Personal Particulars
Full name: Pieter Hubert van der Kamp Date of birth: 14 September 1972 Place of birth: Medemblik, the Netherlands Marital Status: Registered Partnership Children: Four girls (2004, 2007, 2007, 2010) Position: Lecturer, level B
Contact Information

Department of Mathematics and Statistics La Trobe University Victoria 3086, Australia 

Office:  Physical Sciences 2, Room 319  
Phone:  +61 3 9479 1614  
Fax:  +61 3 9479 2466  
Email:  P.vanderKamp at LaTrobe.edu.au 
Education & Career 
19851991  Atheneum  Han Fortmann College  Heerhugowaard 
19911998  Physics (Doctoraal)  Vrije Universiteit  Amsterdam 
19921996  Philosophy  Vrije Universiteit  Amsterdam 
19982002  Mathematics (PhD)  Vrije Universiteit  Amsterdam 
20032004  Research Associate  University of Kent  Canterbury 
20042005  Lecturer A  University of Kent  Canterbury 
20062007  Research Officer A  La Trobe University  Melbourne 
20082010  Research Fellow B  La Trobe University  Melbourne 
20112013  Lecturer B  La Trobe University  Melbourne 
During my studies I specialized in theoretical physics. I wrote my master thesis at the NIKHEF Institute (Amsterdam) under the supervision of Prof. Bert Schellekens. The thesis dealt with quantum states of black holes in string theory. After my studies I started working on a PhD research project in mathematics at the vrije Universiteit. My supervisor was Dr. Jan Sanders. By making use of number theory I was able to make important contributions to the classification of integrable evolution equations. On January 9, 2003 I successfully defended my thesis Integrable Evolution Equations: a Diophantine Approach. In September 2003 I started to work as a research associate at the University of Kent at Canterbury with Prof. Elizabeth Mansfield. I was appointed Lecturer A in April 2004. The research involved the lifting of integrability from the evolution of curvature invariants to the motion of the curve, using the geometric theory of moving frames. In January 2006 I joined the group of Prof. Reinout Quispel at La Trobe university. We study integrable properties and reductions of integrable lattice equations, such as integrals, symplectic structures and growth of degrees.
Awards 
I was awarded the 2003 Stieltjes Prize for the best PhD thesis written in the Stieltjes Institute.
Research Interests 
I enjoy doing research in integrable systems, a broad area at the boundary of physics and mathematics,. I am mainly concerned with algebraic and geometric properties of nonlinear differential equations and difference equations. Keywords are symmetries and invariants.
Publications 
Published:
On Testing Integrability, Peter H. van der Kamp and Jan A. Sanders, Journal of Nonlinear Mathematical Physics 8, 561—574, 2001.
Almost Integrable Evolution Equations, Peter H. van der Kamp and Jan A. Sanders, Selecta Mathematica (N.S.) 8, 705—719, 2002.
The use of padic numbers in calculating symmetries of evolution equations, Peter H. van der Kamp, proceedings `Symmetry in Nonlinear Mathematical Physics 2001’, 151—155, 2002.
On proving integrability, Peter H. van der Kamp, Inverse Problems 18, 405—414, 2002.
Integrable systems and number theory, Peter H. van der Kamp and Jan A. Sanders and Jaap Top, proceedings `Differential Equations and the Stokes Phenomenon 2001’, 171—201, 2002.
Hanging a Carillon in a Broeksystem, Budd, Fokkink, Hek, Van der Kamp, Pik, Rottschäfer, Proceedings ‘FourtyFifth European Study Group with Industry’, 73—90, 2004.
Classification of Integrable Bequations, Peter H. van der Kamp, Journal of Differential Equations 202, 256—283, 2004.
Evolution of curvature invariants and lifting integrability, Elizabeth L. Mansfield and Peter H. van der Kamp, Journal of Geometry and Physics 56, 1294—1325, 2006.
