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Part II — Integrable Systems — Part II | Integrable Systems | Year 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2021 58 Paper 1, Section II 33D Integrable Systems (a) Let U(z, z,¯ λ) and V (z, z,¯ λ) be matrix-valued functions, whilst ψ(z, z,¯ λ) is a vector-valued function. Show that the linear system ∂zψ = Uψ , ∂z¯ψ = V ψ is over-determined and derive a consistency condition on U, V that is necessary for there to be non-trivial solutions. (b) Suppose that 1 λ ∂ u e u 1 ∂ u λeu U = z − and V = − z¯ , 2λ eu λ ∂ u 2 λe u ∂ u − z − z¯ where u(z, z¯) is a scalar function. Obtain a partial differential equation for u that is equivalent to your consistency condition from part (a). (c) Now let z = x + iy and suppose u is independent of y. Show that the trace of (U V )n is constant for all positive integers n. Hence, or otherwise, construct a non-trivial − first integral of the equation d2φ = 4 sinh φ , where φ = φ(x) . dx2 Part II, 2021 List of Questions 2021 59 Paper 2, Section II 34D Integrable Systems (a) Explain briefly how the linear operators L = ∂2 +u(x, t) and A = 4∂3 3u∂ − x x − x − 3∂ u can be used to give a Lax-pair formulation of the KdV equation u +u 6uu = 0 . x t xxx − x (b) Give a brief definition of the scattering data 2 N u(t) = R(k, t) k R , κn(t) , cn(t) n=1 S { } ∈ {− } attached to a smooth solution u = u(x, t) of the KdV equation at time t. [You may assume u(x, t) to be rapidly decreasing in x.] State the time dependence of κn(t) and cn(t), and derive the time dependence of R(k, t) from the Lax-pair formulation. (c) Show that N 2 κn(t)x 1 ∞ ikx F (x, t) = c (t) e− + R(k, t) e dk n 2π n=1 X Z−∞ 3 satisfies ∂tF + 8∂xF = 0. Now let K(x, y, t) be the solution of the equation K(x, y, t) + F (x + y, t) + ∞ K(x, z, t)F (z + y, t) dz = 0 Zx and let u(x, t) = 2∂ φ(x, t), where φ(x, t) = K(x, x, t). Defining G(x, y, t) by G = − x ∂2 ∂2 u(x, t) K(x, y, t), show that x − y − G(x, y, t) + ∞ G(x, z, t)F (z + y, t) dz = 0 . Zx (d) Given that K(x, y, t) obeys the equations (∂2 ∂2)K uK = 0 , x − y − (∂ + 4∂3 + 4∂3)K 3(∂ u)K 6u ∂ K = 0 , t x y − x − x where u = u(x, t), deduce that ∂ K + (∂ + ∂ )3K 3u (∂ + ∂ )K = 0 , t x y − x y and hence that u solves the KdV equation. Part II, 2021 List of Questions [TURN OVER] 2021 60 Paper 3, Section II 32D Integrable Systems (a) Consider the group of transformations of R2 given by gs :(t, x) (t,˜ x˜) = 1 7→ (t, x + st), where s R. Show that this acts as a group of Lie symmetries for the equation ∈ d2x/dt2 = 0. 2 (b) Let (ψ1, ψ2) R and define ψ = ψ1 + iψ2. Show that the vector field ∈ ψ ∂ ψ ∂ generates the group of phase rotations gs : ψ eisψ . 1 ψ2 − 2 ψ1 2 → (c) Show that the transformations of R2 C defined by × 2 gs :(t, x, ψ) (t,˜ x,˜ ψ˜) = (t, x + st, ψ eisx+is t/2) 7→ form a one-parameter group generated by the vector field V = t∂ + x(ψ ∂ ψ ∂ ) = t∂ + ix(ψ∂ ψ∗∂ ) , x 1 ψ2 − 2 ψ1 x ψ − ψ∗ and find the second prolongation Pr(2)gs of the action of gs . Hence find the coefficients { } η0 and η11 in the second prolongation of V , (2) 0 1 00 01 11 pr V = t∂x + ixψ∂ψ +η ∂ψt +η ∂ψx +η ∂ψtt +η ∂ψxt +η ∂ψxx +complex conjugate . (d) Show that the group gs of transformations in part (c) acts as a group of Lie { } 1 2 2 symmetries for the nonlinear Schr¨odingerequation i∂tψ + 2 ∂xψ + ψ ψ = 0. Given that 2 | | aeia t/2 sech(ax) solves the nonlinear Schr¨odingerequation for any a R, find a solution ∈ which describes a solitary wave travelling at arbitrary speed s R. ∈ Part II, 2021 List of Questions 2020 53 Paper 1, Section II 33C Integrable Systems (a) Show that if L is a symmetric matrix (L = LT ) and B is skew-symmetric (B = BT ) then [B, L] = BL LB is symmetric. − − (b) Consider the real n n symmetric matrix × 0 a1 0 0 ......... 0 a1 0 a2 0 ......... 0 0 a2 0 a3 ......... 0 0 0 a ............ 0 L = 3 ........................ 0 ... ... ... ... ... an 2 0 − 0 ... ... ... ... an 2 0 an 1 − − 0 ............ 