亨特 - 萨克斯顿方程(Hunter–Saxton equation)是一个模拟向列型液晶中波动传播的非线性偏微分方程：

:
(u_t + u u_x)_x = \frac{1}{2} \, u_x^2

解析解
亨特 - 萨克斯顿方程有解析解：

u(x,t)=\frac{1}{1-t/2}*(x-t)+1

此外，Maple软件包 TWSolution 给出多个行波解：
;tanh 展开法
f1 := u(x, t) = -1.7045454545454545454+1.1363636363636363636(-(1/4)(6.336ln(tanh(2.3+.88x+1.5t)-1)-6.336ln(tanh(2.3+.88x+1.5t)+1)+9.2928)^(1/3)-(1/4I)sqrt(3)(6.336ln(tanh(2.3+.88x+1.5t)-1)-6.336ln(tanh(2.3+.88x+1.5*t)+1)+9.2928)^(1/3))^2

f2 := u(x, t) = -1.7045454545454545454+1.1363636363636363636(-(1/4)(6.336ln(tanh(2.3+.88x+1.5t)-1)-6.336ln(tanh(2.3+.88x+1.5t)+1)+9.2928)^(1/3)+(1/4I)sqrt(3)(6.336ln(tanh(2.3+.88x+1.5t)-1)-6.336ln(tanh(2.3+.88x+1.5*t)+1)+9.2928)^(1/3))^2

u(x, t) = -1.7045454545454545454 + 0.28409090909090909090 (6.336
ln(tanh(2.3 + 0.88 x + 1.5 t) - 1) - 6.336 ln(tanh(2.3 + 0.88 x + 1.5 t) + 1) + 9.2928)^(2/3)

其中 f1、f2 是重解，因此，只有两个独立的行波解。

;sech arctan 展开法

f[1] := -.20000000000000000000+(1/12)(-(36(1.3arctan(1/sqrt(sech(2+3x+.6t)^2-1))sqrt(sech(2+3x+.6t)+1)sqrt(sech(2+3x+.6t)-1)-2.6sqrt(sech(2+3x+.6t)^2-1)))/sqrt(sech(2+3x+.6t)^2-1))^(2/3)

f[2] := -.20000000000000000000+(1/3)(-(1/4)(-(36(1.3arctan(1/sqrt(sech(2+3x+.6t)^2-1))sqrt(sech(2+3x+.6t)+1)sqrt(sech(2+3x+.6t)-1)-2.6sqrt(sech(2+3x+.6t)^2-1)))/sqrt(sech(2+3x+.6t)^2-1))^(1/3)-(1/4I)sqrt(3)(-(36(1.3arctan(1/sqrt(sech(2+3x+.6t)^2-1))sqrt(sech(2+3x+.6t)+1)sqrt(sech(2+3x+.6t)-1)-2.6sqrt(sech(2+3x+.6t)^2-1)))/sqrt(sech(2+3x+.6*t)^2-1))^(1/3))^2
f[3] := -.20000000000000000000+(1/3)(-(1/4)(-(36(1.3arctan(1/sqrt(sech(2+3x+.6t)^2-1))sqrt(sech(2+3x+.6t)+1)sqrt(sech(2+3x+.6t)-1)-2.6sqrt(sech(2+3x+.6t)^2-1)))/sqrt(sech(2+3x+.6t)^2-1))^(1/3)+(1/4I)sqrt(3)(-(36(1.3arctan(1/sqrt(sech(2+3x+.6t)^2-1))sqrt(sech(2+3x+.6t)+1)sqrt(sech(2+3x+.6t)-1)-2.6sqrt(sech(2+3x+.6t)^2-1)))/sqrt(sech(2+3x+.6*t)^2-1))^(1/3))^2
f[4] := -.20000000000000000000+(1/3)(-(1/4)(187.2+140.4x+28.08t)^(1/3)-(1/4I)sqrt(3)(187.2+140.4x+28.08*t)^(1/3))^2
f[5] := -.20000000000000000000+(1/3)(-(1/4)(187.2+140.4x+28.08t)^(1/3)+(1/4I)sqrt(3)(187.2+140.4x+28.08*t)^(1/3))^2
f[6] := -.20000000000000000000+(1/12)(187.2+140.4x+28.08*t)^(2/3)

