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the studies of nonlinear excitations of two - dimensional ( 2d ) and quasi 2d correlated spin systems are an important issue of modern physics of magnetism , and can be useful for development of general soliton concepts . @xcite the topological textures like localized solitons ( skyrmions @xcite ) or magnetic vortices make an important contribution to the thermodynamics of magnetically ordered systems @xcite or even determine the character of its ordering as in the case of berezinskii - kosterlitz - thouless transition . @xcite last years the interest for two - dimensional solitons has grown since they are frequently realized as ground state in the finite - size mesoscopic magnetic samples , so - called magnetic dots .  @xcite    the most `` famous '' topological solitons are magnetic vortices having @xmath9 topological charge . these vortices are usually related to thermodynamic aspects of soliton physics . @xcite also , they appear in mesoscopic nanostructures .  @xcite easy - plane magnets with continuously degenerated ground state have vortices with the energy being logarithmically divergent as a function of system size . the other and much less studied example of topological solitons are magnetic skyrmions which are present in isotropic or easy - axis magnets . contrary to the above vortices , the latter textures are characterized by nontrivial @xmath0 topological charge and finite energy . it is known that they determine the response functions of 2d magnets at finite temperatures @xcite and take part in long - range order breakdown in isotropic magnets . @xcite the skyrmions form ground state of magnetic nanoparticles with easy - axis anisotropy . @xcite their analysis is more complicated as compared to magnetic vortices and comprises many nontrivial features . an important example of latter features is the problem of a skyrmion stability since due to hobart - derrick theorem the static solitons with finite energy are unstable for wide class of models including standard continuous magnetic models . @xcite    for magnetic vortices , the consideration of lowest possible topological charge @xmath10 is sufficiently as the vortex energy grows with @xmath3 , @xmath11 . because of that it is advantageous for a vortex with @xmath12 to decay for @xmath13 vortices with @xmath10 and the vortices with @xmath7 can be stable in exceptional cases only . @xcite the situation for skyrmions is not that simple . the simplest continuous model for isotropic 2d ferromagnet ( fm ) @xmath14 ^ 2d^2x\ ] ] admits the well - known belvin - polakov ( bp ) solution ,  @xcite which reads @xmath15 where @xmath16 is normalized magnetization @xmath17 @xmath18 is a spin value , @xmath19 is a lattice constant of a 2d fm , @xmath20 is its exchange constant , @xmath21 and @xmath22 are polar coordinates in the @xmath23plane , @xmath24 is an arbitrary constant . solution has the energy @xmath25 so that the state of bp skyrmions with @xmath12 merges or dissociates into several other similar skyrmions with different @xmath26 s ( the only rule that in such process the topological charge should conserve ) without their energy altering . such exact degeneration is related to very high hidden symmetry , stemming from exact integrability of corresponding static model ( see , e.g. ,  @xcite ) . this degeneration should certainly be lifted if we go beyond the model . the most important characteristic here is the parameter @xmath27 , which is appropriate to call the _ topological energy density _ , @xmath28 where @xmath29 is the energy of a soliton with topological charge @xmath13 . if @xmath27 is a growing function of @xmath13 , the most favorable state with a given @xmath13 comprises @xmath3 solitons with unit topological charge otherwise such state is unstable . latter question is especially important for the investigation of general regularities of the highly excited magnet states evolution , ( see , e.g. ref .   