亨特_-_萨克斯顿方程

**亨特 - 薩克斯頓方程**(Hunter–Saxton equation) 是一個模擬向列型液晶中波動傳播的非線性偏微分方程：

　
: $(u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}$

## 解析解

亨特 - 薩克斯頓方程有解析解：

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

此外，Maple 軟體包 TWSolution 給出多個行波解：

**tanh 展開法**

$$
f1:=u(x,t)=-1.7045454545454545454+1.1363636363636363636*(-(1/4)*(6.336*ln(tanh(2.3+.88*x+1.5*t)-1)-6.336*ln(tanh(2.3+.88*x+1.5*t)+1)+9.2928)^{(}1/3)-(1/4*I)*sqrt(3)*(6.336*ln(tanh(2.3+.88*x+1.5*t)-1)-6.336*ln(tanh(2.3+.88*x+1.5*t)+1)+9.2928)^{(}1/3))^{2}
$$

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

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

其中 f1、f2 是重解，因此，只有兩個獨立的行波解。

**sech arctan 展開法**

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

 $f[2]:=-.20000000000000000000+(1/3)*(-(1/4)*(-(36*(1.3*arctan(1/sqrt(sech(2+3*x+.6*t)^{2}-1))*sqrt(sech(2+3*x+.6*t)+1)*sqrt(sech(2+3*x+.6*t)-1)-2.6*sqrt(sech(2+3*x+.6*t)^{2}-1)))/sqrt(sech(2+3*x+.6*t)^{2}-1))^{(}1/3)-(1/4*I)*sqrt(3)*(-(36*(1.3*arctan(1/sqrt(sech(2+3*x+.6*t)^{2}-1))*sqrt(sech(2+3*x+.6*t)+1)*sqrt(sech(2+3*x+.6*t)-1)-2.6*sqrt(sech(2+3*x+.6*t)^{2}-1)))/sqrt(sech(2+3*x+.6*t)^{2}-1))^{(}1/3))^{2}$
 $f[3]:=-.20000000000000000000+(1/3)*(-(1/4)*(-(36*(1.3*arctan(1/sqrt(sech(2+3*x+.6*t)^{2}-1))*sqrt(sech(2+3*x+.6*t)+1)*sqrt(sech(2+3*x+.6*t)-1)-2.6*sqrt(sech(2+3*x+.6*t)^{2}-1)))/sqrt(sech(2+3*x+.6*t)^{2}-1))^{(}1/3)+(1/4*I)*sqrt(3)*(-(36*(1.3*arctan(1/sqrt(sech(2+3*x+.6*t)^{2}-1))*sqrt(sech(2+3*x+.6*t)+1)*sqrt(sech(2+3*x+.6*t)-1)-2.6*sqrt(sech(2+3*x+.6*t)^{2}-1)))/sqrt(sech(2+3*x+.6*t)^{2}-1))^{(}1/3))^{2}$
 $f[4]:=-.20000000000000000000+(1/3)*(-(1/4)*(187.2+140.4*x+28.08*t)^{(}1/3)-(1/4*I)*sqrt(3)*(187.2+140.4*x+28.08*t)^{(}1/3))^{2}$
 $f[5]:=-.20000000000000000000+(1/3)*(-(1/4)*(187.2+140.4*x+28.08*t)^{(}1/3)+(1/4*I)*sqrt(3)*(187.2+140.4*x+28.08*t)^{(}1/3))^{2}$
 $f[6]:=-.20000000000000000000+(1/12)*(187.2+140.4*x+28.08*t)^{(}2/3)$
 $<f[7]:=-1.6666666666666666667*_{C}6+1.6666666666666666667*(-(1/4)*(9.36*Intat(1/sqrt(4*_{a}^{4}-5*_{a}^{2}+1),_{a}=JacobiSN(3+.6*x+_{C}6*t,2))+18.72)^{(}1/3)-(1/4*I)*sqrt(3)*(9.36*Intat(1/sqrt(4*_{a}^{4}-5*_{a}^{2}+1),_{a}=JacobiSN(3+.6*x+_{C}6*t,2))+18.72)^{(}1/3))^{2}$
 $f[8]:=-1.6666666666666666667*_{C}6+1.6666666666666666667*(-(1/4)*(9.36*Intat(1/sqrt(4*_{a}^{4}-5*_{a}^{2}+1),_{a}=JacobiSN(3+.6*x+_{C}6*t,2))+18.72)^{(}1/3)+(1/4*I)*sqrt(3)*(9.36*Intat(1/sqrt(4*_{a}^{4}-5*_{a}^{2}+1),_{a}=JacobiSN(3+.6*x+_{C}6*t,2))+18.72)^{(}1/3))^{2}$
 $f[9]:=-1.6666666666666666667*_{C}6+.41666666666666666668*(9.36*Intat(1/sqrt(4*_{a}^{4}-5*_{a}^{2}+1),_{a}=JacobiSN(3+.6*x+_{C}6*t,2))+18.72)^{(}2/3)$

 