f[8] := -1.6666666666666666667_C6+1.6666666666666666667(-(1/4)(9.36Intat(1/sqrt(4_a^4-5_a^2+1), _a = JacobiSN(3+.6x+_C6t, 2))+18.72)^(1/3)+(1/4I)sqrt(3)(9.36Intat(1/sqrt(4_a^4-5_a^2+1), _a = JacobiSN(3+.6x+_C6t, 2))+18.72)^(1/3))^2
f[9] := -1.6666666666666666667_C6+.41666666666666666668(9.36Intat(1/sqrt(4_a^4-5_a^2+1), _a = JacobiSN(3+.6x+_C6*t, 2))+18.72)^(2/3)

File:Hunter Saxton extended plot1.gif
;综合展开
p[12] := -1.0714+.71429(-.50821((1.1ln(cos(1.3+1.4x+1.5t)+sqrt(cos(1.3+1.4x+1.5t)^2-1.))sqrt((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.))+1.2sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.)))sqrt(cos(1.3+1.4x+1.5t)-1.)sqrt(cos(1.3+1.4x+1.5t)+1.))/(sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.)))^(1/3)-(.88026I)((1.1ln(cos(1.3+1.4x+1.5t)+sqrt(cos(1.3+1.4x+1.5t)^2-1.))sqrt((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.))+1.2sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.)))sqrt(cos(1.3+1.4x+1.5t)-1.)sqrt(cos(1.3+1.4x+1.5t)+1.))/(sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.)))^(1/3))^2

p[13] := -1.0714+.71429(-.50821((1.1ln(cos(1.3+1.4x+1.5t)+sqrt(cos(1.3+1.4x+1.5t)^2-1.))sqrt((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.))+1.2sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.)))sqrt(cos(1.3+1.4x+1.5t)-1.)sqrt(cos(1.3+1.4x+1.5t)+1.))/(sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.)))^(1/3)+(.88026I)((1.1ln(cos(1.3+1.4x+1.5t)+sqrt(cos(1.3+1.4x+1.5t)^2-1.))sqrt((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.))+1.2sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.)))sqrt(cos(1.3+1.4x+1.5t)-1.)sqrt(cos(1.3+1.4x+1.5t)+1.))/(sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.)))^(1/3))^2
:p[16] := -1.0714+.71429(-.50821((1.1ln(sin(1.3+1.4x+1.5t)+sqrt(sin(1.3+1.4x+1.5t)^2-1.))sqrt((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.))+1.2sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.)))sqrt(sin(1.3+1.4x+1.5t)-1.)sqrt(sin(1.3+1.4x+1.5t)+1.))/(sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.)))^(1/3)-(.88026I)((1.1ln(sin(1.3+1.4x+1.5t)+sqrt(sin(1.3+1.4x+1.5t)^2-1.))sqrt((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.))+1.2sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.)))sqrt(sin(1.3+1.4x+1.5t)-1.)sqrt(sin(1.3+1.4x+1.5t)+1.))/(sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.)))^(1/3))^2
:p[17] := -1.0714+.71429(-.50821((1.1ln(sin(1.3+1.4x+1.5t)+sqrt(sin(1.3+1.4x+1.5t)^2-1.))sqrt((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.))+1.2sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.)))sqrt(sin(1.3+1.4x+1.5t)-1.)sqrt(sin(1.3+1.4x+1.5t)+1.))/(sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.)))^(1/3)+(.88026I)((1.1ln(sin(1.3+1.4x+1.5t)+sqrt(sin(1.3+1.4x+1.5t)^2-1.))sqrt((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.))+1.2sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.)))sqrt(sin(1.3+1.4x+1.5t)-1.)sqrt(sin(1.3+1.4x+1.5t)+1.))/(sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.)))^(1/3))^2

