The first-order Euler-Lagrange equations and some of their uses

Abstract

In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.PACS numbers: I. INTRODUCTIONNonlinear field theories are ubiquitous in the description of physical systems from particle physics [1] -[4] to condensed matter systems [5] -[7] and cosmology [8], where any genuine interaction is generally related to the nonlinearity of the underlying field theory. In these theories, one powerful strategy to obtain solutions of physical importance is to reduce the order of the original field equations (the Euler-Lagrange (EL) equations) of the system. The resulting equations of lower order -Bogomolnyi equations, self-duality equations, Bäcklund transformations, etc. -are easier to solve and allow to obtain a large number of relevant solutions with particular characteristics, like solitons, nonlinear waves, vortices, monopoles, instantons, etc. There exist several known methods to achieve this reduction of order, where the best-known one is probably the Bogomolnyi trick [9], [10], [11] of completing a square. To consider an example, let us assume that we have the energy functional of a field theory which for static fields may be expressed as a sum of two terms, E = d d x(A 2 + B 2 ) (typically, A depends on first derivatives, whereas B only depends on the fields). This may trivially be rewritten asIf, in addition, Q is a homotopy invariant (i.e., AB is locally a total derivative), then it does not contribute to the EL equations, and its value only depends on the boundary conditions imposed on the fields. As a consequence, E andĒ lead to the same EL equations. Further,Ē is non-negative, so E obeys the inequality E ≥ |Q| (Bogomolnyi bound) which is saturated by solutions to the reduced-order (usually, first-order) equation A = ±B (Bogomolnyi equation or BPS equation).Recently it has been observed [12] that it can be useful to partly invert the logic of this construction. That is to say, let us assume that we have two functionals (functions of the fields, their derivatives, and possibly also of the coordinates x µ ) A, B which are in some sense "duals" of each other, and which are such that the product AB is locally a total derivative (the integral Q = 2 d d xAB is a homotopy invariant). This automatically implies that the "energy functional" E = d d x(A 2 + B 2 ) is a BPS action, and the "self-duality equations" (BPS equations) A = ±B

BPS Equations of Monopole and Dyon in SU(2) Yang-Mills-Higgs Model, Nakamula-Shiraishi Models, and Their Generalized Versions from the BPS Lagrangian Method

“…The third method called On-Shell method which works by adding and solving auxiliary fields into the Euler-Lagrange equations and assuming the existence of BPS equations within the Euler-Lagrange equations [16,21]. The forth method called First-Order Euler-Lagrange (FOEL) formalism, which is generalization of Bogomolnyi decomposition using a concept of strong necessary condition developed in [22], which works by adding and solving a total derivative term into the Lagrangian [23] (in our opinion the procedure looks similar to the On-Shell method by means that adding total derivative terms into the Lagrangian is equivalent to introducing auxiliary fields in the Euler-Lagrange equations. However, we admit that the procedure is written in a more covariant way).…”

The ϕ4 model with the BPS preserving defect

“…In addition, we prove the existence of the zero mode (moduli space), which has not yet been completely achieved for the vortex model. 2 The BPS φ 4 impurity model…”

Non-topological kink scattering in a two-component scalar field theory model

“…The resolution of these equations shows that there exist one-parameter kink families for different topological sectors. Kink solutions have also been calculated for models coupling the φ 4 and/or sine-Gordon models [94,95,96], models with a real scalar Higgs field and a scalar triplet field [97], models coupled to gravity in warped spacetimes [98,99,100,101], scalar field theories possessing self-dual sectors [102,103,104], etc. In addition, some deformation procedures have been developed, which allows to obtain exact solutions of two-field models from one-field models, see [105].…”