and references therein ) or for the analysis of essentially inhomogeneous magnet states under strong pumping . @xcite latter states can be generated by the ultrafast pulses of magnetic field , see refs . for details . the preceding discussion demonstrates that the problem of obtaining and investigation of the stable skyrmions with higher topological charges is extremely important . @xcite    the present work is devoted to the analysis of skyrmions with higher @xmath0 topological charges in 2d heisenberg ferromagnet with uniaxial anisotropy . we show that there exists a certain range of system parameters ( exchange and anisotropy constants ) , where stable precessional solitons with topological charge @xmath30 exist . it turns out that in wide range of anisotropy constants , the topological energy density @xmath31 of the textures with @xmath30 is lower then that of the textures with @xmath32 . on the other hand , the solitons with @xmath5 and @xmath33 have monotonously growing phase boundary functions @xmath34 , while the case @xmath7 has peculiar nonmonotonous behavior , determining the transition regime from low to high topological charges . this means that the preferable values of soliton topological charge are neither @xmath5 nor high charges , but rather @xmath7 or @xmath8 . we begin with the discrete model of a classical 2d fm with uniaxial anisotropy , described by the following hamiltonian @xmath35.\end{aligned}\ ] ] here @xmath36 is a classical spin vector with fixed length @xmath18 on the site @xmath37 of a 2d square lattice . the summations run over all lattice sites @xmath37 and nearest neighbors @xmath38 , @xmath39 is the exchange integral and the constant @xmath40 describes the anisotropy of spin interaction . in subsequent discussion , we refer to this type of anisotropy as exchange anisotropy ( ea ) . additionally , we took into account single - ion anisotropy ( sia ) with constant @xmath41 . we consider @xmath42axis to be easy magnetization direction so that @xmath43 or @xmath44 . the analysis of real magnetic systems with discreet spins can be performed only numerically . in principle , it can be done by the same method as was described in ref .  , but it is not easy to extract necessary information from the set of numerical data . on the other hand , if we neglect the specific effects of discreteness ( which appear at strong anisotropy only , @xmath45 ) , like the presence of pure collinear structures , see ref .  , the consideration can be simplified using the generalized continuous approximation and classical landau - lifshitz equation . in this approximation we can introduce the smooth function @xmath46 instead of discreet variable @xmath47 . in this case , the classical magnetic energy functional @xmath48 $ ] can be constructed expanding the discrete hamiltonian in power series of magnetization gradients , yielding @xmath49 where @xmath50 contains zeroth and second order contributions to magnetic energy , see , and @xmath51 contains the fourth powers of gradients . the explicit ( and quite cumbersome ) expressions for @xmath50 and @xmath51 had been presented in ref .  . we note here , that single - ion anisotropy enters only @xmath50 , but not @xmath51 and higher terms , while exchange anisotropy enters every term of the expansion .    for isotropic case @xmath52 and @xmath53 , @xmath50 coincides with the energy of the isotropic continuous model bearing bp soliton solutions of the form with degenerate ( with respect to topological charge ) topological energy density , see also . a simple accounting of magnetic anisotropy in @xmath50 generates a model , which is typical example of the models governed by hobbard - derrick theorem - it does not admit static stable soliton solutions . in the model with @xmath54 only , the size of any texture like domain wall , soliton etc ( see , e.g. ref . for details ) is given by the characteristic length @xmath55 @xmath56 in the case of weak anisotropy , @xmath57 , the length scale @xmath58 so that the magnetization varies slowly in a space . now we consider generalized model , including higher powers of gradients . in the expansion , we limit ourselves to the terms of fourth order only as they are playing a decisive role in solitons stabilization , see refs .   for details . the simplest possible generalized model with account for @xmath51 only can in principle admit the above static stable solitons . @xcite simple scaling arguments can demonstrate that . namely , if the soliton core has size @xmath59 , the specific calculations with energy functionals like ( see below ) yield the dependence of the energy @xmath60 on @xmath59 in the form @xmath61 where the first term comes from @xmath51 and the rest come from @xmath50 . if @xmath62 had a minimum ( this occurs if @xmath63 ) , this would mean the existence of a stable static skyrmion . unfortunately , for the real model of magnet @xmath64 so that the first term is negative and the dependence @xmath65 does not have a minimum . we note also that the expansion and the above scale arguments are valid for any symmetry of initial 2d discrete lattice . the only