Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 31E / iii

Table of contents
- Solution iii

Solution

Balancing the time derivative, cubic nonlinearity and third spatial derivative gives

x \sim t^{1/3}

and

u \sim t^{- 1/3}

. Choose

ζ = x / (3 t)^{1/3}

and

u = (3 t)^{- 1/3} w (ζ)

. Then

u_{t} = (3 t)^{- 4/3} (- w - ζ w^{'}), 6 u^{2} u_{x} = (3 t)^{- 4/3} 6 w^{2} w^{'}, u_{xxx} = (3 t)^{- 4/3} w^{'''}, u / t = (3 t)^{- 4/3} 3 w .

Thus the similarity solution is

w^{'''} + 6 w^{2} w^{'} - ζ w^{'} + 2 w = 0.

The extra spherical term changes the coefficient of

w

. This third-order equation cannot be integrated directly into the standard second-order Painlevé II equation of the ordinary modified KdV reduction.

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