Stability of localized modes in -symmetric nonlinear potentials
We report on detailed investigation of the stability of localized modes in the nonlinear Schrödinger equations with a nonlinear parity-time (alias ) symmetric potential. We are particularly focusing on the case where the spatially-dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and -shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, what suggests that the relation between width of the modes and spatial size of the complex potential define the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples.
Nonlinear guided waves Optical instabilities (quantum optics)
Since the introduction of the concept of parity-time- (-) symmetric potentials , non-Hermitian Hamiltonians possessing purely real spectrum have received considerable attention  due to their relevance to the quantum mechanics and optics. In the context of optical applications, it was natural that the concept was generalized to the nonlinear systems , where the existence of localized modes was shown to be possible (we notice that experimental observation of the symmetry in linear optics was recently reported ). As a natural extension of this activity, the existence of stable localized nonlinear modes in nonlinear -symmetric lattices has been recently demonstrated in . Nonlinear gain and loss compensating each other were also addressed recently within the framework of the nonlinear dimer model . A general important property of nonlinear -symmetric systems is that they admit continuous families of localized modes parameterized by the propagation constant, similarly to what happens for the nonlinear Hamiltonian systems. This is in spite of the need to satisfy the balance between dissipation and gain, as this happens in dissipative systems of general type where localized modes appear as attractors, rather than elements of a family of solutions. In this sense, the models with -symmetric potentials occupy a special “place” between the Hamiltonian and dissipative systems.
A striking effect related to the existence and stability of the localized modes in nonlinear -lattices is that they can be stable even in the absence of modulation of conservative part of nonlinearity . The stability, however depends on the relation between the mode width and the period of the potential. More specifically only sufficiently narrow modes were found to be stable. Now we observe, that in the limit when the width of a mode goes to zero and the potential is a smooth function, the behavior of the imaginary part of the potential can be approximated by the respective linear function of the coordinate (recall, that the imaginary part is an odd function). For instance, the purely imaginary nonlinear -potential considered in  can be approximated as for sufficiently narrow modes strongly localized about . This leads to the natural question: is it possible to obtain stable localized modes in a nonlinear -symmetric potential of a general type, provided the widths of the modes are small enough? Here we give an affirmative answer to this question and illustrate stable modes for three qualitatively different case examples: of a periodically varying dissipation and gain, i.e. the lattice case, of dissipation and gain tending to constants at the infinity (see (6) below), and to the linearly increasing gain and losses (the model (23)).
The aim of the present paper is the analytic study of the linear stability and dynamics of solitons in nonlinear -symmetric potentials, giving positive answer to the above question. The main model we are interested in is the complex nonlinear Schrödinger equation :
In the optical applications is the dimensionless electric field propagating along the -direction () with () being the transverse coordinate. We notice that the physical mechanism for both gain and loss are well known. The former one refers to the standard two-photon absorption, which becomes dominating mechanism of losses in semiconductors at sufficiently high intensities, while nonlinear amplification can be realized in electrically-pumped semiconductor optical amplifiers (see e.g. ). We are interested in the special case of a -symmetric nonlinear potential where the and are both real and obeying the relations:
2 Localized modes
We look for stationary localized solutions of eqs. (1)–(2), which can be searched in the form subject to the boundary conditions . Bearing in mind optical applications, we refer to as to the propagation constant. The stationary wave function obeys the equation
Let us calculate how many parameters one has to introduce in order to unambiguously define a localized mode . To this end, let us first agree that we identify the modes and , , which are not distinguishable from physical point of view. Then a specific symmetry of eq. (3) induced by relations (2) suggests that without loss of generality the localized mode can be chosen to be -symmetric, i.e. having even real part and odd imaginary one:
Let be some solution of eq. (3) vanishing as , but not necessarily vanishing as . Then for the nonlinear term in eq. (3) is negligible and in the corresponding limit is described by the linear equation . Thus for the solution behaves as , where and are real constants. For a generic solution the constant is not zero, but let us temporarily restrict ourselves to the case . Eqs. (4) dictate that if the solution represents a localized mode, then must obey and . Now, let us admit nonzero in the asymptotics for . Obviously, this leads just to multiplication of by the factor . Therefore, we can formulate a weaker condition for to represent a localized mode: there must exist such that and . These equations are compatible if and only if
Thus any localized mode can be identified with a solution of the eq. (5) which contains two real unknowns: and . If we fix one of them, typically this is the propagation constant , then eq. (5) results in one or several solutions for the parameter what indicates that eq. (3) admits continuous families of localized modes for fixed and . This feature is typical for -symmetric (linear or nonlinear) potentials and constitutes significant difference compared to the conventional dissipative systems.
