Integrable Systems
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Paper 1 , Section II, 33D
comment(a) Let and be matrix-valued functions, whilst is a vector-valued function. Show that the linear system
is over-determined and derive a consistency condition on that is necessary for there to be non-trivial solutions.
(b) Suppose that
where is a scalar function. Obtain a partial differential equation for that is equivalent to your consistency condition from part (a).
(c) Now let and suppose is independent of . Show that the trace of is constant for all positive integers . Hence, or otherwise, construct a non-trivial first integral of the equation
Paper 2, Section II, 34D
comment(a) Explain briefly how the linear operators and can be used to give a Lax-pair formulation of the equation .
(b) Give a brief definition of the scattering data
attached to a smooth solution of the KdV equation at time . [You may assume to be rapidly decreasing in .] State the time dependence of and , and derive the time dependence of from the Lax-pair formulation.
(c) Show that
satisfies . Now let be the solution of the equation
and let , where . Defining by , show that
(d) Given that obeys the equations
where , deduce that
and hence that solves the equation.
Paper 3, Section II, 32D
comment(a) Consider the group of transformations of given by , where . Show that this acts as a group of Lie symmetries for the equation .
(b) Let and define . Show that the vector field generates the group of phase rotations .
(c) Show that the transformations of defined by
form a one-parameter group generated by the vector field
and find the second prolongation of the action of . Hence find the coefficients and in the second prolongation of ,
complex conjugate .
(d) Show that the group of transformations in part (c) acts as a group of Lie symmetries for the nonlinear Schrödinger equation . Given that solves the nonlinear Schrödinger equation for any , find a solution which describes a solitary wave travelling at arbitrary speed .
Paper 1, Section II, 33C
comment(a) Show that if is a symmetric matrix and is skew-symmetric then is symmetric.
(b) Consider the real symmetric matrix
(i.e. for , all other entries being zero) and the real skew-symmetric matrix
(i.e. for , all other entries being zero).
(i) Compute .
(ii) Assume that the are smooth functions of time so the matrix also depends smoothly on . Show that the equation implies that
for some function which you should find explicitly.
(iii) Using the transformation show that
for . [Use the convention ]
(iv) Deduce that given a solution of equation ( , there exist matrices depending on time such that , and explain how to obtain first integrals for from this.
Paper 2, Section II, 33C
comment(i) Explain how the inverse scattering method can be used to solve the initial value problem for the equation
including a description of the scattering data associated to the operator , its time dependence, and the reconstruction of via the inverse scattering problem.
(ii) Solve the inverse scattering problem for the reflectionless case, in which the reflection coefficient is identically zero and the discrete scattering data consists of a single bound state, and hence derive the 1-soliton solution of .
(iii) Consider the direct and inverse scattering problems in the case of a small potential , with arbitrarily small: . Show that the reflection coefficient is given by
and verify that the solution of the inverse scattering problem applied to this reflection coefficient does indeed lead back to the potential when calculated to first order in
Paper 3, Section II, 32C
comment(a) Given a smooth vector field
on define the prolongation of of arbitrary order .
Calculate the prolongation of order two for the group of transformations of given for by
and hence, or otherwise, calculate the prolongation of order two of the vector field . Show that both of the equations and are invariant under this action of , and interpret this geometrically.
(b) Show that the sine-Gordon equation
admits the group , where
as a group of Lie point symmetries. Show that there is a group invariant solution of the form where is an invariant formed from the independent variables, and hence obtain a second order equation for where .
Paper 1, Section II, C
commentLet be equipped with its standard Poisson bracket.
(a) Given a Hamiltonian function , write down Hamilton's equations for . Define a first integral of the system and state what it means that the system is integrable.
(b) Show that if then every Hamiltonian system is integrable whenever
Let be another phase space, equipped with its standard Poisson bracket. Suppose that is a Hamiltonian function for . Define and let the combined phase space be equipped with the standard Poisson bracket.
(c) Show that if and are both integrable, then so is , where the combined Hamiltonian is given by:
(d) Consider the -dimensional simple harmonic oscillator with phase space and Hamiltonian given by:
where . Using the results above, or otherwise, show that is integrable for .
(e) Is it true that every bounded orbit of an integrable system is necessarily periodic? You should justify your answer.