Closedform expressions for integrals of mKdV and sineGordon maps, Peter H. van der Kamp, O. Rojas and G.R.W. Quispel, J. Phys A.: Math Gen. 40, 12789—12798, 2007.
Lax representation for integrable OΔEs, O. Rojas, Peter H. van der Kamp and G.R.W. Quispel, proceedings `Symmetry and Perturbation Theory 2007', 271—272.
Towards global classifications: a Diophantine approach, Peter H. van der Kamp, proceedings `Symmetry and Perturbation Theory 2007', 278—279. Erratum
Closedform expressions for integrals of traveling wave reductions of integrable lattice equations, Dinh T. Tran, Peter H. van der Kamp, G.R.W. Quispel, J. Phys. A: Math. Theor. 42 (2009) 225201 (20pp).
Global classification of 2component approximately integrable evolution equations, Peter H. van der Kamp, Foundations of Computational Mathematics 9 (2009), 559–597. Maple code
Initial value problems for lattice equations, Peter H. van der Kamp, J. Phys. A: Math. Theor. 42 (2009) 404019.
Sufficient number of integrals for the pth order Lyness equation, Dinh T Tran, Peter H. van der Kamp, G.R.W. Quispel, J. Phys. A: Math. Theor. 43 (2010) 302001.
The staircase method: integrals for periodic reductions of integrable lattice equations, Peter H. van der Kamp and G.R.W. Quispel, J. Phys. A: Math. Theor. 43 (2010) 465207.
Growth of degrees of integrable mappings, Peter H. van der Kamp, Journal of Difference Equations and Applications, iFirst article (2011) 114.
Higher analogues of the discretetime Toda equation and the quotientdifference algorithm, Paul E. Spicer, Frank W. Nijhoff, Peter H. van der Kamp, Nonlinearity 24 (2011), 22292263.
Involutivity of
integrals for sineGordon, modified KdV and potential KdV maps, D. Tran,
Peter H. van der Kamp, G.R.W. Quispel, J. Phys. A 44 (2011), 295206.
Miscellaneous:
Integrable Evolution Equations: a Diophantine Approach, Peter H. van der Kamp, Thesis, Vrije Universiteit, Amsterdam, 2003.
Symmetry condition in terms of Lie brackets, Peter H. van der Kamp, preprint arXiv.org:nlin/0802.1954 (8pp) 2008.
Lax representations for integrable maps, O. Rojas, Peter H. van der Kamp, G.R.W. Quispel. [Here we show that the monodromy matrix is one of the Laxmatrices for the travelling wave reduction]
Track Record 

Field of research 
The field of integrable equations is a broad area at the boundary of physics and mathematics. It was born with a quest for exact solutions to Newton’s equations of motion. Since the Kepler problem, which was solved by Newton himself, quite a few integrable models were found, including Euler, Lagrange and Kowalevsky tops. In the second half of the twentieth century the inverse scattering method was invented. This boosted the development into the realm of partial differential equations, leading to new concepts, e.g. the Lax pair, and exciting applications, e.g. solitons. The field is very much alive and has interactions with geometry, algebra, analysis and more recently also number theory. Its methods and ideas have been applied in condensed matter physics, statistical and quantum mechanics, quantum field theory and string theory. For almost half a century great effort has gone into recognising and classifying integrable equation as well as understanding the nature of integrability and proving integrability. Of course, these questions are intimately related. More recently the field has witnessed a shift of direction towards the discrete setting.
Background 
In the symmetry approach an
equation is called integrable if it has infinitely many generalized
symmetries. It was shown that the existence of a formal symmetry of
sufficiently high order guarantees integrability. In this way the first
complete list, of nonlinear integrable KleinGordon models, was obtained.
Among the many different approaches to recognition and classification of
integrable equations, the symmetry approach has proven to be particularly
successful.
Symbolic method
The first global result, where the order of the equations is not fixed, was
obtained by J.A. Sanders and J.P. Wang who classified homogeneous x and
tindependent scalar equations of positive weight [J.
