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the theme of @xmath0 ( parity  time ) symmetric systems was initiated in the works of bender and collaborators  @xcite as an alternative to the standard quantum theory , where the hamiltonian is postulated to be hermitian . the principal conclusion of these works was that @xmath0-invariant hamiltonians , which are not necessarily hermitian , may still give rise to completely real spectra , thus being appropriate for the description of physical settings . in terms of the schrdinger - type hamiltonians , which include the usual kinetic - energy operator and the potential term , @xmath3 , the @xmath0-invariance admits complex potentials , subject to constraint that @xmath4 . recent developments in optics have resulted in an experimental realization of the originally theoretical concept of the @xmath0-symmetric hamiltonians , chiefly due to the work by christodoulides and co - workers  christo1 ( see also @xcite ) . it has been demonstrated that the controllable imposition of symmetrically set and globally balanced gain and loss may render optical waveguiding arrays a fertile territory for the construction of @xmath0-symmetric complex potentials . the first two such realizations made use of couplers composed of two waveguides with and without loss  @xcite ( so - called passive @xmath5couplers ) , or , in more standard " form , a pair of coupled waveguides , one carrying gain and the other one loss  @xcite . in fact , more general models of linearly coupled active ( gain - carrying ) and passive ( lossy ) intrinsically nonlinear waveguides , without imposing the condition of the gain - loss balance , were considered earlier , and stable solitons were found in them @xcite , including exact solutions @xcite ( see also a brief review in ref . recently , an electronic analog of such settings has also been implemented  @xcite . configurations with a hidden @xmath6 symmetry have been identified also in fine - tuned parameter regions of microwave billiards @xcite . effects of the nonlinearity in a gross - pitaevski equation on the @xmath7 properties of a bose - einstein condensate have been analyzed in @xcite . the possibility to engineer @xmath0-symmetric _ oligomers _  ( coupled complexes of a few loss - and gain - carrying elements ) @xcite , which may include nonlinearity , was an incentive to a broad array of additional studies on both the few - site systems and entire @xmath0-symmetric lattices  @xcite . more recently , nonlinear @xmath0-symmetric systems , incorporating @xmath0-balanced nonlinear terms , have drawn considerable interest  too @xcite-@xcite .    most of the @xmath0-invariant systems considered thus far have been one - dimensional ( 1d ) in their nature , although the stability of solitons in 2d periodic @xmath0-symmetric potentials has also been recently investigated @xcite . actually , 2d arrays of optical waveguides  can be readily built @xcite ( the same is true about other quasi - discrete systems , including electrical ones ) , hence , a natural question is whether @xmath0-symmetric oligomers ( and ultimately lattices built of such building blocks ) can be created in a 2d form . this work aims to make a basic step in this direction , by introducing fundamental 2d plaquettes consisting , typically , of four sites ( in one case , it will be a five - site cross ) . these configurations , illustrated by fig .  [ modes ] , are inspired by earlier works on 2d hamiltonian lattices described by discrete nonlinear schrdinger equations  @xcite , where diverse classes of modes , including discrete solitary vortices  malomed , pelin , have been predicted and experimentally observed  moti , yuri . the plaquettes proposed herein should be straightforwardly accessible with current experimental techniques in nonlinear optics , as a straightforward generalization of the coupler - based setting reported in ref . we start from the well - established hamiltonian form of such plaquettes in the conservative form , gradually turning on the gain - loss parameter ( @xmath2 ) , as the strength of the @xmath0-invariant terms , to examine stationary states supported by the plaquettes , studying their stability against small perturbations and verifying the results through direct simulations . actually , in this work we focus on those ( quite diverse , although , obviously , not most generic ) modes that can be found in an analytical form , while their stability is studied by means of numerical methods . the analytical calculations and the manifestations of interesting features , such as a potential persistence past the critical point of the linear @xmath0 symmetry , are enabled by the enhanced symmetry of the modes that we consider below . it is conceivable that additional asymmetric modes may exist too within these 2d configurations . our principal motivation for studying the above systems stems from the fact that realizations of @xmath0-symmetry e.g. within the realm of nonlinear optics will be inherently endowed with nonlinearity . hence , it is only natural to inquire about the interplay of the above type of linear systems with the presence of nonlinear effects . in addition to this physical argument , there exists an intriguing mathematical one which concerns the existence , stability and dynamical fate of the nonlinear states in the presence of @xmath0-symmetric perturbations . in particular , previous works  @xcite point to the direction that neither the existence , nor the stability of @xmath0-symmetric nonlinear states