difference is in the coefficients before gradient powers . although these coefficients influence the solitons properties , the main feature of these expansions , consisting in the fact that in eq . @xmath66 remains the same . in other words , we did not find any symmetry of 2d fm with nearest - neighbor ferromagnetic interaction , where @xmath67 has a minimum so that stable static soliton can exist . @xcite    in the absence of _ static _ two  dimensional solitons , it is possible to construct stable soliton states with _ stationary dynamics _ , due to the presence of additional integrals of motion for magnetization fields . the purely uniaxial model possesses the exact symmetry with respect to the spin rotation around @xmath68axis so that the energy functional @xmath69 $ ] does not depend explicitly on the variable @xmath70 . this leads to the appearance of an additional integral of motion , @xmath42-projection of total spin . this integral of motion can be conveniently parameterized via integer @xmath71 defining a number of bound magnons in a soliton @xmath71 , see ref .   for details . in continuous approximation it can be written as @xmath72 conservation of @xmath71 leads to the presence of so  called _ precessional _ solitons characterizing by time - independent projection of magnetization onto the easy @xmath68axis and with the precession of the magnetization vector @xmath73 at constant frequency @xmath74 around the @xmath42 axis , @xmath75 which holds instead of ( [ qe2 ] ) in this case . the analogs of such precessional solitons are known to occur in different field - theoretical models , it is enough to note the non - topological coleman s @xmath76-balls ,  @xcite which do not have topological properties as well as @xmath0 topological @xmath76 - lumps ,  @xcite see ref .   for details .    stable dynamical solutions with nonzero @xmath74 correspond to conditional ( for fixed @xmath71 value ) minimum of the energy functional @xmath77 . namely , we may look for an extremum of the expression @xmath78 where @xmath79 is an internal soliton precession frequency , which in this case can be regarded as lagrange multiplier . note , that functional is nothing but the lagrangian of 2d fm magnetization field calculated with respect to specific time dependence . this condition leads to the relation @xcite @xmath80 , which determines the microscopic origin of the precessional frequency @xmath74 . namely , an addition of one extra spin deviation ( bound magnon ) to a soliton changes its energy by @xmath81 . thus , the dependencies @xmath82 and @xmath83 are very important for the problem of a soliton stability . further analysis of above continuous model consists in the solution of differential equations for soliton structure , what is equivalent to the minimization of the functional . as these equations can barely be solved analytically , here we analyze the solitons properties by direct variational method . as we have shown earlier by comparison of variational approach and direct numerical minimization of initial discreet energy on a lattice , @xcite the variational approach gives fairly good results for weak anisotropies @xmath84 , where the generalized continuous description is valid . the continuous models like and are usually parameterized by angular variables so that the energy @xmath29 becomes a functional of these variables , @xmath85 $ ] . having energy functional @xmath29 , we can write the corresponding landau - lifshitz equations and lagrangian .    to apply direct variational method for minimization of the energy @xmath86 we use the trial function @xmath87 where @xmath88 is the mcdonald function with index @xmath3 . @xcite note , that trial function ( [ macd ] ) is based on the interpolative solution , constructed in ref .  . the trial function gives correct asymptotics both for @xmath89 ( corresponding to bp soliton ) and for @xmath90 ( exponential decay with some characteristic scale @xmath91 ) , see refs .   