Paper 2, Section II, C
commentSuppose is a smooth, real-valued, function of which satisfies for all and as . Consider the Sturm-Liouville operator:
which acts on smooth, complex-valued, functions . You may assume that for any there exists a unique function which satisfies:
and has the asymptotic behaviour:
(a) By analogy with the standard Schrödinger scattering problem, define the reflection and transmission coefficients: . Show that . [Hint: You may wish to consider for suitable functions and
(b) Show that, if , there exists no non-trivial normalizable solution to the equation
Assume now that , such that and 0 as . You are given that the operator defined by:
satisfies:
(c) Show that form a Lax pair if the Harry Dym equation,
is satisfied. [You may assume .]
(d) Assuming that solves the Harry Dym equation, find how the transmission and reflection amplitudes evolve as functions of .
Paper 3, Section II, C
commentSuppose is a smooth one-parameter group of transformations acting on , with infinitesimal generator
(a) Define the prolongation of , and show that
where you should give an explicit formula to determine the recursively in terms of and .
(b) Find the prolongation of each of the following generators:
(c) Given a smooth, real-valued, function , the Schwarzian derivative is defined by,
Show that,
for where are real functions which you should determine. What can you deduce about the symmetries of the equations: (i) , (ii) , (iii) ?
Paper 1, Section II, A
commentLet be equipped with the standard symplectic form so that the Poisson bracket is given by:
for real-valued functions on . Let be a Hamiltonian function.
(a) Write down Hamilton's equations for , define a first integral of the system and state what it means that the system is integrable.
(b) State the Arnol'd-Liouville theorem.
(c) Define complex coordinates by , and show that if are realvalued functions on then:
(d) For an anti-Hermitian matrix with components , let . Show that:
where is the usual matrix commutator.
(e) Consider the Hamiltonian:
Show that is integrable and describe the invariant tori.
[In this question , and the summation convention is understood for these indices.]
Paper 2, Section II, A
comment(a) Let be two families of linear operators, depending on a parameter , which act on a Hilbert space with inner product , . Suppose further that for each is self-adjoint and that is anti-self-adjoint. State 's equation for the pair , and show that if it holds then the eigenvalues of are independent of .
(b) For , define the inner product:
Let be the operators:
where are smooth, real-valued functions. You may assume that the normalised eigenfunctions of are smooth functions of , which decay rapidly as for all .
(i) Show that if are smooth and rapidly decaying towards infinity then:
Deduce that the eigenvalues of are real.
(ii) Show that if Lax's equation holds for , then must satisfy the Boussinesq equation:
where are constants whose values you should determine. [You may assume without proof that the identity:
holds for smooth, rapidly decaying
Paper 3, Section II, A
commentSuppose is a smooth one-parameter group of transformations acting on .
(a) Define the generator of the transformation,
where you should specify and in terms of .
(b) Define the prolongation of and explicitly compute in terms of .
Recall that if is a Lie point symmetry of the ordinary differential equation:
then it follows that whenever .
(c) Consider the ordinary differential equation:
for a smooth function. Show that if generates a Lie point symmetry of this equation, then:
(d) Find all the Lie point symmetries of the equation:
where is an arbitrary smooth function.
Paper 1, Section II, A
commentDefine a Lie point symmetry of the first order ordinary differential equation 0. Describe such a Lie point symmetry in terms of the vector field that generates it.
Consider the -dimensional Hamiltonian system governed by the differential equation
Define the Poisson bracket . For smooth functions show that the associated Hamiltonian vector fields satisfy
If is a first integral of , show that the Hamiltonian vector field generates a Lie point symmetry of . Prove the converse is also true if has a fixed point, i.e. a solution of the form .
Paper 2, Section II, A
commentLet and be non-singular matrices depending on which are periodic in with period . Consider the associated linear problem
for the vector . On the assumption that these equations are compatible, derive the zero curvature equation for .
Let denote the matrix satisfying
where is the identity matrix. You should assume is unique. By considering , show that the matrix satisfies the Lax equation
Deduce that are first integrals.
By considering the matrices
show that the periodic Sine-Gordon equation has infinitely many first integrals. [You need not prove anything about independence.]
Paper 3, Section II, A
commentLet be a smooth solution to the equation
which decays rapidly as and let be the associated Schrödinger operator. You may assume and constitute a Lax pair for KdV.
Consider a solution to which has the asymptotic form
Find evolution equations for and . Deduce that is -independent.
By writing in the form
show that
Deduce that are first integrals of KdV.
