The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system.
It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian
and the equations of motion
where is the displacement of the -th particle from its equilibrium position,
and is its momentum (mass ),
and the Toda potential .
Soliton solutions are solitary waves spreading in time with no change to their shape and size and interacting with each other in a particle-like way. The general N-soliton solution of the equation is
where
with
where
and
The Toda lattice is a prototypical example of a completely integrable system. To see this one uses Flaschka's variables
such that the Toda lattice reads
To show that the system is completely integrable, it suffices to find a Lax pair, that is, two operators L(t) and P(t) in the Hilbert space of square summable sequences such that the Lax equation
(where [L, P] = LP - PL is the Lie commutator of the two operators) is equivalent to the time derivative of Flaschka's variables. The choice
where f(n+1) and f(n-1) are the shift operators, implies that the operators L(t) for different t are unitarily equivalent.
The matrix has the property that its eigenvalues are invariant in time. These eigenvalues constitute independent integrals of motion, therefore the Toda lattice is completely integrable.
In particular, the Toda lattice can be solved by virtue of the inverse scattering transform for the Jacobi operator L. The main result implies that arbitrary (sufficiently fast) decaying initial conditions asymptotically for large t split into a sum of solitons and a decaying dispersive part.
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In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.
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