mirrors that of their linear counterparts ( or respects the phase transition of the latter generically ) . the presentation of our results is structured as follows . section [ symmetry ] contains a part of the analytical results , including a detailed analysis of the @xmath8symmetry properties of the nonlinear schrdinger type model , as well as the spectral properties of the linear hamiltonian subsystems . section [ numerics ] is devoted to the existence , stability and dynamics of stationary modes in the nonlinear systems . beside analytical results , it contains a detailed presentation of the numerical findings . in section [ conclu ] we summarize conclusions and discuss directions for future studies . the dynamics of the 2d plaquettes that we are going to consider is described by a multicomponent nonlinear schrdinger equation ( nlse ) @xmath9built over a transposition - symmetric linear @xmath10 hermitian matrix hamiltonian @xmath11 and an additional nonlinear @xmath10 matrix operator , @xmath12 . to understand the symmetry properties of this nlse , we first analyze the associated linear problem @xmath13and check then whether the symmetry is preserved by the nonlinear term , @xmath14 . the analysis can be built , in a part , on techniques developed for other nonlinear dynamical systems with symmetry preservation nonlin-1,nonlin-2,nonlin-3,nonlin-4,nonlin-5,nonlin-6,nonlin-7,nonlin-8 . for the present setups , the time reversal operation @xmath15 can be defined as the combined action of a scalar - type complex conjugation @xmath16 , @xmath17 , and the sign change of time , @xmath18 , in full accordance with wigner s original work which introduced these concepts @xcite . for the linear schrdinger equation ( [ a2 ] ) and its solutions @xmath19this implies @xmath20where the overbar denotes complex conjugation . from the actual form of the gain - loss arrangements in the 2d plaquettes we can conjecture the existence of certain plaquette - dependent parity operators @xmath21 , with @xmath22 , which will render the hamiltonians @xmath8symmetric , @xmath23=0 $ ] . to find an explicit representation of these parity operators @xmath21 , we use the following ansatz , @xmath24=0   \label{a2 - 3}\]]together with the pseudo - hermiticity condition @xmath25the latter follows trivially from eq . ( [ a2 - 3 ] ) , @xmath23=0 $ ] , and the transposition symmetry , @xmath11 . these parity operators @xmath21 will be used to check whether the corresponding nonlinear terms @xmath26 satisfy the same @xmath27symmetry . in contrast to linear setups with the @xmath28symmetry being either exact ( @xmath29=0 $ ] ,  @xmath30 ) or spontaneously broken ( @xmath29=0 $ ] ,  @xmath31 ) , the nonlinear setups considered in the present paper allow for sectors of exact @xmath28symmetry ( @xmath32=0 $ ] ,  @xmath30 ) and of broken @xmath28symmetry ( @xmath31@xmath33@xmath34\neq 0 $ ] ) , as it is common for nonlinear @xmath6-symmetric systems @xcite-@xcite . we start from the 2d plaquette of 0 + 0- type depicted as configuration ( a ) in fig . [ modes ] . this plaquette has only two ( diagonally opposite ) nodes carrying the gain and loss , while the other two nodes bear no such effects . the corresponding dynamical equations for the amplitudes at the four sites of this oligomer are @xmath35where @xmath36 is the above - mentioned gain - loss coefficient , and @xmath37 is a real coupling constant . the nonlinearity coefficients are scaled to be @xmath38 ( we use time @xmath39 as the evolution variable , although in the mathematically equivalent propagation equations for optical waveguides @xmath39 has to be identified with the propagation distance , @xmath40 ) .    denoting @xmath41 , the matrices @xmath42 and @xmath26 in ( [ a1 ] ) take the form of @xmath43to find the parity matrix @xmath21 which renders the linear hamiltonian @xmath8symmetric , @xmath23=0 $ ] , we use the pseudo - hermiticity condition ( [ a2 - 4 ] ) and notice that @xmath44obviously , the following relations should hold : @xmath45=0 \nonumber \\ \mathcal{p}h_{l,1}\mathcal{p}=-h_{l,1 } & \qquad \longrightarrow \qquad & \left\ { \mathcal{p},h_{l,1}\right\ } = 0 . \label{a6}\end{aligned}\]]the first of these conditions together with @xmath22 , @xmath46 reduces the possible form of the parity transformation to one of the three types , @xmath47where @xmath48 are the usual pauli matrices . taking into account that @xmath49 , the anti - commutativity condition in eqs . ( [ a6 ] ) singles out the only possible parity matrix : @xmath50for the linear transformation , i.e. , the matrix interchanging @xmath51 and @xmath52 , as well as @xmath53 and @xmath54 . one then immediately checks that @xmath55hence , in contrast to the linear component @xmath42 , the nonlinear terms @xmath26 corresponding to eqs . ( [ zpzm1 ] ) are not @xmath8symmetric , in the usual matrix sense . rather , the symmetry properties of the nonlinear terms have to be considered in the context of the nonlinear schrdinger equation itself . acting with @xmath56 on eq . ( [ a1 ] ) we observe that @xmath57 ,   \notag \\ i\partial _ { t}(\mathcal{p}\mathbf{t}\mathbf{u } ) & = & h_{l}\mathcal{p}\mathbf{t}\mathbf{u}+h_{nl}(\mathcal{p}\mathbf{t}\mathbf{u})(\mathcal{p}\mathbf{t}\mathbf{u})\label{a13}.