for details . latter exponential asymptotics is absent for solitons in isotropic 2d fm , e.g. for bp soliton . it can be shown that the exponential asymptotics occurs for anisotropic models , where the length of decay is proportional to @xmath55 . it had been demonstrated that due to power - law asymptotics of @xmath92 for isotropic magnet , the integrals defining soliton energy and number of bound magnons ( see below eqs . ) are divergent for @xmath5 . @xcite to avoid this divergency in anisotropic models like , the interpolative ( between bp asymptotics at @xmath89 and exponential one at @xmath93 ) solution had been put forward in ref . . our analysis of continuous model shows that at @xmath5 the above divergence can be cut off by the exponential asymptotics only . for @xmath7 the power - law asymptotics is sufficient for convergence of corresponding integrals , while their derivatives with respect to parameter @xmath94 are divergent . in this case , the divergences in derivatives are also eliminated by exponential asymptotics . at @xmath95 all integrals and their derivatives ( with respect to @xmath94 ) are convergent . this means that the behavior of solitons with @xmath5 and 2 on one side and those with @xmath95 on the other side is different , being determined by the interplay between effects of anisotropy and higher spatial derivatives . in our minimization method , the parameter @xmath96 is variational , while the parameter @xmath59 is kept constant as it is related to @xmath71 , @xmath97 , see , e.g. @xcite . in other words , we minimize the energy @xmath98 with the trial function ( [ macd ] ) over @xmath96 for constant @xmath59 . this approach has the advantage that it also permits to investigate the stability of obtained soliton texture . namely , a soliton is stable if it corresponds to the conditional minimum of the energy at fixed @xmath71 , and it is unstable otherwise .    to proceed further , we introduce following dimensionless variables @xmath99 and express trial function in terms of them . we have @xmath100 then using the equation , we can calculate the number of bound magnons in the soliton @xmath101 in the variables the soliton energy assumes the form @xmath102 where    @xmath103 ,   \nonumber \\ & & \gamma _ 4(z)=\int_0^{\infty } xdx\biggl\{\left ( \delta _ x\theta \right)^2\left ( 1+\kappa \sin ^2\theta \right ) + \theta^{\prime 4 } \left ( 1+\kappa \cos ^2\theta \right ) + \frac{\nu^2\sin ^2\theta } { x^2}% \left ( \frac{\nu^2}{x^{2}}+2\theta ^{\prime 2}\right ) +   \nonumber \\ & & + \delta _ { x}\theta \sin 2\theta \left ( \kappa \theta ^{\prime 2}-\frac{% \nu ^2}{x^2}\right ) \biggr\},\ \ \theta ' = \frac{d\theta}{dx } , \ \delta _ x\theta = \frac{d ^2 \theta}{dx^2 } + \frac 1x\frac{d \theta}{dx}\;. \label{asm7}\end{aligned}\ ] ]    thus , we express the energy and the number of magnons via two parameters , @xmath104 and @xmath42 . it turns out , that initial dimensional variables @xmath96 and @xmath59 enter the problem only in the form of their product @xmath42 . the dependence of @xmath71 and @xmath29 on @xmath42 enters the problem via a few complicated functions @xmath105 and @xmath106 , which can be written only implicitly in the form of integrals . however , in terms of these functions , the dependence on @xmath107 ( [ dimen ] ) turns out to be quite simple . this permits reformulation of the initial variational problem in terms of variables @xmath42 and @xmath71 only . namely , we express @xmath108 and substitute this expression into the dimensionless energy ( [ dimen ] ) . this gives us the expression for the energy of a soliton with given @xmath71 , as a function of variational parameter @xmath42 . then we can find a minimum of @xmath29 with respect to @xmath42 , keeping @xmath71 constant . the result of such numerical minimization in the form of the dependence @xmath109 is shown in fig.[fig : v1 ] for a magnet with purely exchange anisotropy @xmath110 . the curves for higher @xmath40 are qualitatively the same . to justify the applicability of our direct variational approach , earlier we have shown,@xcite that for @xmath5 the dependencies @xmath111 found by variational and numerical minimizations of the energy , are identical at small enough anisotropy , @xmath112 . our analysis shows ( see also fig.[fig : v1 ] ) that topological energy density grows slowly as function of @xmath71 . we note here that this property holds for all anisotropy constants and topological charges . even more interesting is the fact that all curves @xmath109 have threshold points @xmath113 so that solitons exist only at @xmath114 and @xmath115 . these threshold values determine the minimal soliton energy , which is most important characteristic of soliton contribution into magnet thermodynamics and can be observed experimentally . @xcite for instance ( see fig.