The University of SydneySchool of Mathematics and Statistics

Solutions to Integrable Systems

MATH440x: Applied Mathematics Honours Semester 1, 2012

Lecturer: Nalini Joshi (Carslaw 720) http://www.maths.usyd.edu.au/u/nalini/AM4/is.html

1. Consider the modified Korteweg-de Vries equation (mKdV) in the following form (slightlydifferent from lectures)

wt + 6w2wx + wxxx = 0. (1)

(a) Deduce a solitary wave solution of this equation and find the relationship betweenits height and speed.

Solution:

w(x, t) =√csech(

√c(x− c t− x0))

The speed is c and height is√c.

(b) Under the boundary conditions |w| → 0 as |x| → ∞, write down the first conservedquantity for Equation (1), and verify by explicit integration that this is indeedconserved for the solution you found in part (a).

Solution: The conserved quantity∫∞−∞wdx = 2π

√c.

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