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Many Faces of Solitons
Kanehisa Takasaki (Kyoto University)
KdV equation --
Modified KdV equation --
Modified KdV equation vt + vxxx + 6v2vx= 0
The modified KdV equation differs from the original KdV equation
in the last nonlinear term only. This, however, causes several
substantial differences; the superficial similarity is rather
inessential. On the other hand, these two equations are linked at a
deeper level by the so called "Miura transformation".
- 2-soliton solution
Here is an animation of the 2-soliton solution of the
modified KdV equation. The profile and the process
of collision are qualitatively very similar to the
2-soliton solution of the KdV equation. The analytical
expression of the solutions, however, are completely
[profile of 2-soliton solution before collision]
solution: v = 2*(g_x * f - g * f_x)/(f^2 + g^2),
f = 1 - c12*Exp[eta1+eta2]/(4*k1*k2),
g = Exp[eta1]/(2*k1) + Exp[eta2]/(2*k2),
eta1 = k1*(x - x1) - k1^3*t, eta2 = k2*(x - x2) - k2^3*t,
c12 = (k1-k2)^2 / (k1+k2)^2.
parameters: k1 = 2, k2 = 1, x1 = -8, x2 = -2.
- Emergence of solitons from sinusoidal wave
Here is an animation of a numerical solution that
imitates the computation of Zabusky and Kruskal.
The initial value is a sinusoidal wave. The result
is half similar to the KdV equation, but also
considerably different. Unlike the case of the
KdV equation, two trains of solitons emerge from
the top and bottom of the sinusoidal wave.
[two soliton trains developed from sinusoidal initial value]