In fig. 1 (a) we show two families of localized modes plane , where is the energy flow (the integration limits are omitted wherever the integration is over whole real axis), obtained for the nonlinear potential
with . The modes exist only if exceeds certain threshold value. The modes of the lower (upper) curve in fig. 1 (a) can be referred to as fundamental (higher) modes. A typical profile of a fundamental mode is shown in fig. 1 (b). Below we focus on the fundamental modes.
On the other hand, the localized modes can be considered as bifurcating from the limit , where eq. (1) reduces to the conventional nonlinear Schrödinger equation. In fig. 1 (c) we show a branch of fundamental modes on the plane found for a fixed value of . The continuation from the limit will be used below as an approach for analytical investigation of stability of the fundamental modes.
3 Linear stability analysis
Substituting the perturbed solution , where , and the overline stands for the complex conjugation, in eq. (1) and linearizing it around one arrives at the eigenvalue problem , where (hereafter the superscript stands for matrix transposition) and is given by
The mode is unstable if and only if there exists an eigenvalue with positive real part.
For the localized mode is a standard NLS soliton: and . Designating the operator in this case by , we recall that the spectrum of is well known . In particular, the point spectrum of consists of the only eigenvalue which is isolated and has algebraic multiplicity (a.m.) equal to 4 and geometric multiplicity (g.m.) equal to 2. The eigenfunctions corresponding to read
There also exist two generalized eigenfunctions, namely
such that , . Here is obtained by means of differentiation of with respect to .
Let us now assume that and are not equal to zero but satisfy -symmetry relations (2). If at the same time and are small enough, then they can be considered as a perturbation to the operator . Behavior of the spectrum of subject to the perturbation determines linear stability of the localized mode. In particular, it is crucial to understand what happens to the multiple eigenvalue since a generic perturbation of the operator leads to splitting of the eigenvalue into several simple eigenvalues. As a result, unstable eigenvalues can arise in the vicinity of .
The multiplicity of the eigenvalue is related to rotational (i.e. phase) and translational symmetries of the model. The dissipative perturbation introduced by and breaks the translational symmetry and preserves the rotational one. Due to the last fact remains an eigenvalue for . However ceases to be the eigenfunction corresponding to and the only eigenfunction for is given by . Disregarding parity of the functions we observe that if is an eigenvalue of the operator then is also an eigenvalue. Then, recalling eqs. (4) we find that is an eigenvalue, as well. Therefore upon the perturbation exactly two simple eigenvalues arise in the vicinity of and those eigenvalues have opposite signs and either purely real or purely imaginary (see fig. 2).
4 Evans function for -symmetric potentials
The operator can be associated with the Evans function [9, 10], which is an analytic function defined on the whole complex plane except for the points of the continuous spectrum of . An important property of the Evans function is that it has a zero at some if and only if is an eigenvalue of . In addition, the order of that zero is equal to the a.m. of the eigenvalue .
