Description
We refer to the effective electro-magnetic Lagrangian given in Ref.[1] which involves higher order derivatives of the wave vector potential and is constructed so as to include the dependence on the so-called photon quantum nonlinearity parameter.
This Lagrangian makes it possible to describe dispersive effects in the interaction of two counter-propagating light pulses by means of a nonlocal extension of the nonlinear wave equation that is derived from the Heisenberg-Euler Lagrangian.
We discuss the well posedness of higher order Lagrangians for systems with infinite de- grees of freedom and impose that any effective mass arising from the balance between the nonlinear and the dispersive terms remains finite in the limit in which the value of the fine structure constant is set equal to zero.
In the case of a finite amplitude wave impinging on a large amplitude counter-propagating low-frequency wave (the so called cross fields configuration), we show that this higher order derivative Lagrangian leads to Korteveg-de Vries type soliton solutions.
An extension of this procedure so as to include higher order derivatives and higher powers of the fields amplitude could be of interest when searching for novel light soliton solutions, such as solitons with compact support.
References
[1] F. Pegoraro, S.V. Bulanov, Nonlinear waves in a dispersive vacuum described with a high order derivative electromagnetic Lagrangian, Phys. Rev. D, 103, 096012 (2021) .