Diff. Equations, 1998]. Their method is based on the use of a symbolic
calculus, introduced by I.M. Gel'fand and L.A. Dikiĭ [Russ. Math. Surveys,
1975], in which differential objects are represented by polynomials. For
example, differentiation translates into multiplying with a sum of symbols
and the first symmetry condition becomes a divisibility condition between
certain polynomials called Gfunctions. This cleared the path to
using methods and results from algebraic geometry, padic analysis,
and number theory [5].
A conjecture of Fokas
Sanders and Wang formulated an implicit function theorem and used results
from Diophantine approximation theory to perform the classification. Their
exhaustive list contains ten integrable equations and it was proven that
there are no other equations in the possession of symmetries. This result
confirms the first part of Fokas's conjecture ‘If a scalar equation
possesses at least one timeindependent nonLie point symmetry, then it
possesses infinitely many. Similarly for ncomponent equations one
needs n symmetries.’ [Stud. Appl. Mathematics, 1987].
Two component equations
An important problem, which is still open, is the classification of
homogeneous 2component equations. This is the main problem I aimed
to solve in my PhD thesis. A major complication seemed to be the existence
of multicomponent equations with finitely many symmetries. Such equations
are called almost integrable, in the spirit of Fokas's conjecture.
The first example of an almost integrable equation was presented by I.M.
Bakirov. F. Beukers, J.A. Sanders and J.P. Wang used the symbolic method and
padic analysis to show that his 4th order 2component
equation does not have generalised symmetries at any order other than 6
[J. Diff.
Equations, 1998]. The class of equations considered by Bakirov consists
of triangular systems for functions u and v. The equation for
v is a homogeneous linear evolution equation and the equation for u
is an inhomogeneous version of the same equation with a term quadratic in v.
The
Gfunction connected to such equations contains a parameter.
Therefore one has to study sets of zeros. These zeros satisfy an equation
that can be solved for the order: a Diophantine equation. The LechMahler
theorem implies that certain ratios are roots of unity. An algorithm of C.J.
Smyth makes it possible to find these cyclotomic points [Numb. Th.
Millennium, 2001]. This led to a finite list of integrable cases. Each
equation in the list would still be integrable if its quadratic part
contained derivatives of v. However, the list is not complete in this
more general class of so called Bequations.
Most of the research I did during my PhD has been devoted to the
classification and recognition of integrable and almost integrable Bequations
[1,2,3,7].
Although this class of equations seems quite special, the Diophantine
approach I employed and developed provides the key to classifying more
general classes of evolution equations [13].
Connection between geometry and integrability
After my PhD
Elizabeth Mansfield got me interested in the connection between finite
dimensional geometry and integrability. Many integrable equations have been
shown to describe the evolution of curvature invariants associated to a
certain movement of curves in a particular geometric setting. In several
examples the hierarchy of generalized symmetries of a curvature equation has
been translated into a hierarchy of commuting geometric curves. Thus it
seems that assigning to a curve its curvature functions leads to pairs of
equivalent integrable equations.
Curvature functions are constructed, or defined, using a moving frame. This
technique was introduced by Darboux, who studied curves and surfaces in
Euclidean geometry, and was greatly developed by Cartan who used it in the
context of generalizing Klein's Erlangen program. Cartan's intuitive
constructions were made algorithmical by Fels and Olver. Given the action of
a Liegroup on a flat manifold, their method yields a complete set of
invariants, invariant differential operators, and the differential
relations, or syzygies, they satisfy. This approach has lead to new
applications that would not have been envisioned by Cartan, such as computer
vision and numerical schemes that maintain symmetry. The FelsOlver moving
frame method seems to be appropriate to answer the question whether
integrability lifts from curvature evolutions to curve evolutions.
Reductions of integrable
partial difference equations
In 2006 I moved to Melbourne
and started working on discrete equations.
Two main classes may be distinguished: partial
difference equations (PΔE), and ordinary difference equations (OΔE) or
mappings. From a PΔE a mapping can be obtained by traveling wave reduction,
one imposed periodic boundary conditions on a staircase. It is thought, and
in certain cases proven, that if the PΔE is integrable, then the mapping
derived from it is also integrable.
