Behaviour of the extended Toda lattice

• We analyse the behaviour of solutions of the extended Toda lattice.
• We derive PDEs which are asymptotic approximations of the lattice.
• We find similarity solutions of these limiting PDEs.
• We show that in certain cases the PDEs can be transformed to the Boussinesq and/or KdV equations.

We consider the first member of an extended Toda lattice hierarchy. This system of equations is differential with respect to one independent variable and differential-delay with respect to a second independent variable. We use asymptotic analysis to consider the long wavelength limits of the system. By considering various magnitudes for the parameters involved, we derive reduced equations related to the Korteweg-de Vries and potential Boussinesq equations.