##### The area of integrable systems lies at the boundary of mathematics and physics.

##### Integrable systems are universal limiting models of science that are widely applicable. The field grew from observations of astonishingly well-ordered and predictable behaviour in certain models of nonlinear lattices used to describe the thermal properties of metals and led to the theory of solitons and completely integrable systems, one of the most profound advances of twentieth century mathematics. Reductions led to the Painlevé equations, which are canonical representations of integrable models in one dimension.

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Three-dimensional consistency
Local blow up
Blow up space of initial conditions
The max-av composition of fuzzy relations R and S The max-av composition of fuzzy relations R and S
The max-product composition of fuzzy relations R and S The max-product composition of fuzzy relations R and S
The max-min composition of fuzzy relations R and S The max-min composition of fuzzy relations R and S
Fuzzy relation S Fuzzy relation S
Fuzzy relation R Fuzzy relation R
A tritronquée solution of the fourth Painlevé equation A tritronquée solution of the fourth Painlevé equation
An approximation of a tronquée solution of the fourth Painlevé equation An approximation of a tronquée solution of the fourth Painlevé equation
5-cube Orthogonal projection of 5-dimensional hypercube in 2D
Circle pattern Circle pattern corresponding to the discrete analogue
of holomorphic functions z^(4/5).
An algebraic curve with 5 ordinary singularities An algebraic curve with 5 ordinary singularities at (0,1), (0,-1), (1,0), (-1,0) and (0,0).
Initial value space of autonomous q-PIII Initial value space of autonomous q-PIII, wlog, we assume that b>a and d>c. D_i, i=1,2,3,4 are the divisors.
Second invariant curve for autonomous q-PIII in complex projective space These images represent the invariant curve for autonomous q-PIII in complex projective space. They show all 8 base points and potential duplicates.
Invariant curve for autonomous q-PIII in complex projective space These images represent the invariant curve for autonomous q-PIII in complex projective space. They show all 8 base points and potential duplicates.
Algebraic curve with 5 ordinary singularities f(x,y)=(x^4 + y^4 - x^2 - y^2)^2-9 x^2 y^2, an algebraic curve with 5 ordinary
singularities at (0,1), (0,-1), (1,0), (-1,0) and (0,0).
Configuration of the roots of a generalized Okamoto polynomial Configuration of the roots in the complex plane of one of the generalised Okamoto polynomials. These polynomials are associated with rational solutions of the fourth Painlevé equation. Why these roots are so beautifully organised remains a mystery.
Exponentially small waves present in the Frenkel Kontorova Model Exponentially small waves present in the far-field solution of the Frenkel-Kontorova model. The magnitude vanishes on the discrete points for which the solution is defined on
Time evolution of a 2-compacton initial condition While collisions of many kinds of solitary waves have been studied, the case of compactons is still an open question.
Tritronquée-type solutions to a modified Painlevé equation Real and imaginary tritronquée-type solutions to a modified Painlevé equation. The tritronquée-type solution is only valid within the indicated region. In the omitted region, elliptic functions are required to describe the solution behaviour.
Water waves past a submerged source Free-surface profile for potential flow of infinite depth flowing past a submerged source, containing steady waves generated in the downstream flow profile.
Staircase reduction Staircase reduction of the 7-point stencil of the discrete Euler-Lagrange equations of an ABS equation.
Ultra-Discrete KdV equation on a two dimensional domain Propagating from the singularity marked "1" eventually results in a region of singularities trapped by differential points.
Configuration of the bilinear discrete KP equation This figure shows 3 copies of the bilinear discrete Kadomtsev-Petviashvili system. This is a 3-dimensional lattice equation, which can be considered as relating 6 vertices of a cube.
Billiard Trajectories
Double Reflection Theorem
Fomenko Graph
Interval Exchange Transformation
Lightlike Billiard Trajectories
Poncelet Theorem
PonceletDarboux Grid
Tesseract
Tropic curves and their light like tangents
Tropic curves on Confocal Quadrics in the Minkowski Space

Three-dimensional consistency

Local blow up

Blow up space of initial conditions

The max-av composition of fuzzy relations R and S

The max-av composition of fuzzy relations R and S

The max-product composition of fuzzy relations R and S

The max-product composition of fuzzy relations R and S

The max-min composition of fuzzy relations R and S

The max-min composition of fuzzy relations R and S

Fuzzy relation S

Fuzzy relation S

Fuzzy relation R

Fuzzy relation R

A tritronquée solution of the fourth Painlevé equation

A tritronquée solution of the fourth Painlevé equation

An approximation of a tronquée solution of the fourth Painlevé equation

An approximation of a tronquée solution of the fourth Painlevé equation

5-cube

Orthogonal projection of 5-dimensional hypercube in 2D

Circle pattern

Circle pattern corresponding to the discrete analogue
of holomorphic functions z^(4/5).

An algebraic curve with 5 ordinary singularities

An algebraic curve with 5 ordinary singularities at (0,1), (0,-1), (1,0), (-1,0) and (0,0).

Initial value space of autonomous q-PIII

Initial value space of autonomous q-PIII, wlog, we assume that b>a and d>c. D_i, i=1,2,3,4 are the divisors.

Second invariant curve for autonomous q-PIII in complex projective space

These images represent the invariant curve for autonomous q-PIII in complex projective space. They show all 8 base points and potential duplicates.

Invariant curve for autonomous q-PIII in complex projective space

These images represent the invariant curve for autonomous q-PIII in complex projective space. They show all 8 base points and potential duplicates.

Algebraic curve with 5 ordinary singularities

f(x,y)=(x^4 + y^4 - x^2 - y^2)^2-9 x^2 y^2, an algebraic curve with 5 ordinary
singularities at (0,1), (0,-1), (1,0), (-1,0) and (0,0).

Configuration of the roots of a generalized Okamoto polynomial

Configuration of the roots in the complex plane of one of the generalised Okamoto polynomials. These polynomials are associated with rational solutions of the fourth Painlevé equation. Why these roots are so beautifully organised remains a mystery.

Exponentially small waves present in the Frenkel Kontorova Model

Exponentially small waves present in the far-field solution of the Frenkel-Kontorova model. The magnitude vanishes on the discrete points for which the solution is defined on

Time evolution of a 2-compacton initial condition

While collisions of many kinds of solitary waves have been studied, the case of compactons is still an open question.

Tritronquée-type solutions to a modified Painlevé equation

Real and imaginary tritronquée-type solutions to a modified Painlevé equation. The tritronquée-type solution is only valid within the indicated region. In the omitted region, elliptic functions are required to describe the solution behaviour.

Water waves past a submerged source

Free-surface profile for potential flow of infinite depth flowing past a submerged source, containing steady waves generated in the downstream flow profile.

Staircase reduction

Staircase reduction of the 7-point stencil of the discrete Euler-Lagrange equations of an ABS equation.

Ultra-Discrete KdV equation on a two dimensional domain

Propagating from the singularity marked "1" eventually results in a region of singularities trapped by differential points.

Configuration of the bilinear discrete KP equation

This figure shows 3 copies of the bilinear discrete Kadomtsev-Petviashvili system. This is a 3-dimensional lattice equation, which can be considered as relating 6 vertices of a cube.

Billiard Trajectories

Double Reflection Theorem

Fomenko Graph

Interval Exchange Transformation

Lightlike Billiard Trajectories

Poncelet Theorem

PonceletDarboux Grid

Tesseract

Tropic curves and their light like tangents

Tropic curves on Confocal Quadrics in the Minkowski Space