纽厄尔-怀特海德方程

纽厄尔-怀特海德方程（Newell-Whitehead equation)是一个非线性偏微分方程： u t − u x x − α ∗ u + b e t a ∗ u 3 = 0 {\displaystyle u_{t}-u_{xx}-\alpha *u+beta*u^{3}=0} 纽厄尔-怀特海德方程有行波解： u ( x , t ) = 1 / ( β ) {\displaystyle u(x,t)=1/{\sqrt {(}}\beta )} , u ( x , t ) = − 1 / ( β ) {\displaystyle u(x,t)=-1/{\sqrt {(}}\beta )} , u ( x , t ) = − ( 1 / 2 ∗ I ) / ( − β ) − ( 1 / 2 ) ∗ c o t ( C 1 − ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x − ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*cot(_{C}1-(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = − ( 1 / 2 ∗ I ) / ( − β ) + ( 1 / 2 ) ∗ c o t ( C 1 − ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x + ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*cot(_{C}1-(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = − ( 1 / 2 ∗ I ) / ( − β ) − ( 1 / 2 ) ∗ c o t ( C 1 + ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x − ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*cot(_{C}1+(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = − ( 1 / 2 ∗ I ) / ( − β ) + ( 1 / 2 ) ∗ c o t ( C 1 + ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x + ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*cot(_{C}1+(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = − ( 1 / 2 ∗ I ) / ( − β ) + ( 1 / 2 ) ∗ t a n ( C 1 − ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x − ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*tan(_{C}1-(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = − ( 1 / 2 ∗ I ) / ( − β ) − ( 1 / 2 ) ∗ t a n ( C 1 − ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x + ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*tan(_{C}1-(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = − ( 1 / 2 ∗ I ) / ( − β ) + ( 1 / 2 ) ∗ t a n ( C 1 + ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x − ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*tan(_{C}1+(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = − ( 1 / 2 ∗ I ) / ( − β ) − ( 1 / 2 ) ∗ t a n ( C 1 + ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x + ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*tan(_{C}1+(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = ( 1 / 2 ∗ I ) / ( − β ) + ( 1 / 2 ) ∗ c o t ( C 1 − ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x − ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*cot(_{C}1-(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = ( 1 / 2 ∗ I ) / ( − β ) − ( 1 / 2 ) ∗ c o t ( C 1 − ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x + ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*cot(_{C}1-(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = ( 1 / 2 ∗ I ) / ( − β ) + ( 1 / 2 ) ∗ c o t ( C 1 + ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x − ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*cot(_{C}1+(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = ( 1 / 2 ∗ I ) / ( − β ) − ( 1 / 2 ) ∗ c o t ( C 1 + ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x + ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*cot(_{C}1+(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = ( 1 / 2 ∗ I ) / ( − β ) − ( 1 / 2 ) ∗ t a n ( C 1 − ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x − ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*tan(_{C}1-(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = ( 1 / 2 ∗ I ) / ( − β ) + ( 1 / 2 ) ∗ t a n ( C 1 − ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x + ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*tan(_{C}1-(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = ( 1 / 2 ∗ I ) / ( − β ) − ( 1 / 2 ) ∗ t a n ( C 1 + ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x − ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*tan(_{C}1+(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = ( 1 / 2 ∗ I ) / ( − β ) + ( 1 / 2 ) ∗ t a n ( C 1 + ( 1 / 4 ∗ I ) ∗ ( 2 ) ∗ x + ( 3 / 4 ∗ I ) ∗ t ) / ( − β ) {\displaystyle u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*tan(_{C}1+(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )} , u ( x , t ) = − 1 / ( 2 ∗ ( β ) ) + ( 1 / 2 ) ∗ c o t h ( C 1 − ( 1 / 4 ) ∗ ( 2 ) ∗ x − ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=-1/(2*{\sqrt {(}}\beta ))+(1/2)*coth(_{C}1-(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = − 1 / ( 2 ∗ ( β ) ) − ( 1 / 2 ) ∗ c o t h ( C 1 − ( 1 / 4 ) ∗ ( 2 ) ∗ x + ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=-1/(2*{\sqrt {(}}\beta ))-(1/2)*coth(_{C}1-(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = − 1 / ( 2 ∗ ( β ) ) + ( 1 / 2 ) ∗ c o t h ( C 1 + ( 1 / 4 ) ∗ ( 2 ) ∗ x − ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=-1/(2*{\sqrt {(}}\beta ))+(1/2)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = − 1 / ( 2 ∗ ( β ) ) − ( 1 / 2 ) ∗ c o t h ( C 1 + ( 1 / 4 ) ∗ ( 2 ) ∗ x + ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=-1/(2*{\sqrt {(}}\beta ))-(1/2)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = − 1 / ( 2 ∗ ( β ) ) + ( 1 / 2 ) ∗ t a n h ( C 1 − ( 1 / 4 ) ∗ ( 2 ) ∗ x − ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=-1/(2*{\sqrt {(}}\beta ))+(1/2)*tanh(_{C}1-(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = − 1 / ( 2 ∗ ( β ) ) − ( 1 / 2 ) ∗ t a n h ( C 1 − ( 1 / 4 ) ∗ ( 2 ) ∗ x + ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=-1/(2*{\sqrt {(}}\beta ))-(1/2)*tanh(_{C}1-(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = − 1 / ( 2 ∗ ( β ) ) + ( 1 / 2 ) ∗ t a n h ( C 1 + ( 1 / 4 ) ∗ ( 2 ) ∗ x − ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=-1/(2*{\sqrt {(}}\beta ))+(1/2)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = − 1 / ( 2 ∗ ( β ) ) − ( 1 / 2 ) ∗ t a n h ( C 1 + ( 1 / 4 ) ∗ ( 2 ) ∗ x + ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=-1/(2*{\sqrt {(}}\beta ))-(1/2)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = 1 / ( 2 ∗ ( β ) ) − ( 1 / 2 ) ∗ c o t h ( C 1 − ( 1 / 4 ) ∗ ( 2 ) ∗ x − ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=1/(2*{\sqrt {(}}\beta ))-(1/2)*coth(_{C}1-(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = 1 / ( 2 ∗ ( β ) ) + ( 1 / 2 ) ∗ c o t h ( C 1 − ( 1 / 4 ) ∗ ( 2 ) ∗ x + ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=1/(2*{\sqrt {(}}\beta ))+(1/2)*coth(_{C}1-(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = 1 / ( 2 ∗ ( β ) ) − ( 1 / 2 ) ∗ c o t h ( C 1 + ( 1 / 4 ) ∗ ( 2 ) ∗ x − ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=1/(2*{\sqrt {(}}\beta ))-(1/2)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = 1 / ( 2 ∗ ( β ) ) + ( 1 / 2 ) ∗ c o t h ( C 1 + ( 1 / 4 ) ∗ ( 2 ) ∗ x + ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=1/(2*{\sqrt {(}}\beta ))+(1/2)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = 1 / ( 2 ∗ ( β ) ) − ( 1 / 2 ) ∗ t a n h ( C 1 − ( 1 / 4 ) ∗ ( 2 ) ∗ x − ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=1/(2*{\sqrt {(}}\beta ))-(1/2)*tanh(_{C}1-(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = 1 / ( 2 ∗ ( β ) ) + ( 1 / 2 ) ∗ t a n h ( C 1 − ( 1 / 4 ) ∗ ( 2 ) ∗ x + ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=1/(2*{\sqrt {(}}\beta ))+(1/2)*tanh(_{C}1-(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = 1 / ( 2 ∗ ( β ) ) − ( 1 / 2 ) ∗ t a n h ( C 1 + ( 1 / 4 ) ∗ ( 2 ) ∗ x − ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=1/(2*{\sqrt {(}}\beta ))-(1/2)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )} , u ( x , t ) = 1 / ( 2 ∗ ( β ) ) + ( 1 / 2 ) ∗ t a n h ( C 1 + ( 1 / 4 ) ∗ ( 2 ) ∗ x + ( 3 / 4 ) ∗ t ) / ( β ) {\displaystyle u(x,t)=1/(2*{\sqrt {(}}\beta ))+(1/2)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )} </math> Newell--Whitehead equation. [2014-01-25]. （原始内容存档于2019-07-18）.  *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