:p[20] := -1.0714+.71429(-.64030((-1.1arctan(1/sqrt(sec(1.3+1.4x+1.5t)^2-1.))sqrt(sec(1.3+1.4x+1.5t)-1.)sqrt(sec(1.3+1.4x+1.5t)+1.)+1.2sqrt(sec(1.3+1.4x+1.5t)^2-1.))/sqrt(sec(1.3+1.4x+1.5t)^2-1.))^(1/3)-(1.1091I)((-1.1arctan(1/sqrt(sec(1.3+1.4x+1.5t)^2-1.))sqrt(sec(1.3+1.4x+1.5t)-1.)sqrt(sec(1.3+1.4x+1.5t)+1.)+1.2sqrt(sec(1.3+1.4x+1.5t)^2-1.))/sqrt(sec(1.3+1.4x+1.5t)^2-1.))^(1/3))^2
:p[21] := -1.0714+.71429(-.64030((-1.1arctan(1/sqrt(sec(1.3+1.4x+1.5t)^2-1.))sqrt(sec(1.3+1.4x+1.5t)-1.)sqrt(sec(1.3+1.4x+1.5t)+1.)+1.2sqrt(sec(1.3+1.4x+1.5t)^2-1.))/sqrt(sec(1.3+1.4x+1.5t)^2-1.))^(1/3)+(1.1091I)((-1.1arctan(1/sqrt(sec(1.3+1.4x+1.5t)^2-1.))sqrt(sec(1.3+1.4x+1.5t)-1.)sqrt(sec(1.3+1.4x+1.5t)+1.)+1.2sqrt(sec(1.3+1.4x+1.5t)^2-1.))/sqrt(sec(1.3+1.4x+1.5t)^2-1.))^(1/3))^2
:p[29] := -1.0714+.73794((1.1ln(cos(1.3+1.4x+1.5t)+sqrt(cos(1.3+1.4x+1.5t)^2-1.))sqrt((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.))+1.2sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((cos(1.3+1.4x+1.5t)-1.)(cos(1.3+1.4x+1.5t)+1.)))sqrt(cos(1.3+1.4x+1.5t)-1.)sqrt(cos(1.3+1.4x+1.5t)+1.))/(sqrt(cos(1.3+1.4x+1.5t)+1.)sqrt(cos(1.3+1.4x+1.5*t)-1.)))^(2/3)
:p[31] := -1.0714+.73794((1.1ln(sin(1.3+1.4x+1.5t)+sqrt(sin(1.3+1.4x+1.5t)^2-1.))sqrt((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.))+1.2sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5t)-1.))(1.+sqrt(1/((sin(1.3+1.4x+1.5t)-1.)(sin(1.3+1.4x+1.5t)+1.)))sqrt(sin(1.3+1.4x+1.5t)-1.)sqrt(sin(1.3+1.4x+1.5t)+1.))/(sqrt(sin(1.3+1.4x+1.5t)+1.)sqrt(sin(1.3+1.4x+1.5*t)-1.)))^(2/3)
:p[32] := -1.0714+1.1714((-1.1arctan(1/sqrt(csc(1.3+1.4x+1.5t)^2-1.))sqrt(csc(1.3+1.4x+1.5t)-1.)sqrt(csc(1.3+1.4x+1.5t)+1.)+1.2sqrt(csc(1.3+1.4x+1.5t)^2-1.))/sqrt(csc(1.3+1.4x+1.5*t)^2-1.))^(2/3)
:p[33] := -1.0714+1.1714((-1.1arctan(1/sqrt(sec(1.3+1.4x+1.5t)^2-1.))sqrt(sec(1.3+1.4x+1.5t)-1.)sqrt(sec(1.3+1.4x+1.5t)+1.)+1.2sqrt(sec(1.3+1.4x+1.5t)^2-1.))/sqrt(sec(1.3+1.4x+1.5*t)^2-1.))^(2/3)
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;雅可比橢圓函數展开

参考文献

*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