On PT Symmetry Systems: Invariance, Conservation Laws, and Reductions
An analysis of a PT symmetric coupler with “gain in one waveguide and loss in another” is made; a transformation in the PT system and some assumptions results in a scalar cubic Schrödinger equation. We investigate the relationship between the conservation laws and Lie symmetries and investigate a Lagrangian, corresponding Noether symmetries, conserved vectors, and exact solutions via “double reductions.”
Physical systems exhibiting parity-time (PT) symmetry have been the subject of much investigation in recent years and are now extensively considered in diverse areas of physics, namely, quantum field theories, non-Hermitian Anderson models, and complex Lie algebras, just to name a few [1–10]. We know that even a single PT cell can exhibit unconventional features; it follows that one may wish to investigate what new behaviour and properties can be expected from PT symmetric lattices [11, 12].
In optics, it has recently been discovered that there is a class of optical systems, of which elements consist of gain and loss, that can be interpreted as an optics equivalent of the PT symmetry in quantum mechanics [11, 13]. The underlying equations describing the effects of pulse dispersion  have the following form: To analyze the solutions of this equation we make a change of variables where is a constant angle satisfying and is an arbitrary real parameter. As a result of the transformation, (1) becomes The system (4) admits a reduction to the following scalar cubic Schrödinger equation: where is the complex valued dependent variable and .
Equation (5) has a family of stationary soliton solutions; however, we will study the invariance, exact solutions, conservation laws, and double reductions. This will be done by decomposing (5) into real and imaginary parts to obtain the following system of partial differential equations (PDEs); if , then we have the following system: In light of symmetries, conservation laws, and double reduction to exact solutions, we will briefly consider the system of PDEs in (1) but the bulk of the analysis will centre around (5) via the system (6).
2. On the Conservation Laws of (1)
In order to determine conserved densities and fluxes, we resort to the invariance and multiplier approach based on the well-known result that the Euler-Lagrange operator annihilates a total divergence (see ). Firstly, if is a conserved vector corresponding to a conservation law, then along the solutions of the differential equation (, say).
If the system (1) is split into real and imaginary parts with and and replacing with “general” parameters, we get It turns out that the system (8) only admits nontrivial conservation laws (two) for and corresponding to multipliers (see below) and which in turn correspond to time and space translations, respectively. In this case, (8) becomes Thus, (1) has no nontrivial conservation laws even though the system is invariant under time and space translations.
We thus do a detailed study of the special case given in (6) instead.
3. Symmetries, Reductions, and Conservation Laws of (6)
The Lie symmetry approach on differential equations is well known; for details, see, for example, [15, 16]. In this section, we list a summary of these and explore the notion of a “double reduction” in order to obtain symmetry invariant (exact) solutions.
3.1. Symmetries and Reductions
A one parameter Lie group of transformations that leave invariant (6) will be written as a vector field This would be a generator of point symmetries of the system. We get the algebra generated by
3.2. Conservation Laws
In order to determine conserved densities and fluxes, we resort to the invariance and multiplier approach based on the well-known result that the Euler-Lagrange operator annihilates a total divergence (see ). Firstly, if is a conserved vector corresponding to a conservation law, thenalong the solutions of the differential equation (, say).
3.2.1. The Mutiplier Approach
If there exists a nontrivial differential function , called a “multiplier,” such that then is a total divergence; that is, for some (conserved) vector and is the respective Euler-Lagrange operator. Thus, knowledge of each multiplier leads to a conserved vector determined by, inter alia, a homotopy operator. See details and references in [14, 17].
For a system and , , say, we get In each case, is the conserved density.
The lengthy calculations for the system (6) lead to the following multipliers and corresponding conserved vectors.
(i) Consider , The conserved density for the scalar equation is
(ii) Consider . Consider The conserved density for the scalar equation is
(iii) Consider , The conserved density for the scalar equation is
(iv) Consider , The conserved density for the scalar equation is
3.2.2. A Lagrangian Formulation
The system (6) admits a Lagrangian so that the corresponding Lagrangian for the Schrödinger equation (5) is . The Noether symmetries, that is, the one parameter Lie groups of transformations, that leave invariant the functional with zero gauges are the translations with corresponding conserved vectors with density of (5) given by and with density of (5) given by respectively.
3.3. Double Reduction
To demonstrate how one uses symmetries and conservation laws to double reduce PDEs, we recall some definitions and theorems.
Theorem 2 (see ). Suppose that is any Lie-Bäcklund symmetry of (6) and , , are the components of the conserved vector of (6). Then constitute the components of a conserved vector of (6); that is, .
Theorem 3 (see ). Suppose that is a conservation law of the partial differential equation system (6). Then under a contact transformation, there exist functions such that , where are given as in which and .
Theorem 4 (fundamental theorem on double reduction ). Suppose that is a conservation law of the partial differential equation system (6). Then under a similarity transformation of a symmetry of the form (1) for the partial differential equation, there exist functions such that is still a symmetry for the partial differential equation and where and .
Our original system is equivalent to The system (36) can be rewritten as
3.3.1. A Double Reduction of (6) by
We first show that and are associated with using the following version of (6) for : We obtain where Thus, and are both associated with .
We can get a reduced conserved form for the first equation of the first system, from (36), since and are both associated symmetries with .
We consider a linear combination of and , that is, of the form , and transform this generator to its canonical form , where we assume that this generator is of the form .
From , , , and , we obtain We solve (41) using the method of invariance, where the results are summarized as follows: where , , , and are arbitrary functions all dependent on , , and .
By choosing , , , and , we obtain the canonical coordinates where and , since .
From (32), we compute and : where .
From (43), the inverse canonical coordinates are given by We compute the first- and second-order partial derivatives of and from (45): We now apply the formula from (33) with to obtain the reduced conserved form By substituting (45) and (46) into (47), we obtain where the reduced conserved form is also given by The second step of double reduction can be given as or equivalently where is a constant.
Differentiating (50) implicitly with respect to results in The second equation of from (36) is given by After transforming (53) using (45) and (46), we obtain We now substitute (50) and (52) into (54) and simplify.
3.3.2. Case 2: A Reduction of (6) by on
We now show that is associated with using the formula (38).
We obtain where Thus, is associated with .
We again transform the generator to its canonical form .
From , , , and , we obtain The results from solving (60) are summarized as follows: where , , , and are arbitrary functions all dependent on , , and .
By choosing , , , and , we obtain the canonical coordinates From (32), we compute and : where .
From (62), the inverse canonical coordinates are given by We compute the first- and second-order partial derivatives of and from (64): By substituting (64) and (65) into (47), we obtain where the reduced conserved form is also given by The second step of double reduction can be given as or equivalently where is a constant.
Differentiating (68) implicitly with respect to results in or equivalently, after dividing both sides by , The second equation of from (36) is given by After transforming (72) using (64) and (65) and multiplying both sides by , we get the ODE If we substitute (69) into (73) and then integrate both sides with respect to , Using (64), we obtain the final solution to our original system (6) as so that .
We have constructed conservation laws for the scalar cubic Schrödinger equation via the invariance and multiplier approach based on the well-known result that the Euler-Lagrange operator annihilates total divergence. Interestingly enough, the scalar cubic Schrödinger equation admits a Lagrangian resulting in Noether symmetries. Furthermore, two cases of double reduction were successfully performed and exact solutions were calculated and diagrammatically represented.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of New York, NY, USA, Springer, 2nd edition, 1993.