Notable research achievements 
Almost integrable
evolution equations
In my paper with J.A. Sanders [1] we have
presented 7th order Bequations with symmetries at order 11 and
29. The padic method of Skolem was used to prove
that there are no other symmetries of this equation. Therefore it is a
counterexample to the conjecture of Fokas. We also proved the existence of
infinitely many equations that are almost integrable [1].
Using resultants we provided a method to obtain all nth order Bequations
with a symmetry at order m [2]. This
was used to calculate all almost integrable equations of order 3<n<11
with a symmetry of order n<m<n+151. Refinements to the method of
Skolem were made to prove that all these equations have exactly one higher
symmetry, with the exception of the 7th order equations which have
two higher symmetries.
Classification of integrable Bequations
I solved the classification and recognition problems for
Bequations
[7]. Using the algorithm of Smyth the
problem was solved locally at low order n<24. Every zero of
infinitely many Gfunctions could be described in terms of roots of
unity pointing to the zero from 0 and 1, motivating the
introduction of biunit coordinates. These were used to construct, at every
order, a set of integrable Bequations that are not in a lower order
hierarchy. Moreover, biunit coordinates provided the key to establishing
that the hierarchies of symmetries are exhaustive, which was done in
cooperation with Beukers. I also gave formulas for the number of integrable
equations and proved that they are real, up to a complex scaling. The
recognition problem was solved by giving a description of all Bequations
that belong to a lower hierarchy. I gave a procedure to obtain the order of
the hierarchy.
Global classification of 2component approximately integrable evolution
equations
The Lie algebra of pairs of differential polynomials is a graded algebra.
The linear part has total grading 0, the quadratic terms have total grading
1, and so on. Gradings are used to divide the condition for the existence of
a symmetry, [K,S]=0, into a number of simpler conditions: [K,S]=0 modulo
quadratic terms, [K,S]=0 modulo cubic terms, and so on. This has been called
the perturbative symmetry approach, and in the same spirit the idea of an
approximate symmetry was defined. In [13] I
have classified 2component approximately integrable evolution
equations globally, that is, the order of the equations can be arbitrarily
high. This is achieved by applying the techniques developed in the special
case of Bequations, where any approximate symmetry is a genuine
symmetry.
A conjecture of Foursov
Foursov gave a classification of
3rd order symmetrically coupled KDVlike
equations [Inverse Problems, 2000]. One equation appeared to have an
interesting symmetry structure. This equation is linearly equivalent to the
KDVequation coupled to a purely nonlinear equation with parameter q.
Foursov conjectured that for all negative and rational q this equation has a
hierarchy of even order polynomial symmetries. I showed that is the case [4].
In fact, I proved a much stronger statement. There are several infinite sets
of symmetries for any, possibly complex, value of q. These symmetries
are not polynomial and do not necessarily commute.
Lifting integrability
My paper with E.L. Mansfield [8] is based on
the FelsOlver approach to moving frames. We give a method that determines,
from minimal data, the curvature and evolution invariants that are
associated to a curve moving in the geometry defined by the action of a Lie
group. The syzygy satisfied by these invariants is obtained as a zero
curvature relation in the relevant Lie algebra. An invariant motion of the
curve is uniquely associated with a constraint specifying the evolution
invariants as a function of the curvature invariants. The syzygy and this
constraint together determine the evolution of curvature invariants. We
prove that the condition for two curvature evolutions to commute appears as
a differential consequence of the condition that the corresponding curve
evolutions commute. This implies that integrability does not necessarily
lift from the curvature evolution to the curve evolution. However, most
commonly studied integrable curvature equations are homogeneous polynomials
or rational functions of the differential invariants. Since in these classes
the kernel of the differential operator is empty, pairs of integrable
equations result.
