正向Hirota双线性化(p=2)

u → f → D

推广的双线性化(p=3)

D→f

逆向双线性化

f→u

#正向双线性化
restart:
with(PDETools): alias(u = u(x, y, z, t), f = f(x, y, z, t), A = A(x, y, z, t))

#输入方程
eq1 := 3*(Diff((diff(u, x))*(diff(u, y)), x))+diff(u, t, y)+diff(u, x, t)+diff(u, t, t)-(diff(u, z, z))
#(1)代入u=2(A)x (A=lnf 此处先不代入化简,对应于前面的手算图纸）
eq2 = u = 2*(diff(A, x))
eq3 := dsubs(eq2, eq1)
#(2)对x积分
eq4 := int(eq3, x)
#(3)化简(两边同乘f^2)
eq5 := simplify(dsubs(A = ln(f), eq4*f^2))

#简化一下方便肉眼观查(此处也可以调用fafa函数包)
subs({(diff(f, z))^2 = f[z], diff(f, t) = f[t], diff(f, x) = f[x], diff(f, y) = f[y], diff(f, t, t) = f[tt], diff(f, x, t) = f[xt], diff(f, x, x) = f[xx], diff(f, y, t) = f[yt], diff(f, y, x) = f[yx], diff(f, z, z) = f[zz], diff(f, x, x, x) = f[xxx], diff(f, y, x, x) = f[yxx], diff(f, y, x, x, x) = f[yxxx]}, eq5)
#去系数
simplify((1/2)*%)

#逆向双线性化
restart:
with(PDETools): alias(u = u(x, y, z, t), f = f(x, y, z, t), A = A(x, y, z, t))

#输入方程
eq1 := 3*(Diff((diff(u, x))*(diff(u, y)), x))+diff(u, t, y)+diff(u, x, t)+diff(u, t, t)-(diff(u, z, z))
#(1)除以f^2
#(2)对x求导
eq2 := diff(eq1/f^2, x)
#(3)u=2(lnf)x逆运算 f=。。。参见手算图纸
eq3 := f = exp(int((1/2)*u, x))
AA := dsubs(eq3, eq2)

#令Vy=Ux
BB := subs(int((1/2)*(diff(u, y)), x) = (1/2)*v, AA)
#简化一下方便肉眼观查(此处也可以调用fafa函数包)
subs({diff(u, x) = u[x], diff(u, y) = u[y], diff(u, t, t) = u[tt], diff(u, x, t) = u[xt], diff(u, x, x) = u[xx], diff(u, y, t) = u[yt], diff(u, y, x) = u[yx], diff(u, z, z) = u[zz]}, 2*BB)