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the dynamics of magnetic flux propagation in a long josephson junction ( ljj ) has been and continues to be a subject of strong theoretical and practical interest during the last three decades @xcite . the magnetic flux quantum in a ljj is a soliton ( also known as _ fluxon _ ) governed by the well - known sine - gordon ( sg ) equation . the convenient way to prepare a junction with the required properties is to install various inhomogeneities into it . up to now the substantial theoretical work has been devoted to the study of the fluxon motion in the one - dimensional ( 1d ) ljjs with point - like @xcite and spatially extended @xcite inhomogeneities . experimental results on the fluxon scattering on impurities are reported in @xcite spatially inhomogeneous josephson systems with trapped fluxons have been discussed as prospective applications , such as fluxon - based information devices @xcite . real ljjs are always two - dimensional ( 2d ) , or , more precisely , quasi - one - dimensional ( q1d ) , in the sense that they have a finite width in the direction perpendicular to the direction of fluxon propagation . up to now the fluxon dynamics in q1d junctions has been scarcely investigated as compared to the pure 1d case . most of attention has been focused on the various isotropic 2d structures like oscillons and ring kinks @xcite . it is worth mentioning also the case of window junctions @xcite , which can be called inverse in some sense : it studies point or rectangular junctions embedded in a larger two - dimensional superconduncting sample . quasi - one - dimensional solitons , or , in other words , solitonic fronts in infinite ( in both @xmath1 and @xmath2 directions ) samples have been studied in detail by malomed @xcite . several interesting results in the absence of dissipation have been reported including the waves in the q1d sine - gordon equation , travelling along the soliton line @xcite and skyrmion scattering on impurities in the 2d baby scyrme model @xcite . however , in the case of the fluxon dynamics in a ljj the presence of dissipation is unavoidable . it is of interest to investigate the q1d fluxon dynamics in the presence of spatial inhomogeneities when the junction width is finite . we expect that the fluxon transmission in this case will be significantly enhanced comparing to the pure 1d case . for example , in the previously studied case of the lattice acoustic soliton front interaction with mass impurities @xcite it has been shown that the front can _ round _ the point impurity while a 1d lattice soliton gets reflected from it . moreover , the soliton front can overcome even the impurity of the infinite mass . to our knowledge the fluxon interaction with spatial inhomogeneities has not been studied except in @xcite , however this paper deals only with the infinite sample width . thus , in this letter we aim at studying the q1d fluxon interaction with impurities and finding out how this interaction depends on the junction width and other system parameters . in particular , it is planned to find how the retrapping current ( e.g. , the minimal bias current for which the fluxon propagation is still possible ) depends on the junction width . the paper is organized as follows . in the next section , the model is described . section [ sec3 ] is devoted to the fluxon transmission through impurities . in the last section discussion and conclusions are presented . we consider the q1d long josephson junction ( ljj ) subjected to the external time - independent bias . the main dynamical variable is the difference between the phases @xmath3 of the macroscopic wave functions of the superconducting layers of the junction . the time evolution of the phase difference is governed by the perturbed sine - gordon ( sg ) equation of the form @xmath4\sin \phi=-\alpha \phi_t-\gamma,\ ] ] where @xmath5 and the indexes @xmath6 stand for the respective partial derivatives . in this dimensionless equation the spatial variables @xmath1 and @xmath2 are normalised to the josephson penetration depth @xmath0 , the temporal variable @xmath7 is normalised to the inverse josephson plasma frequency @xmath8 @xcite . the bias current @xmath9 is normalised to the critical josephson current of the junction and @xmath10 is the dimensionless dissipation parameter . the function @xmath11 describes the spatial inhomogenity . in the case of point impurities in the general form it reads @xmath12 it is supposed that there are @xmath13 impurities in this junction , positioned at the points @xmath14 , @xmath15 , @xmath16 , with @xmath17 being the `` strength '' or the amplitude of the impurity . only the microshorts ( @xmath18 ) , i.e. , a narrow regions of locally enhanced critical density of the tunnelling superconctucting current will be investigated in this article . however , the size an inhomogeneity in experimental samples is finite @xcite . therefore , we consider also the case of the line microshort of width @xmath19 , stretched along the @xmath2 direction : @xmath20 \,.