[fig : v1 ] ) , at @xmath10 and @xmath116 @xmath117 , which is substantially higher then that expected from eq . @xmath118 . [ fig : v1 ] demonstrates one more unexpected soliton property , namely that at @xmath116 the energy density @xmath111 _ decreases _ with increase of @xmath13 and fixed @xmath71 value . as we will show below , such behavior occurs for many values of @xmath119 , although there can be exceptions . the most important feature , however , is the existence of above threshold energy and bound magnons number values . the behavior of these values at variation of anisotropy constants is quite nontrivial . for example , at @xmath116 ( the value chosen for fig . [ fig : v1 ] ) @xmath120 but @xmath121 . at the same time , the corresponding threshold energies behave monotonically @xmath122 . our extensive analysis of numerical curves @xmath109 for different anisotropies have shown that their overall behavior is dictated primarily by the threshold values : if @xmath123 , then the entire curve @xmath109 lies above corresponding curve @xmath124 in wide interval of @xmath71 s . this shows the importance of the above threshold values for the properties of solitons . the information about these values can be conveniently represented in the form of so - called solitons phase diagram , i.e. the dependence of @xmath125 and @xmath126 on @xmath127 . to obtain the phase diagram , we should pay attention to the details of above minimization procedure . namely , to obtain truly stable soliton texture we should demand that the conditional extremum of the energy @xmath60 is a minimum . so , we keep track not only to the first derivative @xmath128 to be zero but also to the second derivative to be positive ( corresponding to a minimum ) at the point @xmath129 , where @xmath130 . when a soliton approaches the limit of its stability , the modulus of second derivative diminishes , becoming zero at the stability limit @xmath131 , corresponding to above values @xmath132 and @xmath133 . thus the above soliton phase diagram is indeed determined by the instability point @xmath131 ( as a function of ratio @xmath134 ) where both first and second derivatives of the energy are zero . the shape of the above phase diagram is determined by the character of anisotropy and @xmath3 value and is reported on fig . [ fig : v2 ] for exchange anisotropy . first of all , one can see that the dependence @xmath135 is quite complicated and its character change twice , at @xmath136 and at @xmath137 . it is seen from fig.[fig : v2]a that while there is a quite large region of @xmath4 values , where @xmath138 and @xmath139 are smaller then @xmath140 , the region where @xmath141 is approximately three times smaller . we have shown that @xmath142 is always larger then @xmath139 although still smaller then @xmath143 . the energy @xmath135 is growing with @xmath3 at @xmath144 only . [ fig : v2]b demonstrates the divergence of @xmath145 at small @xmath146 as @xmath5 and 2 . this is related to the fact that in bp soliton the integral describing @xmath71 diverges logarithmically as @xmath93 for @xmath10 , see also above . overall , fig . [ fig : v2 ] demonstrates the complicated behavior of solitons phase diagram , which is needed to be understood . to understand better the above complex behavior , it is instructive to obtain the phase diagram analytically . without loss of generality , we shall do so for the case of uniaxial anisotropy only . such analytical treatment is possible since the anisotropy constant @xmath41 in this case enters the problem only via coefficient in the first term of eq . . this permits to obtain the analytic dependencies @xmath147 and @xmath148 in the inverse form @xmath149 . note , that the structure of eq . suggests that the above analytical procedure is applicable for both exchange and uniaxial anisotropy . our analysis shows that this does not change the situation qualitatively . moreover , the quantitative results are close to each other , compare figs . [ fig : v2 ] and [ fig : w4 ] .    as it was shown above , the soliton phase diagram is determined by the instability point @xmath131 . to obtain an equation for latter point , we rewrite the eq . in the form @xmath150 ( function @xmath151 is defined by eq . ) and equate to zero the first and second derivatives of @xmath27 with respect to @xmath42 . this gives the equation for the dependence @xmath152 in the form @xmath153 where primes mean corresponding derivatives . then , the equation for @xmath131 can be obtained by substitution of above @xmath154 into the one of the equations determining zero of first or second energy derivatives with respect to @xmath42 . it turns out that this equation always has real solutions if we formally admit the existence of negative @xmath41 . the dependence permits to obtain the equation for the phase diagram in the implicit form @xmath155 , which reads @xmath156,\nonumber \\ & & q(z)=\frac{\gamma_0(z)}{2\psi(z)}.