Suppose $q(x, t)$ satisfies the $m K d V$ equation

$$q_{t}+q_{x x x}+6 q^{2} q_{x}=0$$

where $q_{t}=\partial q / \partial t$ etc.

(a) Find the 1-soliton solution.

[You may use, without proof, the indefinite integral $\int \frac{d x}{x \sqrt{1-x^{2}}}=-\operatorname{arcsech} x$.]

(b) Express the self-similar solution of the $\mathrm{KKdV}$ equation in terms of a solution, denoted by $v(z)$, of the Painlevé II equation.

(c) Using the Ansatz

$$\frac{d v}{d z}+i v^{2}-\frac{i}{6} z=0$$

find a particular solution of the mKdV equation in terms of a solution of the Airy equation

$$\frac{d^{2} \Psi}{d z^{2}}+\frac{z}{6} \Psi=0$$