\end{aligned}\ ] ] hence , the full nlse system ( [ zpzm1 ] ) remains invariant if we define the @xmath58 transformation of the vectorial wave function obeying this system as follows : @xmath59 this is in full analogy to the condition of exact @xmath27symmetry for the corresponding linear schrdinger equation , nonlinearity matrices @xmath60 of more general type than that in ( [ nl-1 ] ) and ( [ nl-2 ] ) may be envisioned which may produce @xmath61symmetric solutions @xmath62 with less simple time dependence . a detailed analysis of such systems will be presented elsewhere . ] . but the condition of _ spontaneously broken _ @xmath28symmetry ( @xmath29=0 $ ] ,  @xmath31 ) is replaced by the condition of _ completely broken _ @xmath28symmetry ( @xmath31@xmath63@xmath64\neq 0 $ ] ) . in contrast to the present 2d plaquettes , which are mainly motivated by feasible experimental realizations , one can envision more sophisticated setups with  @xmath65 . this will lead to a new type of partial ( or intermediate ) @xmath28symmetry ( to be considered elsewhere ) , which for solutions @xmath66 with broken @xmath28symmetry will keep the nonlinear term @xmath26 explicitly @xmath8symmetric ( @xmath67pseudo - hermitian ) in the matrix sense , but not @xmath28symmetric ( under inclusion of the explicit time reversal @xmath18 ) in the sense of the nlse system .    for configurations ( b ) and ( c ) in fig . [ modes ] , @xmath68 and @xmath26 are still given by eqs . ( [ a3 ] ) and ( [ a5 ] ) , but with @xmath69respectively . hence , relations ( [ a6 ] ) are valid for both configurations ( b ) and ( c ) as well . using ( [ a7 ] ) in @xmath70 , we find a richer variety of parity operators @xmath21 than for configuration ( a ) . configuration ( b ) allows for @xmath71whereas configuration ( c ) may be associated with @xmath72for configuration ( d ) , we have @xmath73and simple computer algebra gives again two possible parity operators : @xmath74 in strong structural analogy to configuration ( b ) . one verifies that @xmath75 holds also for configurations ( b ) , ( c ) and ( d ) , hence all 2d plaquettes considered in the present paper are not @xmath28symmetric in the usual matrix sense . next , we turn to the eigenvalue problems of the linear setups associated with plaquettes ( a ) - ( d ) , i.e. , to solutions of the equation @xmath76from the corresponding characteristic polynomials , @xmath77 , @xmath78 = 0 ,   \notag \\ ( \mathrm{b}):\qquad \qquad & ( e^{2}+\gamma ^{2})\left [ e^{2}-(4k^{2}-\gamma ^{2})\right ] = 0 ,   \notag \\ ( \mathrm{c}):\qquad \qquad & e^{4}-2(2k^{2}-\gamma ^{2})e^{2}+\gamma ^{4}=0 , \notag \\ ( \mathrm{d}):\qquad \qquad & e(e^{2}+\gamma ^{2})\left [ e^{2}-(4k^{2}-\gamma ^{2})\right ] = 0,\end{aligned}\]]we find @xmath79obviously , the matrix hamiltonians @xmath42 for plaquettes ( a ) and ( d ) are not of full rank . for plaquette ( a ) we find @xmath80 , and @xmath42 has a two - dimensional kernel space , @xmath81 . for plaquette ( d ) we find @xmath82 and @xmath83 , where @xmath84 . moreover , we see that the spectrum for plaquette ( d ) , up to the additional eigenvalue @xmath85 , coincides with that for ( b ) . the different eigenvalues of the 4-node plaquettes displayed in eqs . ( [ a15 - 2 ] ) show that these plaquettes are also physically not equivalent . equivalence classes of nonlinear 4-node plaquettes with isospectral linear hamiltonians @xmath86 but different pairwise couplings have been considered , e.g. , in  @xcite . for plaquettes ( a ) , ( b ) and ( d ) an exceptional point ( ep ) occurs at @xmath87 , being associated with a branching of the eigenvalue pair @xmath88 @xmath89 in the case of plaquette ( a ) , all four eigenvalues are involved in the branching at @xmath90 , where @xmath91 . via jordan decomposition ( e.g. , with the help of the corresponding linear algebra tool of mathematica ) we find that @xmath92i.e . , a spectral degeneration of the type @xmath93 in arnold s notation @xcite , or , in other words , a third - order ep with a single decoupled fourth mode . hence , plaquettes of type ( a ) may serve as an easily implementable testground for the experimental investigation of third - order eps ( see e.g. @xcite ) . for plaquettes ( b ) and ( d ) we have second - order eps at @xmath87 , similar as for plaquette ( c ) where a pair of second - order eps occurs at @xmath94 with @xmath95 , @xmath96 .    from the eigenvalues in ( [ a15 - 2 ] ) we read off the @xmath27symmetry content of the four types of plaquettes . the sector of exact @xmath28symmetry ( i.e. , the sector with all eigenvalues purely real , @xmath97 ) corresponds to @xmath98i.e . , for plaquettes ( b ) and ( d ) the @xmath28symmetry is spontaneously broken as soon as the gain - loss coupling is switched on , namely for @xmath99 . in this section , we seek stationary solutions of the type @xmath100constructed over constant vectors @xmath101 . according to ( [ a13 ] ) and ( [ a13 - 2 ] ) such solutions will be @xmath61symmetric provided it holds @xmath102 for some @xmath103 . we will test these symmetry properties for the solutions to be obtained . we note that restricting the explicit analysis to stationary solutions of the type ( [ e ] ) we by construction exclude from this analysis @xmath61violating solutions with @xmath104 which are necessarily non - stationary .    a useful technical tool to facilitate the explicit derivation of stationary solutions @xmath105 are conservation equations of the type @xmath106constructed from eq . ( [ a1 ] ) and its adjoint , where @xmath107 denotes an arbitrary constant matrix . the most simplest of them can be found via eqs . ( [ a2 - 4 ] ) , ( [ a3 ] ) and eq . ( [ a5 ] ) to be @xmath108 \mathbf{u}.   \label{a18 - 2}\end{aligned}\]]for stationary equations @xmath109 the time - dependent phase factors @xmath110 cancel so that the left - hand - sides of these relations vanish , yielding simple algebraic constraints on the right - hand - sides . from eq . ( [ a18 - 2 ] ) we see that for stationary solutions the @xmath6 inner product quantum mechanics ( ptqm ) the @xmath7 inner product was introduced first by znojil in @xcite in 2001 . immediately afterwards , it was interpreted by japaridze as indefinite inner product @xcite in a krein space and generalized by mostafazadeh to the @xmath111metric in the context of pseudo - hermitian hamiltonians @xcite . finally , it was used by bender , brody and jones in 2002 to construct the positive definite @xmath112 inner product @xcite . for oligomer settings ( of plaquettes or other few site configurations ) , it can be employed , e.g. , to derive a simple algebraic constraint or as a criterion of the numerical accuracy of the evolutionary dynamics ( especially since the solutions rapidly acquire very large amplitudes when unstable , as will be seen below ) . it also turned out useful in @xcite . ] will remain conserved ( @xmath113const ) regardless of the violated @xmath114pseudo - hermiticity ,  @xmath115 , characteristic for of our specific nonlinear plaquette couplings ( see eq . ( [ nl-2 ] ) ) . subsequently , we first derive classes of stationary solutions @xmath105 explicitly . then , we analyze the stability of small perturbations over these stationary solutions by the linearization , via ansatz @xmath116 + o(\delta ^{2}),\qquad |\delta |\ll 1,\]]where @xmath117 is the small amplitude of the perturbation . exponents @xmath118 can be defined as wick - rotated eigenvalues from the corresponding @xmath119 perturbation matrix @xmath120 ( see , e.g. , @xcite for more details ) :    @xmath121    where @xmath122\mathbf{u}\]]characterizes the stationary problem , and the elements of the matrix @xmath120 are evaluated at @xmath123 . linear stability is ensured for @xmath124 , whereas @xmath125 corresponds to growing and decaying modes , i.e. , exponential instabilities . substituting ansatz ( [ e ] ) for the stationary solutions in eqs . ( [ a1 ] ) , ( [ a18 - 1 ] ) and ( [ a18 - 2 ] ) we obtain the following algebraic equations : @xmath126@xmath127and @xmath128 \mathbf{u , }   \notag   \label{a31 } \\ 0 & = & \left ( |a|^{2}-|c|^{2}\right ) ( \bar{a}c-\bar{c}a)+\left (    equations can be analyzed via the madelung substitution ( i.e. , via amplitude - phase decomposition ) , @xmath129without loss of generality , we may fix @xmath130 .    for arbitrary phase factors in ( [ aa ] ) , eqs . ( [ a30 ] ) and ( [ a31 ] ) are satisfied by @xmath131 and @xmath132 . using this condition in eq . ( [ zpzm2 ] ) and dividing each equation ( [ zpzm2 ] ) by the phase factor on its left - hand side , one obtains the imaginary parts of the resulting equations : @xmath133 = 2kb\sin \left ( \frac{\phi _ { b}+\phi _ { d}}{2}-\phi _ { a}\right ) \cos \left ( \frac{\phi _ { b}-\phi _ { d}}{2}\right ) ,   \notag \\ \gamma b & = & ka\left [ \sin ( \phi _ { a}-\phi _ { b})+\sin ( \phi _ { c}-\phi _ { b})\right ] = 2ka\sin \left ( \frac{\phi _ { a}+\phi _ { c}}{2}-\phi _ { b}\right ) \cos \left ( \frac{\phi _ { a}-\phi _ { c}}{2}\right ) ,   \notag \\ 0 & = & kb\left [ \sin ( \phi _ { b}-\phi _ { c})+\sin ( \phi _ { d}-\phi _ { c})\right ] = 2kb\sin \left ( \frac{\phi _ { b}+\phi _ { d}}{2}-\phi _ { c}\right ) \cos \left ( \frac{\phi _ { b}-\phi _ { d}}{2}\right ) ,   \notag \\ -\gamma b & = & ka\left [ \sin ( \phi _ { a}-\phi _ { d})+\sin ( \phi _ { c}-\phi _ { d})\right ] = 2ka\sin \left ( \frac{\phi _ { a}+\phi _ { c}}{2}-\phi _ { d}\right ) \cos \left ( \frac{\phi _ { a}-\phi _ { c}}{2}\right ) . \notag \\ & &   \label{zpzm_add}\end{aligned}\]]for @xmath130 the first of these equations implies @xmath134 , hence either @xmath135 ( case 1 ) or @xmath136 ( case 2 ) . in case 1 , we conclude from the third equation that either @xmath137 and @xmath138 ( case 1a ) , or @xmath139 and @xmath140 is arbitrary ( case 1b ) . in case 2 the third equation is satisfied automatically . in all the three cases , the second and the fourth equation are compatible . they give @xmath141returning to the phase - factor divided equations ( [ zpzm2 ] ) and considering their real parts , we find @xmath142 + a^{3 } ,   \notag   \label{a33 } \\ eb & = & ka\left [ \cos ( \phi _ { a}-\phi _ { b})+\cos ( \phi _ { c}-\phi _ { b})\right ] + b^{3 } ,   \notag \\ ea & = & kb\left [ \cos ( \phi _ { b}-\phi _ { c})+\cos ( \phi _ { d}-\phi _ { c})\right ] + a^{3 } ,   \notag \\ eb & = & ka\left [ \cos ( \phi _ { a}-\phi _ { d})+\cos ( \phi _ { c}-\phi _ { d})\right ] + b^{3}.\end{aligned}\]]the pairwise compatibility of the first and third , as well as of the second and fourth equations requires @xmath143for case 1a , these conditions are trivially satisfied , whereas for the remaining cases they lead to further restrictions : @xmath144    in this way the phase angles are fixed for all the three cases and we can turn to the amplitudes . the corresponding equation sets reduce to @xmath145  in the latter two cases ( 1b and 2 ) the amplitudes and phases completely decouple and we have @xmath146case 1a allows for a richer behavior . equating the terms @xmath147 in the upper two equations ( [ a36 ] ) leads to the constraint @xmath148which can be resolved by @xmath149 ( case 1aa ) as well as by @xmath150 ( case 1ab ) . the analysis of these two cases can be completed with the help of the relation @xmath151 from eq . ( [ a32 ] ) .    