MATH5175, 2023, Term 1 Assignment 1
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MATH5175, 2023, Term 1
Assignment 1, 6th March
Due: 27th March
Consider a pair of Lax operators of the form
L = ∂北(2) + iv(x, t)∂北 , M = 4∂北(3) + ia(x, t)∂北(2) + b(x, t)∂北 ,
where, a priori, the functions v, a and b are complex.
a) Find the conditions on the functions a, b and v such that the Lax equation Lt + [L, M] = 0
is satisfied.
b) Show that
a = 6v + f (t), b = 3iv北 _ 
v2 _ f (t)v + g(t)
and therefore
vt = v北北北 + 
v v2北 + g(t)v北 . (1)
c) In the remainder, we focus on real functions v and a so that the functions f and g are also real. Find a coordinate transformation
x\ = x + h(t), t\ = t
which transforms (1) into the mKdV equation
vt\ = v北\ 北\ 北\ + 
v2 v北\ . (2)
d) From now on, we set f (t) = g(t) = 0 and x\ = x, t\ = t so that the Lax pair λψ = ψ北北 + ivψ北 , ψt = 4ψ北北北 + 6ivψ北北 + ╱3iv北 _ 
v2、ψ北 (3)
is compatible modulo the mKdV equation (2). Note that, in general, ψ is complex. For
W = |ψ北 |2 _ |ψ|2
is constant, that is, W北 = Wt = 0 modulo the Lax pair (3). In order to simplify the calculations, it may be worth formulating (3)2 as a first-order equation on use of (3)1 .
e) According to Frobenius’ theorem, if v is analytic then the solution ψ of the Lax pair (3) is (locally) uniquely determined by the initial values ψ(x0 , t0 ) and ψ北 (x0 , t0 ) at some given point (x0 , t0 ). Explain why, for λ = 1, this implies that there exist solutions ψ of (3) such that W = 0. In this case, we may introduce a real function ϕ according to
eio = ψ北 (4)
Show that (3)1 becomes
ϕ北 + 2 sin ϕ + v = 0. (5)
f) On elimination of ψ北 and v by means of the relations (4) and (5) respectively, the remaining linear equation (3)2 becomes
= eio ╱iϕ北北 + 
ϕ北(2) + cos 2ϕ _ 2i sin 2ϕ + 3、. (6)
Verify that the compatibility condition [(ln ψ)北]t = [(ln ψ)t]北 associated with the pair (4), (6) gives rise to the modified mKdV (m2 KdV) equation
ϕt = ϕ北北北 + 
ϕ北(3) + 6ϕ北 sin2 ϕ . (7)
Note that you have just derived a Miura-type transformation in the sense that if ϕ is a solution of the m2 KdV equation (7) then v as given by (5) satisfies the mKdV equation (2). You may want to check this directly using computer algebra.
g) Show that the quantity
厂 = eio
constitutes a complex density of the m2 KdV equation by determining the associated con-
servation law
∂ ∂
∂t ∂x
Find two real densities and their associated fluxes.
h) Find two linearly independent solutions ψ1 and ψ2 of the linear equation (3)1 for the trivial solution v = 0 of the mKdV equation and λ = 1 so that the general solution of (3)1 is given by
ψ = T1 (t)ψ1 + T2 (t)ψ2 .
Use the linear equation (3)2 to determine the functions T1 and T2 . State the constraints on the (complex) constants of integration which guarantee that W = 0.
i) Show that the remaining constants of integration may be chosen in such a manner that
π
2
Carefully explain why the latter equality holds. By construction, ϕ is a solution of the m2 KdV equation. Discuss its behaviour as 2x + 8t → 土&. This type of solution is known as an ‘antikink’ . Present a plot of the antikink ϕ . Show that the quantity ϕ北(2) has the typical sech2 profile of a soliton.
j) This is a bonus question which allows you to make up for marks that you may have lost in the previous questions so that you may still get full marks.
The m2 KdV equation (7) is invariant under (x, t) → (x\ , t\ ) = _(x, t) so that application of this invariance to the solution (8) leads to another solution ϕ\ (‘kink’) of the m2 KdV equation. Show that the associated solution v\ of the mKdV equation (2) generated by the Miura-type transformation (5) once again encodes a soliton in that v\2 is of sech2 shape.
In summary, in this assignment, we have generated solutions of both the mKdV and the m2 KdV equations by merely solving simple linear differential equations (effectively ODEs) having constant coefficients. The map v |→ v\ constitutes a particular B¨acklund transformation.
2023-04-19