\end{aligned}\ ] ]    the dependence @xmath157 , obtained by inversion of eq . , is reported on fig . [ fig : w3a ] . it is seen that while the entire curves @xmath157 at @xmath5 and 2 lie at @xmath43 , the curves for @xmath95 lie in this range only partially . since for uniaxial anisotropy @xmath41 can be only positive , only those parts of the curves with @xmath95 where @xmath43 correspond to physically realizable case . this clarifies the reason why the dependence @xmath158 for skyrmions with @xmath95 begins from finite @xmath71 values , see fig . [ fig : v2]b . this different behavior is related to the different kind of convergence of corresponding integrals at @xmath5 and 2 as well as for @xmath95 , see discussion above . the dependence @xmath159 can be easily recalculated to the solitons phase diagram @xmath160 and @xmath161 . this diagram is shown on figs . [ fig : w4]a , b . it is seen the qualitative coincidence with the numerical curves from figs.[fig : v2]a , b . the details of behavior can now be better seen then from above numerical curves . our analysis shows that aforementioned different character of convergence of integrals influences asymptotics of the phase diagram curves at small anisotropies . this influence can be seen from figs . [ fig : v2]a and [ fig : w4 ] , where the dependence @xmath162 at @xmath5 and 2 is nonanalytical at @xmath163 , while for @xmath95 it is analytical . the reason for such behavior can be seen from fig . [ fig : w3a ] , showing that small @xmath164 correspond to small @xmath42 for @xmath5 and 2 , while for @xmath33 all values of @xmath164 including limiting case @xmath165 correspond to finite @xmath42 . this means that at @xmath6 the asymptotic analysis of the curves @xmath166 can be done simply by taylor expansion at small @xmath164 , which yields simple monotonic behavior of @xmath167 and @xmath168 .    at @xmath10 and 2 the situation is not so simple and requires more complicated analysis . such analysis can be performed on the base of eqs . , and is quite cumbersome . the main idea is that functions @xmath169 , @xmath170 and @xmath171 in at small @xmath42 can be represented via exponential integral functions ,  @xcite which , in turn , may be expanded in asymptotic series . this procedure for @xmath10 gives following parametric dependence @xmath172    the asymptotics is shown by dashed line on fig . [ fig : w4]a . in the main logarithmic approximation this gives simple relations @xmath173 which coincide with the results obtained in ref .   by direct numerical modeling of the discreet model , giving @xmath174 and @xmath175 . these expressions give us an idea about behavior of energy and bound magnons number at small @xmath41 . namely we see that the energy has the ( approximate ) square root singularity , while magnons number @xmath71 obeys inverse square root law . the asymptotics is shown on fig . [ fig : w4]a by dashed line .    for @xmath7 the same approximation gives @xmath176 this also yields the divergent @xmath177 and square root peculiarity in the energy @xmath178 . however , the coefficient before @xmath179 in the energy is much smaller then that at @xmath5 , which makes it almost invisible in the scale of fig . [ fig : w4 ] . the details of this behavior are reported on the inset to fig . [ fig : w4]a . this shows the similarities and differences between cases @xmath5 and @xmath7 , which can also be seen from fig . [ fig : w4 ] . namely , if the topological energy densities for skyrmions with @xmath5 and 2 have both square - root nonanalyticity , for @xmath7 this nonanalyticity reveals only as a small cusp in a close vicinity of @xmath52 . other peculiarity of @xmath7 case is nonmonotonous behavior of @xmath180 ( decreasing at @xmath181 with subsequent increase ) . our analysis shows , that the above peculiarities of the skyrmion with @xmath7 are due to interplay between exponential asymptotics of @xmath92 , necessary to cut - off the divergence at @xmath5 and 2 as well as power - law asymptotics , which is sufficient for the convergence at @xmath95 . in this paper we present a comprehensive theoretical study of the localized topological solitons ( skyrmions ) , stabilized by precessional spin dynamics , for the classical 2d ferromagnet with easy - axial anisotropy on a square lattice . our efforts were directed primarily towards the study of the role of higher @xmath182 topological charges @xmath30 on the above solitons properties . our main conclusion