Toda field theory
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In the study of field theory and partial differential equations, a Toda field theory is derived from the following Lagrangian:
![\mathcal{L}=\frac{1}{2}\left[\left({\partial \phi \over \partial t},{\partial \phi \over \partial t}\right)-\left({\partial \phi \over \partial x}, {\partial \phi \over \partial x}\right)\right ]-{m^2 \over \beta^2}\sum_{i=1}^r n_i e^{\beta \alpha_i(\phi)}.](../../../math/d/2/8/d28e4d76c170ecb4f5ffa0076a360593.png)
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 describes 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.
