亨特 - 薩克斯頓方程

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：

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

亨特 - 薩克斯頓方程 Maple 動畫

解析解

亨特 - 薩克斯頓方程有解析解^[1]：

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

此外，Maple軟體包 TWSolution 給出多個行波解：

tanh 展開法

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

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

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

其中 f1、f2 是重解，因此，只有兩個獨立的行波解。

Hunter Saxton equation Maple TWSolution travelling wave solution1

Hunter Saxton equation travelling wave solution3 with Maple TWSolution package

sech arctan 展開法

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

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

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  f1
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  f2
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  f3
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  f4
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  f5
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  f6

File:Hunter Saxton extended plot1.gif

綜合展開

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

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

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

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

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

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

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

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

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

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

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

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  File:Hunter Saxton eq traveling wave plot 12.gif
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  File:Hunter Saxton eq traveling wave plot 13.gif
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  File:Hunter Saxton eq traveling wave plot 16.gif
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  File:Hunter Saxton eq traveling wave plot 17.gif
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  File:Hunter Saxton eq traveling wave plot 18.gif
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  File:Hunter Saxton eq traveling wave plot 19.gif
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  File:Hunter Saxton eq traveling wave plot 20.gif
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  File:Hunter Saxton eq traveling wave plot 21.gif
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  File:Hunter Saxton eq traveling wave plot 29.gif
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  File:Hunter Saxton eq traveling wave plot 31.gif
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  File:Hunter Saxton eq traveling wave plot 32.gif
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  File:Hunter Saxton eq traveling wave plot 33.gif

雅可比橢圓函數展開

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  Hunter Saxton traveling wave Jacobi function plot 35.gif
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  Jacobi function plot 35
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  Hunter Saxton traveling wave Jacobi function plot 36.gif
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  Hunter Saxton traveling wave Jacobi function plot 41.gif
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  Hunter Saxton traveling wave Jacobi function plot 43.gif
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  Hunter Saxton traveling wave Jacobi function plot 46.gif