林家翘-钱学森方程

林-钱方程（Lin-Tsien equation）是林家翘-钱学森和创立的描述可压缩流体中物体跨音速运动的非线性偏微分方程^[1]^[2]

2u_{tx}+u_{x}u_{xx}-u_{yy}=0.\,

行波解

利用Maple中的软件包TWS_solution可得林-钱方程的多种行波解^[3]

u(x,y,t)=TWS_sol := {u(x, y, t) = _C1+(-(1/2)*ln(tanh(_C3+_C4*x+_C5*y+_C6*t)+1)+(1/2)*ln(tanh(_C3+_C4*x+_C5*y+_C6*t)-1))*_C2}
g[2] := {u(x, y, t) = _C1+arctan(1/\sqrt(csc(_C3+_C4*x+_C5*y+_C6*t)^2-1))*_C2}
g[2] := {u(x, y, t) = _C1+(-(1/2)*ln(coth(_C3+_C4*x+_C5*y+_C6*t)+1)+(1/2)*ln(coth(_C3+_C4*x+_C5*y+_C6*t)-1))*_C2}
g[2] := {u(x, y, t) = _C1+arctanh(1/\sqrt(1+csch(_C3+_C4*x+_C5*y+_C6*t)^2))*_C2}

找到了一个新的可积（3 + 1）维泛化; 见论文 ^[4]的系统（40）

行波图

参考文献

^ William F. Ames. Nonlinear partial differential equations in engineering, Vol. 18 （页面存档备份，存于互联网档案馆）. Academic Press, p. 173, 1965.
^ C.C. Lin, E.Reissner,and H.S.Tsien, On Two Dimensional Non-steady Motion of a Slender Body in a Compressible Fluid, 钱学森文集 p513 上海交通大学出版社
^ Graham W. Griffiths Willian Schiesser, Traveling wave analysis of Partial Differential Equations p436
^ A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376, https://arxiv.org/abs/1401.2122 （页面存档备份，存于互联网档案馆）

D. Zwillinger. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 138, 1997
C. Rogers, William F. Ames, Nonlinear boundary value problems in science and engineering （页面存档备份，存于互联网档案馆）. Academic Press, p. 373, 1989.

*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
A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376, https://arxiv.org/abs/1401.2122 （页面存档备份，存于互联网档案馆）

[1] William F. Ames. Nonlinear partial differential equations in engineering, Vol. 18 （页面存档备份，存于互联网档案馆）. Academic Press, p. 173, 1965.

[2] C.C. Lin, E.Reissner,and H.S.Tsien, On Two Dimensional Non-steady Motion of a Slender Body in a Compressible Fluid, 钱学森文集 p513 上海交通大学出版社

[3] Graham W. Griffiths Willian Schiesser, Traveling wave analysis of Partial Differential Equations p436

[4] A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376, https://arxiv.org/abs/1401.2122 （页面存档备份，存于互联网档案馆）

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