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：

亨特 - 萨克斯顿方程 Maple 动画

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

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

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

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

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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

综合展开

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

${\displaystyle }$

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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

雅可比橢圓函數展开

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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

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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2. *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
3. 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
4. 王东明著 《消去法及其应用》 科学出版社 2002
5. *何青 王丽芬编著 《Maple 教程》 科学出版社 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

維基教科書中的相關電子教程：亨特 - 萨克斯顿方程