The majority of natural phenomena are linear. That is why periodic cyclicity, like the succession of seasons, is so common in Nature. And yet non-linearities are important, after all, all forces and interactions are ultimately due to non-linearities in the equations of Quantum Field Theory. If deviations from linearity are small they can be taken into account by perturbation theory which has plenty of applications, from planet motion to particle accelerators.
When interactions are strong, and cannot be regarded as a small perturbation on top of linear evolution, quantitative description of the system at hand becomes difficult and in most cases impossible. Indeed, many unsolved problems in physics are such because they are highly non- linear. To give a few examples, the quark confinement or hydrodynamic turbulence are intrinsically non-linear phenomena and their full theoretical description is lacking to this day despite obvious technological importance and a decades-long effort.
Solvable models are set apart by full analytic control one can gain over all non-linearities in the problem. Solvable models often capture essential, universal aspects of non-linear dynamics. A famous example is Onsager's exact solution of the two-dimensional Ising model which played fundamental role in the theory of critical phenomena. Another prominent case is Bethe's exact solution of the Heisenberg model for a chain of interacting atomic spins, underlying much of the physics of strongly-coupled electrons in solids.
The model studied by Bethe is an integrable system having a plethora of hidden symmetries, not obvious from the model's definition. It is these additional symmetries that give us full analytic control of its highly non-linear, strongly-interacting dynamics. Integrability has numerous applications in strongly-correlated condensed-matter systems, disordered models far from equilibrium, String Theory and Quantum Gravity. It underlies the theory of solitons and has inspired important developments in pure mathematics.
Current research at Nordita is focussed on a relationship between Quantum Field Theory, String Theory and integrable models. Integrability has shed light on the inner working of the Holographic Duality that relates gauge interactions with String Theory and Quantum Gravity. Progress in this area resulted in the first ever exact solution of an interacting Quantum Field Theory in 3+1 dimensions, and has a potential to boost our understanding of gauge interactions in general.