\ ] ] here @xmath21 is the heavyside function . and finally , the rectangular impurity of the finite size in both @xmath1 and @xmath2 directions @xmath22 \left [ \theta\left ( y+\frac{d_y}{2}\right ) + \theta \left ( \frac{d_y}{2}-y\right ) \right ] \ ] ] will be considered as well . it should be noted that the impurity strength @xmath17 has different meanings in all three cases ( [ 2])-([4 ] ) . for the point impurity ( [ 2 ] ) setting @xmath23 does not automatically yield the pure 1d case studied before in @xcite . this case can be retained in the strip impurity case if @xmath24 . the 1d finite - size impurity case @xcite is retained in the same way . we choose the boundary conditions along the @xmath2 direction in the von neumann form : @xmath25 the boundary conditions along the @xmath1 axis are periodic : @xmath26 , where @xmath27 is the junction length , @xmath28 . in order to get an idea about the character of the fluxon dynamics , the numerical integration of the q1d sg equation ( [ 1 ] ) has been performed . the josephson phase and its space derivatives are discretized in the following way : @xmath29 , + @xmath30 , while the @xmath31-function is approximated by the kronecker @xmath31-symbol . the resulting set of ode s with the boundary conditions were integrated using the 4th order runge - kutta scheme . details of the fluxon propagation through the two identical point impurities ( [ 2 ] ) placed along the @xmath2 axis at @xmath32 and @xmath33 are given in fig . it is important to mention that the dissipation in eq . ( [ 1 ] ) is crucial and the soliton interaction with impurities differs from the dissipationless case where the complex resonant behaviour occurs either for the @xmath31-like @xcite or for the finite - size @xcite obstacles . far away from the impurities the fluxon exists as only one attractor of the system with the velocity , predefined by the damping parameter and external bias . therefore , contrary to the non - dissipative case , the transmission consists of only two possible scenarios : trapping and passage . for the sake of better visualization the derivative @xmath34 is plotted on the @xmath35 plane for the different time moments and for three different dissipation values : @xmath36 , @xmath37 and @xmath38 . without loss of generality the topological charge is assumed to be @xmath39 ( soliton ) throughout the paper . the initial conditions are taken in the form of the approximate ( invariant in the @xmath2 direction ) soliton solution , placed at the beginning of the junction and having with the equilibrium velocity . the fluxon interaction with the rectangular impurity is presented in fig . [ fig2 ] for @xmath36 and @xmath38 . in this case , the countour plot of the function @xmath34 is plotted on the @xmath35 plane . the fluxon retrapping current was computed numerically as well , showing its steady decay when the junction width was increased . these results will be discussed in detail in the next subsection . the following conclusions can be drawn from these results . the fluxon shape experiences certain changes after passing the impurities , namely the redistribution of the josephson phase along the fluxon line in the @xmath2 direction , as well as the slight bending of the fluxon shape in the same direction . these distortions eventually die out after some time , for @xmath37 and @xmath36 this happens quite soon after the passage through the impurity . for smaller dissipation ( @xmath38 , as shown in figs . [ fig1]i - l ) the oscillations of the josephson phase along the @xmath2 direction seem to survive for much longer time , comparable with the fluxon propagation time along the junction . the numerical simulations for thinner junctions , @xmath40 ( not shown in the paper ) , demonstrate that these shape distortions are much weaker and are hardly noticeable . thus , when studying the q1d fluxon interaction with impurities one can assume with the high degree of certainty that the fluxon is an almost hard rod at least if @xmath41 . due to finiteness of the junction in the @xmath2 direction and the boundary conditions ( [ 5 ] ) , the straight soliton front is the energetically most favourable solution and thus it is not possible to observe the arc - like solitons reported in @xcite .      similarly to fluxon propagation in the 1d ljj @xcite , there must exist two characteristic values of the bias current , @xmath42 and @xmath43 , @xmath44 . moreover , the current - voltage characteristics of the ljj with impurities have hysteretic nature @xcite . if @xmath45 , the pinning on the impurity is not possible and there exists only one attractor that corresponds to fluxon propagation . this happens because the bias current is too strong for the fluxon to get trapped on the impurity . in the interval @xmath46 , at least two attractors coexist : one corresponds to fluxon pinning on the impurity ( there can be several different pinned fluxons if the impurity has finite length @xcite ) and another one to fluxon propagation . if @xmath47 , the only possible regime is fluxon pinning on the impurity . the value of @xmath42 is defined only by the properties of the impurity , and can be obtained directly from the 1d analog . contrary , for the retrapping current the dimensionality of the junction and its width are crucial .    