In the study of field theory
and partial differential equations
, a Toda field theory
is derived from the following Lagrangian
Here x and t are spacetime coordinates, (,) is the Killing form of a real r-dimensional Cartan algebra of a Kac-Moody algebra over , αi is the ith simple root in some root basis, ni is the Coxeter number, m is the mass (or bare mass in the quantum field theory version) and β is the coupling constant.
Then a Toda field theory is the study of a function φ mapping 2 dimensional Minkowski space satisfying the corresponding Euler-Lagrange equations.
If the Kac-Moody algebra is finite, it's called a Toda field theory. If it is affine, it is called an affine Toda field theory (after the component of φ which decouples is removed) and if it is hyperbolic, it is called a hyperbolic Toda field theory.
Toda field theories are integrable models and their solutions describe solitons.
The sinh-Gordon model is the affine Toda field theory with the generalized Cartan matrix
and a positive value for β after we project out a component of φ which decouples.
The sine-Gordon model is the model with the same Cartan matrix but an imaginary β.