林家翹-錢學森方程

林-錢方程（Lin-Tsien equation）是林家翹-錢學森和創立的描述可壓縮流體中物體跨音速運動的非線性偏微分方程^[1]^[2]

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

行波解

利用Maple中的軟體包TWS_solution可得林-錢方程的多種行波解^[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}

找到了一個新的可積（3 + 1）維泛化; 見論文 ^[4]的系統（40）

行波圖

Lin-Tsien equation traveling wave plot

Lin Tsien eq TWS extended coth

Lin Tsien eq TWS extended csch

Lin-Tsien nlpde animation

Lin-Tsien nlpde 3d plot

Lin Tsien eq extended csc arctan

參考文獻

^ William F. Ames. Nonlinear partial differential equations in engineering, Vol. 18 （頁面存檔備份，存於網際網路檔案館）. 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, 錢學森文集 p513 上海交通大學出版社
^ 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 （頁面存檔備份，存於網際網路檔案館）

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 （頁面存檔備份，存於網際網路檔案館）. Academic Press, p. 373, 1989.

*谷超豪《孤立子理論中的達布變換及其幾何應用》上海科學技術出版社
*閻振亞著《複雜非線性波的構造性理論及其應用》科學出版社 2007年
李志斌編著《非線性數學物理方程的行波解》科學出版社
王東明著《消去法及其應用》科學出版社 2002
*何青王麗芬編著《Maple 教程》科學出版社 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 （頁面存檔備份，存於網際網路檔案館）

[1] William F. Ames. Nonlinear partial differential equations in engineering, Vol. 18 （頁面存檔備份，存於網際網路檔案館）. 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, 錢學森文集 p513 上海交通大學出版社

[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 （頁面存檔備份，存於網際網路檔案館）

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