0 a 0 n 1 − (i.e. L = L = a for 1 i n 1, all other entries being zero) and the real n n i,i+1 i+1,i i 6 6 − × skew-symmetric matrix 0 0 a1 a2 0 ......... 0 0 0 0 a2 a3 ......... 0 a a 0 0 0 ......... 0 − 1 2 0 a a 0 ............ 0 B = − 2 3 ........................ 0 ............... 0 an 2 an 1 − − 0 ............ 0 0 0 0 ............ a a 0 0 n 2 n 1 − − − (i.e. B = B = a a for 1 i n 2, all other entries being zero). i,i+2 − i+2,i i i+1 6 6 − (i) Compute [B, L]. (ii) Assume that the aj are smooth functions of time t so the matrix L = L(t) also dL depends smoothly on t. Show that the equation dt = [B, L] implies that daj = f(aj 1, aj, aj+1) dt − for some function f which you should find explicitly. 1 1 (iii) Using the transformation aj = 2 exp[ 2 uj] show that duj 1 uj+1 uj 1 = e e − ( ) dt 2 − † for j = 1, . n 1. [Use the convention u = , a = 0, u = , a = 0.] − 0 −∞ 0 n −∞ n (iv) Deduce that given a solution of equation ( ), there exist matrices U(t) t R 1† { } ∈ depending on time such that L(t) = U(t)L(0)U(t)− , and explain how to obtain first integrals for ( ) from this. † Part II, 2020 List of Questions [TURN OVER] 2020 54 Paper 2, Section II 33C Integrable Systems (i) Explain how the inverse scattering method can be used to solve the initial value problem for the KdV equation u + u 6uu = 0 , u(x, 0) = u (x) , t xxx − x 0 including a description of the scattering data associated to the operator L = ∂2+u(x, t), u − x its time dependence, and the reconstruction of u via the inverse scattering problem. (ii) Solve the inverse scattering problem for the reflectionless case, in which the reflection coefficient R(k) is identically zero and the discrete scattering data consists of a single bound state, and hence derive the 1-soliton solution of KdV. (iii) Consider the direct and inverse scattering problems in the case of a small potential u(x) = q(x), with arbitrarily small: 0 < 1. Show that the reflection coefficient is given by e 2ikz R(k) = ∞ − q(z) dz + O(2) 2ik Z−∞ and verify that the solution of the inverse scattering problem applied to this reflection coefficient does indeed lead back to the potential u = q when calculated to first order in .[Hint: you may make use of the Fourier inversion theorem.] Part II, 2020 List of Questions 2020 55 Paper 3, Section II 32C Integrable Systems (a) Given a smooth vector field ∂ ∂ V = V (x, u) + φ(x, u) 1 ∂x ∂u on R2 define the prolongation of V of arbitrary order N. Calculate the prolongation of order two for the group SO(2) of transformations of R2 given for s R by ∈ u u cos s x sin s gs = − , x u sin s + x cos s and hence, or otherwise, calculate the prolongation of order two of the vector field 2 3 V = x∂ + u∂ . Show that both of the equations u = 0 and u = (1 + u ) 2 are − u x xx xx x invariant under this action of SO(2), and interpret this geometrically. (b) Show that the sine-Gordon equation ∂2u = sin u ∂X∂T s admits the group g s R, where { } ∈ X esX gs : T e sT 7→ − u u as a group of Lie point symmetries. Show that there is a group invariant solution of the form u(X, T ) = F (z) where z is an invariant formed from the independent variables, and hence obtain a second order equation for w = w(z) where exp[iF ] = w. Part II, 2020 List of Questions [TURN OVER] 2019 57 Paper 3, Section II 32C Integrable Systems Suppose ψs : (x, u) (˜x, u˜) is a smooth one-parameter group of transformations → acting on R2, with infinitesimal generator ∂ ∂ V = ξ(x, u) + η(x, u) . ∂x ∂u (a) Define the nth prolongation Pr(n) V of V , and show that n (n) ∂ Pr V = V + ηi , ∂u(i) i=1 where you should give an explicit formula to determine the ηi recursively in terms of ξ and η. (b) Find the nth prolongation of each of the following generators: ∂ ∂ ∂ V = ,V = x ,V = x2 . 1 ∂x 2 ∂x 3 ∂x (c) Given a smooth, real-valued, function u = u(x), the Schwarzian derivative is defined by, 3 2 uxuxxx 2 uxx S = S[u] := 2− .
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