Mathematics Tripos Papers

A4.22

Nonlinear Waves and Integrable Systems
Part II, 2003

Let Φ+(t),Φ−(t)\Phi^{+}(t), \Phi^{-}(t) denote the boundary values of functions which are analytic inside and outside a disc of radius 12\frac{1}{2} centred at the origin. Let CC denote the boundary of this disc.

Suppose that Φ+,Φ−\Phi^{+}, \Phi^{-}satisfy the jump condition

Φ+(t)=tt2−1Φ−(t)+t3−t2+1t2−t,t∈C.\Phi^{+}(t)=\frac{t}{t^{2}-1} \Phi^{-}(t)+\frac{t^{3}-t^{2}+1}{t^{2}-t}, \quad t \in C .

(a) Show that the associated index is 1 .

(b) Find the canonical solution of the homogeneous problem, i.e. the solution satisfying

X(z)∼z−1,z→∞X(z) \sim z^{-1}, \quad z \rightarrow \infty

(c) Find the general solution of the Riemann-Hilbert problem satisfying the above jump condition as well as

Φ(z)=O(z−1),z→∞\Phi(z)=O\left(z^{-1}\right), \quad z \rightarrow \infty

(d) Use the above result to solve the linear singular integral problem

(t2+t−1)ϕ(t)+t2−t−1πi∮Cϕ(τ)τ−tdτ=2(t3−t2+1)(t+1)t,t∈C.\left(t^{2}+t-1\right) \phi(t)+\frac{t^{2}-t-1}{\pi i} \oint_{C} \frac{\phi(\tau)}{\tau-t} d \tau=\frac{2\left(t^{3}-t^{2}+1\right)(t+1)}{t}, \quad t \in C .