亨特 - 萨克斯顿方程

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：

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

亨特 - 萨克斯顿方程 Maple 动画

解析解

亨特 - 萨克斯顿方程有解析解[1]：

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

此外，Maple软件包 TWSolution 给出多个行波解：

tanh 展开法

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

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

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

其中 f1、f2 是重解，因此，只有两个独立的行波解。

Hunter Saxton equation Maple TWSolution travelling wave solution1

Hunter Saxton equation travelling wave solution3 with Maple TWSolution package

sech arctan 展开法

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

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

f1

f2

f3

f4

f5

f6

File:Hunter Saxton extended plot1.gif

综合展开

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

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

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

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

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

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

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

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

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

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

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

File:Hunter Saxton eq traveling wave plot 12.gif

File:Hunter Saxton eq traveling wave plot 13.gif

File:Hunter Saxton eq traveling wave plot 16.gif

File:Hunter Saxton eq traveling wave plot 17.gif

File:Hunter Saxton eq traveling wave plot 18.gif

File:Hunter Saxton eq traveling wave plot 19.gif

File:Hunter Saxton eq traveling wave plot 20.gif

File:Hunter Saxton eq traveling wave plot 21.gif

File:Hunter Saxton eq traveling wave plot 29.gif

File:Hunter Saxton eq traveling wave plot 31.gif

File:Hunter Saxton eq traveling wave plot 32.gif

File:Hunter Saxton eq traveling wave plot 33.gif

雅可比橢圓函數展开

Hunter Saxton traveling wave Jacobi function plot 35.gif

Jacobi function plot 35

Hunter Saxton traveling wave Jacobi function plot 36.gif

Hunter Saxton traveling wave Jacobi function plot 41.gif

Hunter Saxton traveling wave Jacobi function plot 43.gif

Hunter Saxton traveling wave Jacobi function plot 46.gif

参考文献

1. Anders Samuelsen Nordli On the Hunter-Saxton equation，Norwegian University of Science and Technology，p38, June 2012

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