Closed form expressions
for integrals of highdimensional mappings
The staircase method takes advantage of Lax matrices of a
PΔE
which depend on a variable k, usually called the spectral parameter,
to construct a so called monodromy matrix whose trace yields integrals of motion for
the travelling wave reduction. With O. Rojas and G.R.W. Quispel [9]
we have provided closed form expression for the integrals of travelling wave
reductions of the sineGordon and mKDV equations. Then, with Dinh Tran, we
developed a systematic method to obtain closed form expressions, based on
noncommutative Vietaexpansion. In [12] we applied this method
to maps of the AdlerBobenkoSuris classification and in [15] it was
applied to the nth order Lynessequation.
Functional independence and
involutivity
We expressed the integrals in terms
of multisums of products, novel combinatorial objects which have
interesting properties. In [19] we have
established the Poisson commutation relations between our multisums of
products. This enabled us to prove involutivity of the integrals for certain
lattice reductions. A paper on the functional independence is in the making.
Generalized sreduction
In [14] we showed how to
construct wellposed, or nearly wellposed, initial value problems for lattice equations defined on
arbitrary stencils (as opposed to the ‘standard’ lattice equations defined on a
square). At the same time we have expanded the notion of sreduction to
include all possible periodicities. In [18] we develop a method to construct wellposed
initial value problems for systems of lattice equations (and apply it to a novel
quotientquotientdifference equation).
Dimensional
reduction
In [16] we prove directly, and
in full generality, that the staircase method provides integrals for mappings,
or correspondences, obtained as travelling wave reductions of integrable partial
difference equations. We apply the method to a variety of equations, and systems
of equations. Then we use symmetries of the lattice equations to dimensionally
reduce the mappings obtained as travelling wave reductions. Taking dimensional
reduction into account, our results support the idea that the staircase method
provides sufficiently many integrals for the mappings to be completely
integrable (in the sense of LiouvilleArnold). We also consider reduction on
quadgraphs that differ from the lattice of integers points in the plane.
Growth of degrees
In
[17] we study mappings obtained as periodic
reductions of the lattice KortewegDe Vries equation. For small periodicities we
establish upper bounds on the growth of the degree of the numerator of their
iterates. These upper bounds appear to be exact. We conjecture a similar growth
for periodicities in all directions and have considered the growth in projective
space as well.
Integrable lattice equations
from elliptic orthogonal polynomials
The discrete time Toda equation and the QuotientDifference equation arise in
the standard theory of orthogonal polynomials, see for example [Papageorgiou,
Grammaticos and Ramani, 1995]. By considering twovariable orthogonal
polynomials on elliptic curve, we construct higher order analogues of these
important equations in [18].
Teaching Experience 
In Amsterdam, during my
PhD, I gave exercise classes for several modules including calculus, linear
algebra and discrete mathematics.
In Canterbury I developed projects for
both first and third year students. I was the convener of first year Maple computer classes and of
third year Complex Analysis. I attended the
Associate Teacher
Accreditation Programme.
In Melbourne I have given tutorials and
practice classes in logic, linear algebra, vector calculus, mechanics and
partial differential equations. I have taught the following courses: the Epart of the course
Calculus, Functions and Economic applications; second year Linear Algebra; third
year Advanced Calculus and
Curvature; and a fourth year honours course in Number Theory. I have supervised
two honours students, Dinh Tran and Ritu Taneja, and two PhD students, Omar
Rojas and Dinh Tran.
Family 
Fieke Waninka Maria 1492004  Kris Willemina Bette
2132007
Seona Deïrdre Monika 
Gwyn Huberte Franka 1762010 
We speak Dutch at home. Renske and ik gebruikten katoenen luiers, eerst volgden we de driehoekig gevouwen methode, later vouwden we de luier volgens de door ons zelf ontwikkelde hamermethode, die geschikt is voor grotere babies.
Sport 
I love to play
canoe polo. Also I
enjoy running: in 2009 I ran the
Puffing Billy race, a
half marathon,
and a
marathon.
Links 
Averaging by Jan A. Sanders
This website was last updated on 10 November 2011.