Maple/非线性偏微分方程

非线性偏微分方程

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：

$(u_t+u{u}_x{)}_x=\frac{1}{2}{u}_x^2$

以下是利用Maple的软件包TWSolution求亨特 - 萨克斯顿方程的行波解

Maple TWSolution

restart;with(PDEtools, TWSolutions, declare): sys := {diff(u(x, t), x, t)+(diff(u(x, t), x))^2+u(x, t)*(diff(u(x, t), x, x)) = (1/2)*(diff(u(x, t), x))^2}: > > TWS := TWSolutions(sys, extended); > > > _C1 := 1.2; _C3 := 2.3; _C4 := .88; _C5 := 1.5; _C2 := .88;

> f1 := op(TWS[1]);

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

>f2 := op(TWS[2])

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

> f3 := op(TWS[3]); $u(x,t)=-1.7045454545454545454+0.28409090909090909090(6.336ln(tanh(2.3+0.88x+1.5t)-1)-6.336ln(tanh(2.3+0.88x+1.5t)+1)+9.2928{)}^{(}2/3)$

> with(plots);

> animate(complexplot3d, [f[1], x = -15-I .. 15+I, title = "Hunter Saxton nlpde animation", titlefont = [TIMES, ROMAN, 18], labels = [x, "", ""]], t = 0 .. 30);

Hunter Saxton equation Maple TWSolution travelling wave solution1

> animate(complexplot3d, [f[2], x = -15-I .. 15+I, title = "Hunter Saxton nlpde animation", titlefont = [TIMES, ROMAN, 18], labels = [x, "", ""]], t = 0 .. 30);

> animate(complexplot3d, [f[3], x = -15-I .. 15+I, title = "Hunter Saxton nlpde animation", titlefont = [TIMES, ROMAN, 18], labels = [x, "", ""]], t = 0 .. 30);

Hunter Saxton equation travelling wave solution3 with Maple TWSolution package