玻恩-英费尔德方程

玻恩-英费尔德方程（Born-Infeld equation）是马克斯·玻恩和利奥波德·英费尔德创建的非线性偏微分方程： ( 1 − u t 2 ) u x x + 2 u x u t u x t − ( 1 + u x 2 ) u t t = 0 {\displaystyle \displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0} 玻恩-英费尔德方程有许多行波解 u [ 1 ] := 2 ∗ a t a n ( ( ( a 0 + b 1 ) / ( a 0 − b 1 ) ) ∗ t a n ( ( 1 / 2 ) ∗ ( a 0 2 − b 1 2 ) ∗ η ) ) + ( 1 / 2 ) ∗ π {\displaystyle u[1]:=2*atan({\sqrt {(}}(a0+b1)/(a0-b1))*tan((1/2)*{\sqrt {(}}a0^{2}-b1^{2})*\eta ))+(1/2)*\pi } u [ 2 ] := 2 ∗ a t a n ( ( ( a 0 + b 1 ) / ( a 0 − b 1 ) ) ∗ c o t ( ( 1 / 2 ) ∗ ( a 0 2 − b 1 2 ) ∗ η ) ) + ( 1 / 2 ) ∗ π {\displaystyle u[2]:=2*atan({\sqrt {(}}(a0+b1)/(a0-b1))*cot((1/2)*{\sqrt {(}}a0^{2}-b1^{2})*\eta ))+(1/2)*\pi } u [ 3 ] := 2 ∗ a t a n ( ( a 1 2 + b 1 2 ) ∗ t a n ( ( 1 / 2 ) ∗ ( a 1 2 + b 1 2 ) ∗ η ) / a 1 + b 1 / a 1 ) {\displaystyle u[3]:=2*atan({\sqrt {(}}a1^{2}+b1^{2})*tan((1/2)*{\sqrt {(}}a1^{2}+b1^{2})*\eta )/a1+b1/a1)} u [ 4 ] := 2 ∗ a t a n ( ( a 1 2 + b 1 2 ) ∗ c o t ( ( 1 / 2 ) ∗ ( a 1 2 + b 1 2 ) ∗ η ) / a 1 + b 1 / a 1 ) {\displaystyle u[4]:=2*atan({\sqrt {(}}a1^{2}+b1^{2})*cot((1/2)*{\sqrt {(}}a1^{2}+b1^{2})*\eta )/a1+b1/a1)} u [ 5 ] := 2 ∗ a t a n ( ( a 1 2 + b 1 2 ) ∗ ( c o t h ( ( a 1 2 + b 1 2 ) ∗ e t a ) + c s c ( ( a 1 2 + b 1 2 ) ∗ e t a ) ) / a 1 + b 1 / a 1 ) {\displaystyle u[5]:=2*atan({\sqrt {(}}a1^{2}+b1^{2})*(coth({\sqrt {(}}a1^{2}+b1^{2})*eta)+csc({\sqrt {(}}a1^{2}+b1^{2})*eta))/a1+b1/a1)} u [ 6 ] := 2 ∗ a t a n ( ( a 1 2 + b 1 2 ) ∗ ( c o s h ( ( a 1 2 + b 1 2 ) ∗ η ) + 1 ) / ( a 1 ∗ s i n h ( ( a 1 2 + b 1 2 ) ∗ η ) ) + b 1 / a 1 ) {\displaystyle u[6]:=2*atan({\sqrt {(}}a1^{2}+b1^{2})*(cosh({\sqrt {(}}a1^{2}+b1^{2})*\eta )+1)/(a1*sinh({\sqrt {(}}a1^{2}+b1^{2})*\eta ))+b1/a1)} u [ 7 ] := 2 ∗ a t a n ( ( a 1 2 + b 1 2 ) ∗ s i n h ( ( a 1 2 + b 1 2 ) ∗ η ) / ( a 1 ∗ ( c o s h ( ( a 1 2 + b 1 2 ) ∗ η ) + 1 ) ) + b 1 / a 1 ) {\displaystyle u[7]:=2*atan({\sqrt {(}}a1^{2}+b1^{2})*sinh({\sqrt {(}}a1^{2}+b1^{2})*\eta )/(a1*(cosh({\sqrt {(}}a1^{2}+b1^{2})*\eta )+1))+b1/a1)} u [ 8 ] := 2 ∗ a t a n ( ( ( a 0 + b 1 ) / ( b 1 − a 0 ) ) ∗ c o t ( ( b 1 2 − a 0 2 ) ∗ η + s e c ( h ∗ ( b 1 2 − a 0 2 ) ∗ η ) ) ) + ( 1 / 2 ) ∗ π {\displaystyle u[8]:=2*atan({\sqrt {(}}(a0+b1)/(b1-a0))*cot({\sqrt {(}}b1^{2}-a0^{2})*\eta +sec(h*{\sqrt {(}}b1^{2}-a0^{2})*\eta )))+(1/2)*\pi } u [ 9 ] := 2 ∗ a t a n ( ( ( a 0 + b 1 ) / ( b 1 − a 0 ) ) ∗ s i n h ( ( b 1 2 − a 0 2 ) ∗ η ) / ( c o s h ( ( b 1 2 − a 0 2 ) ∗ e t a ) + 1 ) ) {\displaystyle u[9]:=2*atan({\sqrt {(}}(a0+b1)/(b1-a0))*sinh({\sqrt {(}}b1^{2}-a0^{2})*\eta )/(cosh({\sqrt {(}}b1^{2}-a0^{2})*eta)+1))} 2 ∗ a t a n ( ( ( a 0 + b 1 ) / ( b 1 − a 0 ) ) ∗ ( c o s h ( ( b 1 2 − a 0 2 ) ∗ η ) + 1 ) / s i n h ( ( b 1 2 − a 0 2 ) ∗ η ) ) {\displaystyle 2*atan({\sqrt {(}}(a0+b1)/(b1-a0))*(cosh({\sqrt {(}}b1^{2}-a0^{2})*\eta )+1)/sinh({\sqrt {(}}b1^{2}-a0^{2})*\eta ))} Born Infeld equation animation1 Born Infeld equation animation3 Born Infeld equation animation4 Born Infeld equation animation5 Born Infeld equation animation6 Born Infeld equation animation7 Born Infeld equation animation8 Born Infeld equation animation9 Born Infeld equation animation10 D.B. Fairlie and J.A. Mulvey Integrable Generalisations of the 2-dimensional Born Infeld Equation,University of Durham, 1993 Yuanxi Xie and Jiashi Tang,New Explicit Exact Solutions of the Born Infeld Equation,2005 *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