is that the interplay between high topological charges , effects of lattice discreteness ( in the form of higher powers of magnetization gradients ) and uniaxial magnetic anisotropy makes many unexpected peculiarities into solitons properties as compared to those in the simplest isotropic continuous model with @xmath183 , containing bp solitons . similar to previous studies , it turns out that the presence of even weak anisotropy makes solitons dynamic , i.e. , those with nonzero precession frequency for any number @xmath71 of bound magnons . the minimal consideration of discreteness ( via higher degrees of gradients of magnetization ) yields the existence of some threshold value of both soliton energy ( topological energy density , which is more appropriate characteristic for solitons with @xmath30 ) and the number of bound magnons . similar to the problem of cone state vortices ,  @xcite the instability is related to the joint action of discreteness and anisotropy . as a result , the critical values of bound magnons @xmath184 and soliton energy @xmath185 is present and nonanalytic dependencies of above threshold values on the anisotropy constant appear at @xmath5 and @xmath7 . it was shown earlier ,  @xcite that the variational minimization of @xmath50 ( i.e. energy , incorporating only squares of magnetization gradients ) gives no threshold for soliton existence , i.e. the soliton exists everywhere up to @xmath186 . this means , that mapping of the initial discrete model even for small anisotropy @xmath57 on the simplest continuum model is wrong and to get the correct description of solitons in 2d fm we have to consider at least fourth powers of magnetization gradients . this seemingly paradoxical result is actually due to the fact that the terms with @xmath187 are scale - invariant ( so that the corresponding energy has a saddle point ) while the ( stable ) soliton size is determined by the fourth derivatives as well as by a magnetic anisotropy . our analysis show , that higher ( then fourth ) powers of magnetization gradients do not change the situation qualitatively , rather , in the range of above - studied @xmath188 these terms make only a small quantitative contribution to the solitons phase diagram . this means , that solitons with @xmath5 can be well studied within the model .    in summary , we have shown the existence of stable topological @xmath189 solitons ( skyrmions ) in the 2d ferromagnet with uniaxial anisotropy on a square lattice . since for any 2d lattice symmetry the structure of corresponding energy functional is the same , we may speculate that such textures exist in any 2d magnet with uniaxial anisotropy . we also show that due to different character of the energy and magnons number integrals convergence at @xmath190 and @xmath191 determines the features of so - called solitons phase diagram . the main nontrivial and unexpected result of the paper is that while the solitons with @xmath5 and @xmath33 have monotonously growing phase boundary functions @xmath34 , the case @xmath7 has peculiar nonmonotonous behavior , determining the transition regime from low to high topological charges . this means that the designated value of soliton topological charge , which is expected for highly excited state of fm , is neither @xmath5 nor high charges , but rather @xmath7 or @xmath8 .                              c.  e. zaspel , t.  e. grigereit , and j.  e. drumheller , phys . lett . * 74 * , 4539 ( 1995 ) ; k. subbaraman , c.  e. zaspel , and j.  e. drumheller , phys . lett . * 80 * , 2201 ( 1998 ) ; c.  e. zaspel and j.  e. drumheller , int . j. mod b * 10 * , 3649 ( 1996 ) .
we show that stable localized topological soliton textures ( skyrmions ) with @xmath0 topological charge @xmath1 exist in a classical 2d heisenberg model of a ferromagnet with uniaxial anisotropy . for this model 
 the soliton exist only if the number of bound magnons exceeds some threshold value @xmath2 depending on @xmath3 and the effective anisotropy constant @xmath4 . 
 we define soliton phase diagram as the dependence of threshold energies and bound magnons number on anisotropy constant . 
 the phase boundary lines are monotonous for both @xmath5 and @xmath6 , while the solitons with @xmath7 reveal peculiar nonmonotonous behavior , determining the transition regime from low to high topological charges . 
 in particular , the soliton energy per topological charge ( topological energy density ) achieves a minimum neither for @xmath5 nor high charges , but rather for intermediate values @xmath7 or @xmath8 .