Integrable models are mathematical models that can be solved exactly, meaning their solutions can be written in closed form and do not require numerical approximation. These models have special properties that make them solvable, such as an infinite number of conserved quantities.
Integrable models have been key in understanding physical systems and phenomena, from classical mechanics to quantum field theory. They provide insights into the underlying structure and symmetries of a system, and their solutions can be used to make predictions and test theories.
Some well-known examples of integrable models include the harmonic oscillator, the Ising model in statistical mechanics, and the nonlinear Schrödinger equation in quantum mechanics. Other examples can be found in condensed matter physics, high energy physics, and even in biological systems.
The study of integrable models continues to be a vibrant area of research in physics, with new developments in both theory and applications. Some current frontiers include the study of integrability in quantum gravity, the use of integrable models in understanding strongly correlated systems, and the application of integrability techniques in studying non-equilibrium systems.
Integrable models have deep connections to various areas of mathematics, including algebraic geometry, representation theory, and differential geometry. These connections have led to new insights and techniques for solving integrable models and have also contributed to advances in pure mathematics.