as result we obtain the following set of stationary solutions : @xmath152    from eq . ( [ a31 ] ) , it can also be seen that either @xmath131 or if @xmath153 , then @xmath154 must be true . here , we use the information available so far to check the @xmath28symmetry content of the solutions ( [ a39 - 1aa ] ) - ( [ a39 - 2 ] ) explicitly . for stationary solutions @xmath155 , @xmath156 the @xmath28symmetry condition ( [ a13 - 2 ] ) implies @xmath157taking into account that @xmath16 acts as complex conjugation , we see from the explicit structure of @xmath158 in eq . ( a8 ) that a stationary solution is @xmath28symmetric if , with @xmath130 , it has @xmath138 and @xmath159 ( up to a common phase shift ) . additionally , the amplitudes have to coincide pairwise : @xmath131 , @xmath132 . for eqs . ( [ a39 - 1aa ] ) - ( [ a39 - 2 ] ) this means that all case-1 stationary solutions with @xmath138 are @xmath28symmetric in their present form . the case-2 mode becomes explicitly @xmath8symmetric after a global @xmath160 multiplication by a phase factor : @xmath161 ^{t } ,   \notag \\ \mathcal{p}\mathcal{t}\mathbf{v}_{0 } & = & \mathbf{v}_{0},\end{aligned}\]]where @xmath162 has to be chosen in eq . ( [ a39 - 2 ] ) . we note that this procedure is effectively equivalent to a redefinition of the original phase constraint : @xmath163 at the very beginning of the calculations in eq . ( [ aa ] ) .    the linear stability analysis was performed numerically . subsequently we present corresponding graphical results . the plaquettes ( b ) - ( d ) can be analyzed in a similar way . for brevity s sake , in fig . [ figzpzm1 ] we present only the basic numerical results , by means of the following symbols :    [ figzpzm1 ]    * case 1aa with @xmath164  blue circles ; * case 1aa with @xmath165  red crosses ; * case 1ab  green stars ; * case 2  black squares ; * case 1b is not depicted explicitly because it corresponds to point configurations without gain - loss ( @xmath166 ) and to exceptional point configurations @xmath90 . figure  [ figzpzm1 ] presents the mode branches ( their amplitudes , phases , and also their stability ) over the gain - loss parameter @xmath2 , starting from the conservative system at @xmath166 . the same symbols are used in fig . [ figzpzm2 ] , which displays typical examples of the spectral plane @xmath167 for stability eigenvalues @xmath168 of the linearization ; recall that the modes are unstable if they give rise to @xmath169 . explicitly we observe the following behavior .    * case 1aa with @xmath164  blue circlesaccording to fig . [ figzpzm2 ] , the present solution is stable . notice that , although featuring a phase profile , it can not be characterized as a vortex state ( the same is true for some other configurations carrying phase structure ) . interestingly , the relevant configuration is generically stable bearing two imaginary pairs of eigenvalues . * case 1aa with @xmath165  red crosses.obviously , this kind of solutions as well as the previous one exist up to the exceptional point @xmath90 of the @xmath0-symmetry breaking in the linear system , where the two branches collide and disappear ( leave the stationary regime and become nonstationary ) . as seen in fig . figzpzm2 , the present branch has two eigenvalue pairs which are purely imaginary for small @xmath2 , but become real ( rendering the configuration unstable ) at @xmath170 and then @xmath171 , respectively . ultimately , these pairs of unstable eigenvalues collide at the origin of the spectral plane with those of the previous branch ( blue circles ) . * case 1ab  green stars.this stationary solution has a number of interesting features . firstly , it is the only one among the considered branches which has two unequal amplitudes . secondly , it exists past the critical point @xmath90 of the linear system , due to the effect of the nonlinearity ( the extension of the existence region for nonlinear modes was earlier found in 1d couplers @xcite and oligomers  @xcite ) . furthermore , this branch has three non - zero pairs of stability eigenvalues , two of which form a quartet for small values of the gain - loss parameter , while the third is imaginary ( i.e. , the configuration is unstable due to the real parts of the eigenvalues within the quartet ) . at @xmath172 , the eigenvalues of the complex quartet collapse into two imaginary pairs , rendering the configuration stable , in a narrow parametric interval . at @xmath173 , the former imaginary pair becomes real , destabilizing the state again , while subsequent bifurcations of imaginary pairs into real ones occur at @xmath174 and @xmath175 ( at the latter point , all three non - zero pairs are real ) . shortly thereafter , two of these pairs collide at @xmath176 and rearrange into a complex quartet , which exists along with the real pair past that point . * case 2  black squares . in contrast to all other branches , this one is _ always _ unstable . one of the two nonzero eigenvalue pairs is always real ( while the other is always imaginary ) , as seen in fig . this branch also terminates at the exceptional point @xmath90 , as relation @xmath177 can not hold at @xmath178 . this branch collides with the two previous ones via a very degenerate bifurcation ( that could be dubbed a double saddle - center " bifurcation ) , which involves 3 branches instead of two as in the case of the generic saddle - center bifurcation , and two distinct eigenvalue pairs colliding at the origin of the spectral plane .    