far from the impurity , the fluxon kinetic energy is proportional to the junction width and equals @xmath48 $ ] . here , @xmath49^{-1/2}$ ] is the equilibrium fluxon velocity in the spatially homogeneous ljj @xcite . in the non - relativistic fluxon case ( @xmath50 ) , one gets @xmath51 . by substituting the ansatz + @xmath52/\sqrt{1-v^2}\}$ ] into eq . ( [ 1 ] ) , where @xmath53 is the coordinate of the fluxon center of mass , one can obtain the newtonian equation of motion for the fluxon center .    since the fluxons under consideration are extended objects in the @xmath2 direction , the equation for the center of mass dynamics as well as the impurity potential should depend on @xmath2 . taking into account the numerical simulation in the previous section , we consider the fuxon as an absolutely rigid rod . we also mention that the impurity function @xmath11 can be factorised in the cases ( [ 3])-([5 ] ) , therefore its center of mass dynamics can be effectively projected on the @xmath1 axis and the respective equation of motion can be written as @xmath54 where the center of mass coordinate @xmath55 depends only on time . the impurity potential @xmath56 now can be calculated from the respective 1d problem by simply taking away the @xmath2-dependent part in eqs . ( [ 2])-([4 ] ) . the fluxon mass @xmath57 has to be rescaled depending on the type of the impurity . in the pure 1d case @xmath58 . this assumption works well only if the impurity is consists of lines and/or rectangles . if it has a more complex shape , for example , like triangle , the projection on the 1d problem becomes more complicated . * point impurity . * in the point impurity case ( [ 2 ] ) , we consider the line of @xmath13 identical equidistant impurities with @xmath59 , @xmath16 which are placed symmetrically with respect to the central line @xmath60 . each impurity creates the potential @xmath61 . in order to project the problem on the 1d picture it is necessary to rescale the fluxon mass from @xmath58 to @xmath62 , thus within the kinematic approach the retrapping current can be found as a root of the energy balance equation @xmath63 , where @xmath64\simeq 4wv_\infty^2/n+{\cal o}(v_\infty^4)$ ] is the fluxon kinetic energy . in the non - relativistic case one gets @xmath65 . the correction of the order @xmath66 can be taken into account with the help of the method , developed in @xcite . the modification of this method for the q1d case is straightforward , therefore we describe only the main steps . the improved energy balance relation equates the fluxon energy at @xmath67 and its losses due to the dissipation , @xmath68 , with the maximal height of the potential barrier @xmath69 : @xmath70 where @xmath71 , @xmath72 are the extrema of the potential @xmath69 , @xmath73 , @xmath74 . equation ( [ en - bal ] ) is universal and will be used for all types of impurities . the energy loss due to the dissipation equals + @xmath75 . inserting this correction term @xmath68 into the improved energy balance equation and keeping the terms up to the order @xmath66 , one gets the final corrected expression for the retrapping current : @xmath76 it appears that the @xmath77 relation is obtained simply by dividing the impurity strength by the factor @xmath78 , moreover , the @xmath66 correction does not depend on the junction width at all . from this expression one can clearly see that if only the @xmath79 term is taken into account , the retrapping current disappears as @xmath80 , thus , in the infinitely wide junction a fluxon always passes the impurity . the second term in eq . ( [ 4a ] ) does not depend on @xmath81 , and , therefore it may lead to the wrong conclusion that @xmath43 does not tend to zero as @xmath80 . however , it should be noted that this term has been derived under the assumption of @xmath81 being finite . the numerical simulations confirm the main suggestion of this letter : the retrapping current decays with the growth of the junction width . the approximate expression ( [ 4a ] ) appears to be in good agreement with the numerical data for smaller values of @xmath81 , while for larger @xmath81 the numerical and analytic data diverge ( however , the agreement remains to be satisfactory ) . all this is presented in fig . [ fig1]a where the analytic results are given by solid lines and the numerical data are shown by markers . in the case of two and three impurities , placed along the line in the @xmath2 direction , the retrapping current virtually does not depend on the distance @xmath82 