Before proceeding with explicit definition of the Evans function for the case at hand, we recall that perturbation of the operator leads to that the eigenvalue has a.m. . Hence the Evans function corresponding to the perturbed operator has a zero of the second order at : , (hereafter stays for th partial derivative with respect to evaluated at ). Without loss of generality one can assume that for all . Then the stability of the stationary mode is determined by the sign of : if then necessarily has exactly one positive and one negative zero what corresponds to instability; vice versa, if then both roots of the Evans function lie on the imaginary axis, and the solution is stable. These considerations are illustrated in fig. 2
In order to define the Evans function for the operator given by eq. (7) we rewrite the eigenvalue problem in the form of the system of four first-order ODEs , where and
For the sake of simplicity we suppose that , where is the small parameter while . Then the coefficients become also dependent on . Obviously, now substitution is equivalent to substitution what in turn is equivalent to and . As a result, and are even while and are odd in .
Now we choose four solutions , , such that and are linearly independent in and vanish as while and are linearly independent and vanish as . Then the Evans function is given as -independent determinant : . In order for to be unambiguously defined and to depend analytically on and , we explicitly fix the choice of the solutions setting
(recall that and are given by eqs. (9)). Requiring this choice to be consistent with parity of the coefficients we find that
where , , and stays for the element-wise multiplication of the matrices. Eqs. (12) imply that the Evans function is even with respect to both its arguments: . Respectively, the second derivative of the Evans function at can be searched in the form of the following expansion:
where is to be determined.
Subject to the perturbation, i.e. for , the eigenfunction disappears; the solutions become unbounded and no longer correspond to any eigenfunction of the operator . However, the eigenfunction defined above persists and depends smoothly on . As a result, for all the following equalities hold:
Since has a.m.=2 in the spectrum of the perturbed operator, Lemma 3.3 from  can be applied. It states that for all and . Bearing in mind these facts and differentiating straightforwardly the Evans function one arrives at the following expression for the second derivative of the Evans function at :
Let us now recall that is an even function of . It means that where stands for the th partial derivative with respect to evaluated at . Calculating the second derivative with respect to and evaluating it at one arrives at
Using parity of the coefficients and the symmetries of the solutions given by eqs. (12) one can recognize that r.h.s. of eq. (4) generically is not equal to zero, i.e. we can set in eqs. (13) and obtain:
We failed to obtain expressions for the mixed derivatives and which would allow for efficient analytical or numerical computation of . However, using the information given by (16) one can use another representation for derivatives of the Evans function  which fits better for analytical and numerical investigation. To this end, we recall that apart from the eigenfunction there exists a generalized eigenfunction such that . The adjoint operator for reads . There also exists the adjoint eigenfunction such that . It is found in  that for any the second derivative of the Evans function at can be found as , where represents standard inner product of vector-valued functions. Taking into account eqs. (16) one can construct an expansion for with respect to the small parameter , . To this end, we firstly write down an expansion for the stationary mode itself. Eq. (3) dictates that this expansion acquires the following specific form: , , where the coefficients and solve the equations
and . Notice that is an even function of while is odd. Next, using eqs. (10) and the definition of (i.e. the equation ) we obtain an expansion for the generalized eigenfunction:
where the coefficient is an odd function of solving the equation
Using the definition the adjoint eigenfunction and requiring the derivative to satisfy constrains (16) we observe that expansion for must have the form:
Then is given as , where is a particular solution of the equation
The solvability condition for eq. (21) (i.e. orthogonality of its r.h.s. to ) is automatically provided as long as eq. (17b) holds. At the same time, the coefficient should be chosen to satisfy the solvability condition of the equation with respect to :
Respectively, we require r.h.s. of eq. (4) to be orthogonal to what yields , where
and . Finally, using eqs. (18) and (20), we can rewrite eq. (13) in the form: where the coefficient is given as . Functions and can be computed numerically from the linear equations (19) and (21), what gives an algorithm for obtaining the coefficient . For given positive value of corresponds to the situation when the modes are unstable for small . Vice versa, negative implies stability of the modes for small .
