Paper 2, Section II, 34D

Integrable Systems | Part II, 2021

(a) Explain briefly how the linear operators L=x2+u(x,t)L=-\partial_{x}^{2}+u(x, t) and A=4x33uxA=4 \partial_{x}^{3}-3 u \partial_{x}- 3xu3 \partial_{x} u can be used to give a Lax-pair formulation of the KdV\mathrm{KdV} equation ut+uxxx6uux=0u_{t}+u_{x x x}-6 u u_{x}=0.

(b) Give a brief definition of the scattering data

Su(t)={{R(k,t)}kR,{κn(t)2,cn(t)}n=1N}\mathcal{S}_{u(t)}=\left\{\{R(k, t)\}_{k \in \mathbb{R}},\left\{-\kappa_{n}(t)^{2}, c_{n}(t)\right\}_{n=1}^{N}\right\}

attached to a smooth solution u=u(x,t)u=u(x, t) of the KdV equation at time tt. [You may assume u(x,t)u(x, t) to be rapidly decreasing in xx.] State the time dependence of κn(t)\kappa_{n}(t) and cn(t)c_{n}(t), and derive the time dependence of R(k,t)R(k, t) from the Lax-pair formulation.

(c) Show that

F(x,t)=n=1Ncn(t)2eκn(t)x+12πR(k,t)eikxdkF(x, t)=\sum_{n=1}^{N} c_{n}(t)^{2} e^{-\kappa_{n}(t) x}+\frac{1}{2 \pi} \int_{-\infty}^{\infty} R(k, t) e^{i k x} d k

satisfies tF+8x3F=0\partial_{t} F+8 \partial_{x}^{3} F=0. Now let K(x,y,t)K(x, y, t) be the solution of the equation

K(x,y,t)+F(x+y,t)+xK(x,z,t)F(z+y,t)dz=0K(x, y, t)+F(x+y, t)+\int_{x}^{\infty} K(x, z, t) F(z+y, t) d z=0

and let u(x,t)=2xϕ(x,t)u(x, t)=-2 \partial_{x} \phi(x, t), where ϕ(x,t)=K(x,x,t)\phi(x, t)=K(x, x, t). Defining G(x,y,t)G(x, y, t) by G=G= (x2y2u(x,t))K(x,y,t)\left(\partial_{x}^{2}-\partial_{y}^{2}-u(x, t)\right) K(x, y, t), show that

G(x,y,t)+xG(x,z,t)F(z+y,t)dz=0G(x, y, t)+\int_{x}^{\infty} G(x, z, t) F(z+y, t) d z=0

(d) Given that K(x,y,t)K(x, y, t) obeys the equations

(x2y2)KuK=0(t+4x3+4y3)K3(xu)K6uxK=0\begin{aligned} \left(\partial_{x}^{2}-\partial_{y}^{2}\right) K-u K &=0 \\ \left(\partial_{t}+4 \partial_{x}^{3}+4 \partial_{y}^{3}\right) K-3\left(\partial_{x} u\right) K-6 u \partial_{x} K &=0 \end{aligned}

where u=u(x,t)u=u(x, t), deduce that

tK+(x+y)3K3u(x+y)K=0\partial_{t} K+\left(\partial_{x}+\partial_{y}\right)^{3} K-3 u\left(\partial_{x}+\partial_{y}\right) K=0

and hence that uu solves the KdV\mathrm{KdV} equation.

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