by means of direct simulations , we have also examined the dynamics of the modes belonging to different branches in fig . [ stabzpzm ] . the stable blue - circle branch demonstrates only oscillations under perturbations . this implies that , despite the presence of the gain - loss profile , none of the perturbation eigenmodes grows in this case . nevertheless , the three other branches ultimately manifest their dynamical instability , which is observed through the growth of the amplitude at the gain - carrying site [ b , in fig . [ modes](a ) ] at the expense of the lossy site ( d ) . that is , the amplitude of the solution at the site with the gain grows , while the amplitude of the solution at the dissipation site loses all of its initial power . depending on the particular solution , passive sites ( the ones without gain or loss , such as a and c ) may be effectively driven by the gain ( as in the case of the black - square - branch , where the site a is eventually amplified due to the growth of the amplitude at site b ) or by the loss ( red - cross and green - star branches , where , eventually , the amplitudes at both a and c sites lose all of their optical power ) . we now turn to the generalized ( not exactly @xmath56-symmetric ) configurationsymmetry , spontaneously broken @xmath61symmetry and completely broken @xmath61symmetry see the discussion of eqs . ( [ a13 ] ) and ( [ a13 - 2 ] ) . ] featuring the alternation of the gain and loss along the plaquette in panel ( b ) of fig . [ modes ] . indeed , the absence of @xmath0-symmetry in this case is mirrored in the existence of imaginary eigenvalues in the linear problem of eqs . ( [ a15 - 2 ] ) , as soon as @xmath99 . the corresponding nonlinear solutions ( with @xmath104 ) are not covered by the stationary solution ansatz ( [ e ] ) . stationary solutions ( with @xmath179 ) solely belong to dynamical regimes below the concrete @xmath61thresholds . apart from the two @xmath61symmetry violating solutions , there should exist at least two stationary solutions which we construct in analogy to [ cf . ( [ zpzm2 ] ) ] from @xmath180    substituting the madelung representation ( [ aa ] ) and setting @xmath181 ( for illustration purposes , we focus here only on this simplest case ) , we obtain @xmath182further , fixing @xmath183 , eqs . ( [ phi1 ] ) and ( [ phi2 ] ) yield @xmath184obviously , the solution terminates at point @xmath90 . similar to what was done above , the continuation of this branch and typical examples of its linear stability are shown in figs . [ figpmpm2 ] and [ figpmpm3 ] , respectively . from here it is seen that the blue - circle branch , which has a complex quartet of eigenvalues , is always unstable . in fact , the gain - loss alternating configuration is generally found to be more prone to the instability . the red - cross branch is also unstable via a similar complex quartet of eigenvalues . this quartet , however , breaks into two real pairs for @xmath185 , and , eventually , the additional imaginary eigenvalue pair becomes real too at @xmath186 , making the solution highly unstable with three real eigenvalue pairs . the manifestation of the instability is shown in fig .  [ stabpmpm ] , typically amounting to the growth of the amplitudes at one or more gain - carrying sites . [ figpmpm3 ] we now turn to the plaquette in fig . [ modes](c ) , which involves parallel rows of gain and loss . in this case , the stationary equations are @xmath187 in this case too , we focus on symmetric states of the form of @xmath181 [ see eq . ( [ aa ] ) ] , which gives rise to two solutions displayed in fig . figppmm , represented by the following analytical solutions : @xmath188 @xmath189the analysis demonstrates that the branch with the upper sign in eq . ( + -1 ) is always unstable ( through two real pairs of eigenvalues ) , as shown by blue circles in fig . [ figppmm ] . on the other hand , the branch denoted by the red crosses , which corresponds to the lower sign in eq . ( [ + -1 ] ) is stable up to @xmath190 , and then it gets unstable through a real eigenvalue pair . the black - squares branch with the upper sign in eq .  ( + -3 ) is always stable , while the green - star branch with the lower sign in eq . ( [ + -3 ] ) is always unstable . at the linear-@xmath0-symmetry breaking point @xmath191 , we observe a strong degeneracy , since all the three pairs of eigenvalues for two of the branches ( in the case of the blue circles , two real and one imaginary , and in the case of red crosses one real and two imaginary ) collapse at the origin of the spectral plane . on the other hand , the black - squares branch is always stable with three imaginary eigenvalue pairs , while the green - star branch has two imaginary and one real pair of eigenvalues . between the latter two , there is again a collision of a pair at the origin at the critical condition , @xmath191 . direct simulations , presented for @xmath192 in fig .  [ stabppmm ] , demonstrate the stability of the lower - sign black - squares branch , while the instability of the waveform associated with the blue circles and the green stars leads to the growth and decay of the amplitudes at the sites carrying , respectively , the gain and loss . notice that at the parameter values considered here , the red - cross branch is also dynamically stable as shown in the top right panel of fig . [ stabppmm ] . [ figppmm ] lastly , motivated by the existence of known  cross "- shaped discrete - vortex modes in 2d conservative lattices , in addition to the fundamental discrete solitons  @xcite , we have also examined the five - site configuration , in which the central site does not carry any gain or loss , while the other four feature a @xmath0-balanced distribution of the gain and loss , as shown in panel ( d ) of fig . [ modes ] . seeking for stationary states with propagation constant , @xmath193 [ instead of @xmath194 in eq . ( [ e ] ) , as in this case we reserve label @xmath194 for one of the sites of the 5-site plaquette in fig . [ modes](d ) ] , we get : @xmath195similarly as before , we use the madelung decomposition @xmath196 , cf . ( [ aa ] ) , and focus on the simplest symmetric solutions with @xmath197 . without the loss of generality , we set @xmath138 , reducing the equations to    @xmath198    we report here numerical results for parameters @xmath199 ( smaller @xmath193 yields similar results but with fewer solution branches ) . we have identified five different solutions in this case , see figs . [ figpmzpm1 ] and figpmzpm2 for the representation of the continuation of the different branches , and for typical examples of their stability ( the latter is shown for @xmath200 , @xmath201 and @xmath202 ) . there are two branches ( green stars and black squares ) that only exist at @xmath203 colliding and terminating at that point . one of them has three real eigenvalue pairs and one imaginary pair , while the other branch has two real and two imaginary pairs . two real pairs and one imaginary pair of green stars collide with two real pairs and one imaginary pair of black squares , respectively , while the final pairs of the two branches ( one imaginary for the green stars and one real for the black squares ) collide at the origin of the spectral plane . these collisions take place at @xmath204 , accounting for the saddle - center bifurcation at the point where those two branches terminate . on the other hand , there exist two more branches ( red crosses and magenta diamonds in fig .  [ figpmzpm1 ] ) , which collide at @xmath205 . one of these branches ( the less unstable one , represented by magenta diamonds ) bears only an instability induced by an eigenvalue quartet , while the highly unstable branch depicted by the red crosses has four real pairs ( two of which collide on the real axis and become complex at @xmath206 ) . last but not least , the blue circles branch does not terminate at @xmath207 , but continues to larger values of the gain - loss parameter , @xmath208 . it is also unstable ( as the one represented by the magenta diamonds ) due to a complex quartet of eigenvalues . the dynamics of the solutions belonging to these branches is shown in fig . [ stabpmzpm ] . for the branches depicted by black squares and green stars ( recall that they disappear through the collision and the first saddle - center bifurcation at @xmath204 ) , the perturbed evolution is fairly simple : the amplitudes grow at the gain - carrying sites and decay at the lossy ones , while the central passive site ( c ) stays almost at zero amplitude . for the other branches , the amplitudes also grow at the two gain - carrying sites and decay at the lossy elements , while the passive site may be drawn to either the growth or decay . [ figpmzpm1 ]    [ figpmzpm2 ] in the present work , we have proposed generalizations of the one - dimensional @xmath0-symmetric nonlinear oligomers into two - dimensional plaquettes , which may be subsequently used as fundamental building blocks for the construction of @xmath0-symmetric two - dimensional lattices . in this context , we have introduced four basic types of plaquettes , three of which in the form of four - site squares . the final one was in the form of the five - site cross , motivated by earlier works on cross - shaped ( alias rhombic or site - centered ) vortex solitons in the discrete nonlinear schrdinger equation . our analysis was restricted to modes which could be found in the analytical form , while their stability against small perturbations was analyzed by means of numerical methods . even within the framework of this restriction , many effects have been found , starting from the existence of solution branches that terminate at the critical points of the respective linear @xmath0-symmetric systems  e.g. , in the settings corresponding to plaquettes ( a ) and ( c ) in fig . [ modes ] . the bifurcation responsible for the termination of the pair of branches may take a complex degenerate form [ such as the one in the case of setting ( a ) ] . other branches were found too , that continue to exist , due to the nonlinearity , past the critical points of the underlying linear systems . in addition , we have identified cases [ like the gain - loss alternating pattern ( b ) or the cross plaquette of type ( d ) ] when the @xmath0 symmetry is broken immediately after the introduction of the gain - loss pattern . the spectral stability of the different configurations was examined . most frequently , the stationary modes are unstable , although stable branches were found too [ e.g. , in settings ( a ) and ( c ) ] . we have also studied the perturbed dynamics of the modes . the evolution of unstable ones typically leads to the growth of the amplitudes at the gain - carrying sites and decay at the lossy ones . it was interesting to observe that the passive sites , without gain or loss , might be tipped towards growth or decay , depending on the particular solution ( and possibly on specific initial conditions ) . the next relevant step of the analysis may be to search for more sophisticated stationary modes ( that plausibly can not be found in an analytical form ) , produced by the _ symmetry breaking _ of the simplest modes considered in this work , cf . ref . the difference of such modes from the @xmath0-symmetric ones considered in the present work is the fact that modes with the unbroken symmetry form a continuous family of solutions , with energy @xmath194 depending on the solution s amplitude , see eq . ( [ e ] ) . this feature , which is generic to conservative nonlinear systems , is shared by @xmath0-symmetric ones , due to the automatic " balance between the separated gain and loss . on the other hand , the breaking of the symmetry gives rise to the typical behavior of systems with competing , but not explicitly balanced , gain and loss , which generate a single or several _ attractors _ , i.e. , _ isolated _ solutions with a single or several values of the energy , rather than a continuous family . a paradigmatic example of the difference between continuous families of solutions in conservative models and isolated attractors in their ( weakly ) dissipative counterparts is the transition from the continuous family of solitons in the usual nlse to a pair of isolated soliton solutions , one of which is an attractor ( and the other is an unstable solution playing the role of the separatrix between attraction basins , the stable soliton and the stable zero solution ) in the complex ginzburg - landau equation , produced by the addition of the cubic - quintic combination of small dissipation and gain terms to the nlse @xcite . as concerns the systems considered in the present work , in the context of the breaking of the @xmath0 symmetry it may also be relevant to introduce a more general nonlinearity , which includes @xmath0-balanced cubic gain and loss terms , in addition to their linear counterparts ( cf . @xcite and @xcite ) . nevertheless , it should also be noted that the issue of potential existence of isolated solutions versus branches of solutions in @xmath0-symmetric systems is already starting to be addressed in the relevant literature ( including in plaquette - type configurations ) , as in the very recent work of  @xcite . moreover , the present work may pave the way to further considerations of two - dimensional @xmath0-symmetric lattice systems , and even three - dimensional ones . in this context , the natural generalization is to construct periodic two - dimensional lattices of the building blocks presented here , and to identify counterparts of the modes reported here in the infinite lattices , along with new modes which may exist in that case . on the other hand , in the three - dimensional realm , the first step that needs to be completed would consist of the examination of a @xmath0-symmetric cube composed of eight sites , and the nonlinear modes that it can support . this , in turn , may be a preamble towards constructing full three - dimensional @xmath0-symmetric lattices . these topics are under present consideration and will be reported elsewhere . ug thanks holger cartarius and eva - maria graefe for useful discussions . pgk gratefully acknowledges support from the national science foundation under grant dms-0806762 and cmmi-1000337 , as well as from the alexander von humboldt foundation and the alexander s. onassis public benefit foundation . pgk and bam also acknowledge support from the binational science foundation under grant 2010239 . 99 c. m. bender and s. boettcher , phys . rev . lett . * 80 * , 5243 ( 1998 ) ; c. m. bender , s. boettcher and p. n. meisinger , j. math . phys . * 40 * , 2201 ( 1999 ) . z. h. musslimani , k. g. makris , r. el - ganainy and d. n. christodoulides , phys . lett . * 100 * , 030402 ( 2008 ) ; k. g. makris , r. el - ganainy , d. n. christodoulides and z. h. musslimani , phys . a * 81 * , 063807 ( 2010 ) . b. a. malomed and h. g. winful , phys . rev . e * 53 * , 5365 ( 1996 ) ; h. sakaguchi and b. a. malomed , physica d * 147 * , 273 ( 2000 ) ; w. j. firth and p. v. paulau , eur . j. d * 59 * , 13 ( 2010 ) ; p. v. paulau , d. gomila , p. colet , n. a. loiko , n. n. rosanov , t. ackemann , and w. j. firth , opt . express * 18 * , 8859 ( 2010 ) ; a. marini , d. v. skryabin , and b. a. malomed , _ ibid_. * 19 * , 6616 ( 2011 ) ; p. v. paulau , d. gomila , p. colet , b. a. malomed , and w. j. firth , phys . rev . e * 84 * , 036213 ( 2011 ) . m. znojil , rendic . circ . mat . palermo , ser . ii , suppl . * 72 * ( 2004 ) , 211 - 218 , math - ph/0104012 . g. s. japaridze , j. phys . a : math . theor . * 35 * , 1709 - 1718 ( 2002 ) , quant - ph/0104077 . a. mostafazadeh , j. math . phys . * 43 * , 205 - 214 ( 2002 ) , math - ph/0107001 . c. m. bender , d. c. brody , and h. f. jones , phys . lett . * 89 * , 270401 ( 2002 ) , quant - ph/0208076 .
we introduce four basic two - dimensional ( 2d ) plaquette configurations with onsite cubic nonlinearities , which may be used as building blocks for 2d @xmath0-symmetric lattices . for each configuration 
 , we develop a dynamical model and examine its @xmath1symmetry . 
 the corresponding nonlinear modes are analyzed starting from the hamiltonian limit , with zero value of the gain - loss coefficient , @xmath2 . 
 once the relevant waveforms have been identified ( chiefly , in an analytical form ) , their stability is examined by means of linearization in the vicinity of stationary points . 
 this reveals diverse and , occasionally , fairly complex bifurcations . 
 the evolution of unstable modes is explored by means of direct simulations . in particular , 
 stable localized modes are found in these systems , although the majority of identified solutions is unstable .