| Abstract: |
| Max equation is a piecewise linear type of difference equation constructed from operators `max`, `min`, `+` and `-`. Such equations are studied intensively in recent years after the discovery of ultradiscrete soliton equations, for example, ultradiscrete Lotka-Volterra equation (= box and ball system), ultradiscrete Burgers equation (= elementary cellular automaton of rule number 184), and so on. We discuss the initial value problem for various integrable or non-integrable max equations. Some has conserved quantity or Lyapunov function and asymptotic behavior of solutions can be obtained from those quantities. We can show exact explicit solutions to some equations using formulas on max operators and classify them by their complexity. We review these results for 1+1D max equations including filter type and higher order type. |
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