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]

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

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

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

参考文献

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）

• 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.

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