A multisymplectic approach to defects in integrable classical field theory

Abstract

We introduce the concept of multisymplectic formalism, familiar in covariant field theory, for the study of integrable defects in 1 + 1 classical field theory. The main idea is the coexistence of two Poisson brackets, one for each spacetime coordinate. The Poisson bracket corresponding to the time coordinate is the usual one describing the time evolution of the system. Taking the nonlinear Schrödinger (NLS) equation as an example, we introduce the new bracket associated to the space coordinate. We show that, in the absence of any defect, the two brackets yield completely equivalent Hamiltonian descriptions of the model. However, in the presence of a defect described by a frozen Bäcklund transformation, the advantage of using the new bracket becomes evident. It allows us to reinterpret the defect conditions as canonical transformations. As a consequence, we are also able to implement the method of the classical r matrix and to prove Liouville integrability of the system with such a defect. The use of the new Poisson bracket completely bypasses all the known problems associated with the presence of a defect in the discussion of Liouville integrability. A by-product of the approach is the reinterpretation of the defect Lagrangian used in the Lagrangian description of integrable defects as the generating function of the canonical transformation representing the defect conditions.

Time-dependent defects in integrable soliton equations

“…Furthermore we implement the classical r-matrix method to establish the Liouville integrability of the resulting defect systems. Our results extend the results of [13,14] from the situation of the defect being fixed to the situation of the defect moving with time.…”

“…In this section, we will establish the integrability of the defect system both by constructing an infinite set of conserved densities and by implementing the classical r-matrix method. This analysis is based on an extension of the results of [13,14] from the situation of the defect being fixed to the situation of the defect moving with time.…”

“…Canonical properties of BTs to integrable nonlinear evolution equations have been established in [27,28]. Recently, by introducing a new Poisson bracket (called equal-space bracket), it was shown in [14] that a defect condition described by a frozen BT can be interpreted naturally as a canonical transformation of the system. As a consequence, the classical r-matrix approach [22][23][24] can be implemented to establish Liouville integrability for the defect system with a defect at a fixed location.…”

Jumps and twists in affine Toda field theories

“…[15]). A similar equation involving the derivative with respect to the space coordinate would naturally emerge if one would start the whole analysis considering the time component V of the Lax pair as the fundamental object (see also some recent relevant results [30]). Equivalently one would end up with such an equation considering the corresponding lattice system in discrete time.…”

Integrability of generalised type II defects in affine Toda field theory

“…While these defects are integrable they do not necessarily describe the same systems as the momentum conserving defects found in the Lagrangian set-up. Some attempt to reconcile this Hamiltonian approach and the Lagrangian approach to defects is made in [39,40]. The type I and type II Lagrangians are rewritten as Hamiltonians with second class constraints in [6].…”