VAAbstractIn this paper VA is generalized for solutions with possible elliptic structure. Our analyticalconsiderations have allowed dening the soliton prole for which the radial-symmetric shape isstable. Small deviations from such a shape lead to oscillatory solutions having two independenteigenfrequencies. The behavior of such solutions in a potential with large amplitude is studiednumerically. Also it is considered the resonant case when the frequency of the time variation (timemanaged) potential is close to one of the eigenfrequencies. In this situation even stable (at staticpotential) solution leads to a weak time decay of the soliton state.1I. INTRODUCTIONThe objective of the present work is to construct families of soliton solutions and iden-tify their stability boundaries in the 2D attractive model cubic-quintic nonlinearity (CQN)with the full 2D lattice subject to the time-periodic modulation 1 + (,2) cos(t),and being the amplitude and frequency of its temporal modulation. The dynamics of solitons inCQN diers considerably from pure cubic nonlinear case.Normally symmetric variationalapproximation (VA) helps one to understand what may happen to the 2D soliton underthe action of the weak management", with ,2 1, in the near-critical situation, i.e.,for 1 `, `max 1. For cubic nonlinearity such a model was considered in [32]. In thenormalized form, the respective 2D Gross-Pitaevskii equation for mean-eld wave function(r. . t) in cubic-quintic nonlinearity case isiJJt = 12 J2Jr2 + J2J2+q ||2+i||5\01(t)\ (r. ). (1)where t is time, (r. ) are coordinates in the 2D plane (scaled so that the OL period is:), \0 is the depth of the lattice, 1(t) = 1 + (,2) cos(t),and being the amplitudeand frequency of its temporal modulation. The coecient q < 0 ( further we normally useq = 1) in front of the cubic nonlinear term in Eq. (1) implies that the nonlinearity isattractive. Actually, \0 is measured in units of the recoil energy of atoms trapped in theOL. For a typical case of atoms of 71i loaded in the OL with the period on the order of jm,a characteristic value of the scaled frequency, 2 (see below), corresponds, in physicalunits, to the modulation rate on the order of several KHz. Here, where is the frequency, is the amplitude, \0 is the optical potential, t is the time, and (r. ) are coordinates in 2Dof the OL. The quasiperiodic (QP) lattice potential of depth 2\0 is taken as [6, 24, 25]\ (r. ) = \01Xa=1cos(k(a)r), (2)with the set of wave vectors k(a)= {cos (2:(: 1),1) . sin (2:(: 1),1)} and 1 = 5 or1 7. Here, following Ref. [24], we focus on the basic case of the Penrose-tiling potential,corresponding to 1 = 5. The 2D prole of such QP potential is displayed below in Fig. 1.2FIG. 1: (Color on line.) Quasiperiodic potential \ (r, j) in Eq.(2) for 1 = 5.II. THE ASYMMETRIC VARIATIONAL APPROXIMATIONVariational methods have been quite useful in many problems of nonlinear optics andBEC [4, 5, 13, 3335]. To apply the VA to the present model, we notice that Eq. (1) canbe derived from Lagrangian 1 = R + Ldr, with densityL = i2 (tt) 12|a|2+ |j|2 + q2 ||4+ i3 ||6+\0\ (r. )1(t) ||2. (3)where the asterisk stands for the complex conjugation. For isotropic potential the simplestisotropic ansatz [4] and [5] for the soliton can be used. However for asymmetric potential dueto insensitive to the particular orientation of wave vectors k(a), this isotropic approximationis too coarse and has to be generalized. Such approach for asymmetric potential we deneas an anisotropic ansatz that has to be written asans (r. . t) = (t) expic(t) + i2(/1(t)r2+/2(t)2) 12r2\21(t) +2\21(t). (4)3where 1(t) = 1 + .2 cos(.t) and all variables (t). c(t). /1,2(t). and \1,2(t) (amplitude,phase, radial chirp, and soliton width, respectively) are real. It is considered case q = 1.The substitution of the ansatz in Eq. (3) and calculation of the integrals yield the eectiveLagrangian,1)) = ` c 14\1t2`/1 14\22`/2 14 `\21 14\21`/21 14 `\22 14\22`/22(5)+`\01 (t)5e2W21116 +2` e2(W1(t))2115132\01 (t)5e2W22132 e2W22115132 e2W21132+2` e2W22125132\01 (t)5e2W22132 e2W12125132 e2W21132 +`24: \2\1 i`36:2\21\22 .where the overdot stands for the time derivative, and ` :2\1\2. In case \1 = \2 =\ and /1 = /2 = / the Lagrangian (5) can be simplied to standard isotropic Lagrangian[4] and [5].The rst Euler-Lagrange equation following from eective Lagrangian (5), o1e,oc = 0(o,oc stands for the variational derivative), is tantamount to the conservation of the normof the wave function, which is the single dynamic invariant of Eq. (1). Indeed, the norm ofansatz (4) isZ Z |ans(r. )|2drd = :2\1\2 `. (6)The second Euler-Lagrange equation, o1e,o/1,2 = 0, reduces to the well-known expressionfor the chirp in terms of the time derivative of the width [34, 35], /i =\i,\i (i = 1. 2).Using this relation, the next variational equation, J1e,J (\2i ) = 0, can be cast in thefollowing equations for \1,24\1 =1\13 \1\01 (t) :220e2W12116 +\1e2W12115132\01 (t) :211520e2W22132 e2W22115132 e2W12132(7)\1e2W12115132\01 (t) :220e2W22132 e2W22115132 e2W12132 \1e2W22125132\01 (t) :212520e2W22132 e2W12125132 e2W12132(8)\1e2W22125132\01 (t) :220e2W22132 e2W12125132 e2W12132 `2\12\2: +2`2i3\13\22:2(9)and\2 =1\23 \2e2W12115132\01 (t) :220e2W22132 e2W22115132 e2W12132 \2e2W12115132\01 (t) :211520e2W22132 e2W22115132 e2W12132(10)\2e2W22125132\01 (t) :220e2W22132 e2W12125132 e2W12132 +\2e2W22125132\01 (t) :212520e2W22132 e2W12125132 e2W12132(11)`2\22\1: +2`2i3\23\12:2. (12)where 1i) = cos(5i,,). In Eq. (7) (and similarly for Eq. (10)) in a free potential case(\0 = 0) and i = 0 the right hand side can be written as \13 h1 `, `max(\1,\2)i.Here `max 2: is the well-known VA prediction [35] (at i = 0) for the critical (maximum)norm in the 2D space, which separates collapsing solutions at ` `max, i.e., ones with\(t) 0.(t) at t tcollapse [for \0 = 0 and initial conditions \(t = 0) = \0and\(t = 0) = 0, the collapse time predicted is tcollapse = \20`, `max112], andnoncollapsing ones at `