File:Hunter Saxton extended plot1.gif

**綜合展開**

$$
p[12]:=-1.0714+.71429*(-.50821*((1.1*ln(cos(1.3+1.4*x+1.5*t)+sqrt(cos(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt((cos(1.3+1.4*x+1.5*t)-1.)*(cos(1.3+1.4*x+1.5*t)+1.))+1.2*sqrt(cos(1.3+1.4*x+1.5*t)+1.)*sqrt(cos(1.3+1.4*x+1.5*t)-1.))*(1.+sqrt(1/((cos(1.3+1.4*x+1.5*t)-1.)*(cos(1.3+1.4*x+1.5*t)+1.)))*sqrt(cos(1.3+1.4*x+1.5*t)-1.)*sqrt(cos(1.3+1.4*x+1.5*t)+1.))/(sqrt(cos(1.3+1.4*x+1.5*t)+1.)*sqrt(cos(1.3+1.4*x+1.5*t)-1.)))^{(}1/3)-(.88026*I)*((1.1*ln(cos(1.3+1.4*x+1.5*t)+sqrt(cos(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt((cos(1.3+1.4*x+1.5*t)-1.)*(cos(1.3+1.4*x+1.5*t)+1.))+1.2*sqrt(cos(1.3+1.4*x+1.5*t)+1.)*sqrt(cos(1.3+1.4*x+1.5*t)-1.))*(1.+sqrt(1/((cos(1.3+1.4*x+1.5*t)-1.)*(cos(1.3+1.4*x+1.5*t)+1.)))*sqrt(cos(1.3+1.4*x+1.5*t)-1.)*sqrt(cos(1.3+1.4*x+1.5*t)+1.))/(sqrt(cos(1.3+1.4*x+1.5*t)+1.)*sqrt(cos(1.3+1.4*x+1.5*t)-1.)))^{(}1/3))^{2}
$$

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

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

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

 

　
: $p[20]:=-1.0714+.71429*(-.64030*((-1.1*arctan(1/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt(sec(1.3+1.4*x+1.5*t)-1.)*sqrt(sec(1.3+1.4*x+1.5*t)+1.)+1.2*sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))^{(}1/3)-(1.1091*I)*((-1.1*arctan(1/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt(sec(1.3+1.4*x+1.5*t)-1.)*sqrt(sec(1.3+1.4*x+1.5*t)+1.)+1.2*sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))^{(}1/3))^{2}$

　
: $p[21]:=-1.0714+.71429*(-.64030*((-1.1*arctan(1/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt(sec(1.3+1.4*x+1.5*t)-1.)*sqrt(sec(1.3+1.4*x+1.5*t)+1.)+1.2*sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))^{(}1/3)+(1.1091*I)*((-1.1*arctan(1/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt(sec(1.3+1.4*x+1.5*t)-1.)*sqrt(sec(1.3+1.4*x+1.5*t)+1.)+1.2*sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))^{(}1/3))^{2}$

　
: $p[29]:=-1.0714+.73794*((1.1*ln(cos(1.3+1.4*x+1.5*t)+sqrt(cos(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt((cos(1.3+1.4*x+1.5*t)-1.)*(cos(1.3+1.4*x+1.5*t)+1.))+1.2*sqrt(cos(1.3+1.4*x+1.5*t)+1.)*sqrt(cos(1.3+1.4*x+1.5*t)-1.))*(1.+sqrt(1/((cos(1.3+1.4*x+1.5*t)-1.)*(cos(1.3+1.4*x+1.5*t)+1.)))*sqrt(cos(1.3+1.4*x+1.5*t)-1.)*sqrt(cos(1.3+1.4*x+1.5*t)+1.))/(sqrt(cos(1.3+1.4*x+1.5*t)+1.)*sqrt(cos(1.3+1.4*x+1.5*t)-1.)))^{(}2/3)$

　
: $p[31]:=-1.0714+.73794*((1.1*ln(sin(1.3+1.4*x+1.5*t)+sqrt(sin(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt((sin(1.3+1.4*x+1.5*t)-1.)*(sin(1.3+1.4*x+1.5*t)+1.))+1.2*sqrt(sin(1.3+1.4*x+1.5*t)+1.)*sqrt(sin(1.3+1.4*x+1.5*t)-1.))*(1.+sqrt(1/((sin(1.3+1.4*x+1.5*t)-1.)*(sin(1.3+1.4*x+1.5*t)+1.)))*sqrt(sin(1.3+1.4*x+1.5*t)-1.)*sqrt(sin(1.3+1.4*x+1.5*t)+1.))/(sqrt(sin(1.3+1.4*x+1.5*t)+1.)*sqrt(sin(1.3+1.4*x+1.5*t)-1.)))^{(}2/3)$

　
: $p[32]:=-1.0714+1.1714*((-1.1*arctan(1/sqrt(csc(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt(csc(1.3+1.4*x+1.5*t)-1.)*sqrt(csc(1.3+1.4*x+1.5*t)+1.)+1.2*sqrt(csc(1.3+1.4*x+1.5*t)^{2}-1.))/sqrt(csc(1.3+1.4*x+1.5*t)^{2}-1.))^{(}2/3)$

　
: $p[33]:=-1.0714+1.1714*((-1.1*arctan(1/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt(sec(1.3+1.4*x+1.5*t)-1.)*sqrt(sec(1.3+1.4*x+1.5*t)+1.)+1.2*sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))/sqrt(sec(1.3+1.4*x+1.5*t)^{2}-1.))^{(}2/3)$

　
: $<p[34]:=-1.0714+1.1714*((-1.1*arctan(1/sqrt(sech(1.3+1.4*x+1.5*t)^{2}-1.))*sqrt(sech(1.3+1.4*x+1.5*t)-1.)*sqrt(sech(1.3+1.4*x+1.5*t)+1.)+1.2*sqrt(sech(1.3+1.4*x+1.5*t)^{2}-1.))/sqrt(sech(1.3+1.4*x+1.5*t)^{2}-1.))^{(}2/3)$

　
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**雅可比橢圓函數展開**

## 參考文獻

1. \* 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
2. \* 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
3. 李志斌編著 《非線性數學物理方程的行波解》 科學出版社
4. 王東明著 《消去法及其應用》 科學出版社 2002
5. \* 何青 王麗芬編著 《Maple 教程》 科學出版社 2010 ISBN 9787030177445
6. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
7. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
8. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
9. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
10. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
11. Dongming Wang, Elimination Practice,Imperial College Press 2004
12. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
13. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759