Handbook of mathematics for engineers and scienteists part 195

Handbook of mathematics for engineers and scienteists part 195. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 1326 Nonlinear Mathematical Physics Equations . Monge-Ampere Equations d2w 2 d2w d2w dxdy dx2 dy2 Homogeneous Monge-Ampere equation. 1 . General solution in parametric form w tx p t y t x p t y t 0 where t is the parameter and p p t and t are arbitrary functions. 2 . Solutions involving one arbitrary function w x y p Cix C2y C3x C4y C5 w x y Cix C2 y p y C3x C4y C5 x C5y C2y where C1 . C9 are arbitrary constants and p p z is an arbitrary function. w x y Cix C2y C3 p C4- Ci x 77 C7x C8y C9 C3 d2w X2 d2w d2w 2. I J A. dxdy J dx2 dy2 Nonhomogeneous Monge-Ampere equation. 1 . General solution in parametric form for A a2 0 x P - A X - p fi w fi A A - .p Q 2p fi - 2 A X 2a y 2a W 4a where f3 and A are the parameters p p P and A are arbitrary functions. 2 . Solutions A . . _ _ _ _ w x y x Cix C2y p Cix C2y C3x C4y C2 w x y Ciy2 C2xy -1- C2 - A x2 C3y C4x C5 4Ci w x y fC2y2 C3y - A x3 3C x2 C4y C5x C6 x Ci y 4C2 J i2C2 w x y A Cix - C2y2 Ü3 3 2 Crx C5y C6 3CiC2 where C1 . C6 are arbitrary constants and p p z is an arbitrary function. . Higher-Order Equations 1327 . Higher-Order Equations . Third-Order Equations dw d3w dw dt dx3 dx Korteweg-de Vries equation. It is used in many sections of nonlinear mechanics and physics. 1 . Suppose w x t is a solution of the Korteweg-de Vries equation. Then the function w1 C2w Cx 6CiC2t C3 C33t C4 C2 where Ci . C4 are arbitrary constants is also a solution of the equation. 2 . One-soliton solution w xt __a-_ 2 cosh2 2 y a x - at - b where a and b are arbitrary constants. 3 . Two-soliton solution d 2 w x t -2 - ln 1 Bie31 B2ee2 ABiB2e31 e2 dx2 2 3 3 . I ai a 2 di a1x - a t 02 a2x - a21 A ----------- 1 2 a1 a2 where B1 B2 a1 and a2 are arbitrary constants. 4 . N-soliton solution d2 t t w x t -2 r -On det I C x t k dx2 t J Here I is the N X N identity matrix and C x t the N X N symmetric matrix with entries V pm t pn t r z Cmn X t --------------exp - pm pn xI Pm pn where the normalizing factors pn t .