5 Discussion of the results and Conclusion
Let us now turn to the results of the stability analysis of the particular examples (see fig. 3). We start by recalling the results for the case and reported earlier in . Physically, this case corresponds to the pure dissipative nonlinear lattice where domains with the nonlinear gain alternate with the nonlinear dissipation. This case was already discussed in , and in particular it was shown that for sufficiently small the nonlinear modes become stable if the propagation constant exceeds a threshold value . The developed here approach based on the analysis of the Evans function allows us to compute numerically , as this is shown in fig. 3 (d) (the black curve 1) from which one observes that the coefficient changes its sign at which corroborates results reported in the panel (a). This critical value corresponds to narrow modes, whose widths are of order of the period of the dissipative lattice.
Next we consider potential (6) [see also fig. 1] which is characterized by only one domain with nonlinear gain and one domain with nonlinear dissipation. From fig. 3 (b) we observe that the domains of existence and stability are similar to those obtained for the sin-shaped potential: in particular, there exists the threshold for the stability of the modes. Quantitatively, now the stable modes can be broader than in the case of periodic potential: threshold value of propagation constant is remarkably lower than in the previous case. More specifically, using the approach based on the Evans function we obtain [the blue curve in panel (d)]: . This result agrees with fig. 3 (b) and with fig. 1 (d) where dependence of perturbation growth rate on is shown.
From the results for sin– and tanh– shaped potentials we can conjecture that (i) the most stable modes are localized on the scale where the dissipative potential can be approximated by the linear function, i.e. where on the width of the mode; and (ii) properly introduced gain and dissipation from the both sides of the mode enhances its stability. These arguments readily lead to the model
where all modes in the limit should be stable. This is indeed what happens, as one can see from fig. 3 (c): the instability threshold disappears completely. This is also confirmed by analysis based on the Evans function [the red curve 3 in the panel (d)].
Now from panel (c) we observe that both lines, i.e. upper border of existence domain (black curve) and the upper border of stability domain (red curve), follow the parabolic scaling law . This law can be understood from the simple scaling arguments as follows. If is a solution of eq. (23), then with and is a solution of the complex NLS equation without any parameter: . This observation as well as the fact that a narrow mode “feels” only the local dissipative term, allows one to make further conclusions about the behavior of the curves in fig. 3. Since the mode widths tend to zero at (and bounded), and all the examples of were chosen to have the same slope at : , the existence curves in the panels (a) and (b) tend at to the parabola shown in the panel (c). This conjecture was supported by numerical simulations up to where for the linear, –, and –shape potentials we found , and , respectively.
In fig. 4 we present two examples of evolution of unstable (panel a) and stable (panel b) modes. These results, obtained by direct integration of eq. (23) [i.e. of the particular case of the model (1) which is mostly “exposed” to eventual nonlinear instabilities] confirm the results of linear stability analysis. Thus, the modes belonging to the stability domains propagate undistorted over indefinitely long distances, even if they are strongly perturbed initially.
To conclude, we have investigated fundamental modes in imaginary -symmetric nonlinear potentials. Such potentials allow for existence of localized modes which are stable at least in the limit when the mode is narrow enough. The stability was established both as the linear stability, on the basis of the Evans-function analysis, and using direct numerical study of the mode evolution (notice that while the direct propagation ensures also the nonlinear stability of the modes in a finite domain, the nonlinear nature of the perturbation may introduce new features of the nonlinear stability of the solutions on the whole real axis). Although our analysis was performed for nonlinear potentials, the established symmetry properties have more general character and the approach can be applied also for linear -potentials, as well as to the cases where both linear and nonlinear -symmetric potentials are present. In the latter case stability of the modes may change dramatically. For example, in presence of a periodic linear -potential broad small-amplitude modes are expected to be stable as long as imaginary part of the linear potential is below a critical value. This situation is in contrast to the one shown in fig. 3 (a) where broad modes are unstable. As another interesting question, we would like to mention the exploration of asymmetric nonlinear modes similar to ones reported in , although their existence may require more sophisticated nonlinearity landscapes.
Acknowledgements.DAZ and VVK were supported by FCT (Portugal) under the grants No. SFRH/BPD/64835/2009 and PEst-OE/FIS/UI0618/2011. VVK was partially supported by the program Acções Integradas Luso-Espanholas No. E-27/10.
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