − u x y 2 ) + B u x x + 2 C u x y + D u y y + E = 0 , {\displaystyle L[u]=A(u_{xx}u_{yy}-u_{xy}^{2})+Bu_{xx}+2Cu_{xy}+Du_{yy}+E=0,} 這裡的A,B,C,D,E為一階變量x,y,ux和uy唯一的非獨立函數。 根據齊次蒙日-安培方程： s y s = ( U [ x , t ] ) 2 − U [ x , x ] ∗ U [ t , t ] = 0 ; {\displaystyle sys={(U[x,t])^{2}-U[x,x]*U[t,t]=0};} 其對應的解析解為： u [ 1 ] := C 1 ∗ y 2 + C 2 ∗ x ∗ y + C 2 2 ∗ x 2 / ( 4 ∗ C 1 ) + C 3 ∗ y + C 4 ∗ x + C 5 {\displaystyle u[1]:=C1*y^{2}+C2*x*y+C2^{2}*x^{2}/(4*C1)+C3*y+C4*x+C5} u [ 2 ] := ( C 2 ∗ y 2 + C 3 ∗ y + C 3 2 / ( 4 ∗ C 2 ) ) / ( x + C 1 ) + C 4 ∗ y + C 5 ∗ x + C 6 {\displaystyle u[2]:=(C2*y^{2}+C3*y+C3^{2}/(4*C2))/(x+C1)+C4*y+C5*x+C6} u [ 3 ] := ( C 1 ∗ x + C 2 ∗ y + C 3 ) ( k + 1 ) / ( C 4 ∗ x + C 5 ∗ y + C 6 ) k + C 7 ∗ x + C 8 ∗ y + C 9 {\displaystyle u[3]:=(C1*x+C2*y+C3)^{(}k+1)/(C4*x+C5*y+C6)^{k}+C7*x+C8*y+C9} u [ 4 ] := ( C 1 ∗ x + C 2 ∗ y + C 3 ) k + C 4 ∗ x + C 5 ∗ y + C 6 {\displaystyle u[4]:=(C1*x+C2*y+C3)^{k}+C4*x+C5*y+C6} u [ 5 ] := − ( C 1 ∗ x + C 2 ∗ y + C 3 ) k + C 4 ∗ x + C 5 ∗ y + C 6 {\displaystyle u[5]:=-(C1*x+C2*y+C3)^{k}+C4*x+C5*y+C6} u [ 6 ] := − ( C 1 ∗ x + C 2 ∗ y + C 3 ) ( k + 1 ) / ( C 4 ∗ x + C 5 ∗ y + C 6 ) k + C 7 ∗ x + C 8 ∗ y + C 9 {\displaystyle u[6]:=-(C1*x+C2*y+C3)^{(}k+1)/(C4*x+C5*y+C6)^{k}+C7*x+C8*y+C9} u [ 7 ] := C 1 ∗ x 2 + 2 ∗ C 1 ∗ x ∗ a + C 1 ∗ a 2 + C 2 ∗ x ∗ y + C 2 ∗ x ∗ b + C 2 ∗ a ∗ y + C 2 ∗ a ∗ b + C 3 ∗ y 2 + 2 ∗ C 3 ∗ y ∗ b + C 3 ∗ b 2 + C 5 ∗ x + C 6 ∗ y + C 7 {\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} u [ 8 ] := C 1 ∗ ( C 1 ∗ y 2 + C 2 ∗ x ∗ y + ( 1 / 4 ) ∗ C 2 2 ∗ x 2 / C 1 + C 3 ∗ y + C 4 ∗ x + C 5 ) ∗ ( C 2 ∗ x + C 3 ∗ y + C 4 ) + C 5 ∗ x + C 6 ∗ y + C 7 {\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} u [ 9 ] := C 1 ∗ ( ( C 2 ∗ y 2 + C 3 ∗ y + ( 1 / 4 ) ∗ C 3 2 / C 2 ) / ( x + C 1 ) + C 4 ∗ y + C 5 ∗ x + C 6 ) ∗ ( C 2 ∗ x + C 3 ∗ y + C 4 ) + C 5 ∗ x + C 6 ∗ y + C 7 {\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} u [ 10 ] := C 1 ∗ ( ( C 1 ∗ x + C 2 ∗ y + C 3 ) ( k + 1 ) / ( C 4 ∗ x + C 5 ∗ y + C 6 ) k + C 7 ∗ x + C 8 ∗ y + C 9 ) ∗ ( C 2 ∗ x + C 3 ∗ y + C 4 ) + C 5 ∗ x + C 6 ∗ y + C 7 {\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} Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p775-776 CRC PRESS *谷超豪 《孤立子理论中的达布变换及其几何应用》 上海科学技术出版社 *阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年 李志斌编著 《非线性数学物理方程的行波解》 科学出版社 王东明著 《消去法及其应用》 科学出版社 2002 *何青 王丽芬编著 《Maple 教程》 科学出版社 2010 ISBN 9787030177445 Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000 Saber Elaydi,An Introduction to Difference Equationns, Springer 2000 Dongming Wang, Elimination Practice,Imperial College Press 2004 David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004 George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759