By writing a differential equation for (with real), show that these first integrals are trivial when is even.
Paper 1, Section II, D
commentWhat does it mean for an evolution equation to be in Hamiltonian form? Define the associated Poisson bracket.
An evolution equation is said to be bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways, i.e.
for Hamiltonian operators and functionals . By considering the sequence defined by the recurrence relation
show that bi-Hamiltonian systems possess infinitely many first integrals in involution. [You may assume that can always be solved for , given .]
The Harry Dym equation for the function is
This equation can be written in Hamiltonian form with
Show that the Harry Dym equation possesses infinitely many first integrals in involution. [You need not verify the Jacobi identity if your argument involves a Hamiltonian operator.]
Paper 2, Section II, D
commentWhat does it mean for to describe a 1-parameter group of transformations? Explain how to compute the vector field
that generates such a 1-parameter group of transformations.
Suppose now . Define the th prolongation, , of and the vector field which generates it. If is defined by show that
where and are functions to be determined.
The curvature of the curve in the -plane is given by
Rotations in the -plane are generated by the vector field
Show that the curvature at a point along a plane curve is invariant under such rotations. Find two further transformations that leave invariant.
Paper 3, Section II, D
commentWhat is meant by an auto-Bäcklund transformation?
The sine-Gordon equation in light-cone coordinates is
where and is to be understood modulo . Show that the pair of equations
constitute an auto-Bäcklund transformation for (1).
By noting that is a solution to (1), use the transformation (2) to derive the soliton (or 'kink') solution to the sine-Gordon equation. Show that this solution can be expressed as
for appropriate constants and .
[Hint: You may use the fact that const.]
The following function is a solution to the sine-Gordon equation:
Verify that this represents two solitons travelling towards each other at the same speed by considering constant and taking an appropriate limit.
Paper 1, Section II, D
commentLet be an evolution equation for the function . Assume and all its derivatives decay rapidly as . What does it mean to say that the evolution equation for can be written in Hamiltonian form?
The modified KdV (mKdV) equation for is
Show that small amplitude solutions to this equation are dispersive.
Demonstrate that the mKdV equation can be written in Hamiltonian form and define the associated Poisson bracket ,} on the space of functionals of u. Verify that the Poisson bracket is linear in each argument and anti-symmetric.
Show that a functional is a first integral of the mKdV equation if and only if , where is the Hamiltonian.
Show that if satisfies the mKdV equation then
Using this equation, show that the functional
Poisson-commutes with the Hamiltonian.
Paper 2, Section II, D
comment(a) Explain how a vector field
generates a 1-parameter group of transformations in terms of the solution to an appropriate differential equation. [You may assume the solution to the relevant equation exists and is unique.]
(b) Suppose now that . Define what is meant by a Lie-point symmetry of the ordinary differential equation
(c) Prove that every homogeneous, linear ordinary differential equation for admits a Lie-point symmetry generated by the vector field
By introducing new coordinates
which satisfy and , show that every differential equation of the form
can be reduced to a first-order differential equation for an appropriate function.
Paper 3, Section II, D
commentLet and be real matrices, with symmetric and antisymmetric. Suppose that
Show that all eigenvalues of the matrix are -independent. Deduce that the coefficients of the polynomial
are first integrals of the system.
What does it mean for a -dimensional Hamiltonian system to be integrable? Consider the Toda system with coordinates obeying
where here and throughout the subscripts are to be determined modulo 3 so that and . Show that
is a Hamiltonian for the Toda system.
Set and . Show that
Is this coordinate transformation canonical?
By considering the matrices
or otherwise, compute three independent first integrals of the Toda system. [Proof of independence is not required.]
Paper 1, Section II, D
commentConsider the coordinate transformation
Show that defines a one-parameter group of transformations. Define what is meant by the generator of a one-parameter group of transformations and compute it for the above case.
Now suppose . Explain what is meant by the first prolongation of . Compute in this case and deduce that
Similarly find .
Define what is meant by a Lie point symmetry of the first-order differential equation . Describe this condition in terms of the vector field that generates the Lie point symmetry. Consider the case
where is an arbitrary smooth function of one variable. Using , show that generates a Lie point symmetry of the corresponding differential equation.
Paper 2, Section II, D
Let be a smooth function that decays rapidly as and let denote the associated Schrödinger operator. Explain very briefly each of the terms appearing in the scattering data