between them . for instance , the markers that correspond to @xmath83 ( @xmath84 ) , @xmath85 ( @xmath86 ) , @xmath87 ( @xmath88 ) at @xmath89 and @xmath90 are almost indistinguishable . one may consider the excentrically placed ( i.e. , lying away from the @xmath91 axis ) impurity . the kinematic approach does not distinguish the impurities , placed at the different positions . the numerical simulations show that the difference between the retrapping currents in both the cases is very small . for example , @xmath92 for the centrally placed impurity , @xmath93 , at @xmath94 ( the rest of the parameters are as in fig . [ fig1]a ) and @xmath95 for the impurity at @xmath96 , @xmath97  . * line impurity . * in the case of the impurity line ( [ 3 ] ) , the same energy balance expression holds as for the point impurity but the fluxon mass is assumed to be @xmath98 . the rest of the calculation procedure for the retrapping current is the same as for the point impuritiy . as a result , we obtain the the final expression for the retrapping current : @xmath99 numerically computed @xmath43 appears to be in good correspondence with the approximate expression ( [ 4b ] ) as shown in fig . similarly to the point impurity case , the discrepancy between the analytical and numerical results increases at larger @xmath81 . in the limit @xmath100 , the effective 1d picture is restored because the impurity strip crosses the whole junction in the @xmath2 direction . thus , the retrapping current attains the value @xmath101 for the 1d soliton case ( shown by the thick horizontal line in fig . [ fig3]b ) . thus , we observe that if the strip impurity consitutes , for example , about @xmath102 of the junction width , the retrapping current is about @xmath103 less than the respective 1d value . * rectangular impurity . * finally , we consider the rectangular impurity ( [ 4 ] ) . in the point - particle description , the fluxon `` feels '' this impurity as the potential @xmath104 $ ] @xcite . in this case , the energy balance reads @xmath105=4\mu d_y \tanh ( d_x/2)$ ] , from which the expression for the threshold current can be easily computed . for large @xmath106 it can be complemented by the correction @xmath66 computed for the 1d case in @xcite . the fluxon mass should be renormalized as @xmath107 . in this limit it is assumed that the impurity creates the step - like potential @xmath108 $ ] , which has a minimum at @xmath109 and a maximum at @xmath110 . the energy losses due to the dissipation @xmath111+{\cal o}(1)\}$ ] should be substituted in the energy balance equation ( [ en - bal ] ) . the final approximation reads @xmath112.\ ] ] note that at variance with the point and line impurities , the second correction depends on the both junction and the impurity width . in the 1d limit ( @xmath100 ) , the formula ( [ 9 ] ) retains the form , found in @xcite . in the limit @xmath113 , the second term decays to zero , but much slower than the decay law @xmath114 of the first term . once again , the second term is a correct approximation only for finite values of @xmath81 .    in the opposite limit , @xmath115 , one can use the following approximation : @xmath116\simeq d_x \mbox{sech}^2 x + { \cal o}(d_x^2)$ ] . thus , the effective potential @xmath69 is approximately the same as for the line impurity up to the coefficient @xmath117 . therefore the formula ( [ 4b ] ) can be modified accordingly and finally the retrapping current reads @xmath118.\ ] ] the results of numerical simulations given in fig . [ fig4 ] demonstrate satisfactory correspondence with the analytical approximations ( [ 9])-([10 ] ) . the deviations from the numerical results appear to be stronger as compared to the point and line impurities , mostly because we were unable to provide the @xmath66 correction that works equally well in both limits @xmath119 and @xmath120 and also due to higher degree of the fluxon deformation during the interation with the impurity . it should be mentioned that even for @xmath121 one can consider the `` long '' @xmath106 limit and it works fairly well . indeeed , the markers that correspond to @xmath122 ( @xmath123 ) , @xmath124 ( @xmath125 ) and @xmath126 ( @xmath88 ) seem to be almost indistinguishable in fig . to summarize , we have shown that quasi - one - dimensional fluxon passage across microshorts is significantly enhanced in comparison with the purely one - dimensional case . the retrapping threshold current decays with the junction width approximately as @xmath127 , according to the kinematic approach . the numerical simulations support this dependence for intermediate ( several @xmath0 ) values of @xmath81 . with the increase of @xmath81 , the discrepancy between the numerical results and the kinematic approximation becomes more distinct . the reason of this discrepancy lies in the fact that for @xmath128 the fluxon can not be longer considered as an completely rigid object and its deformation the @xmath2 direction should be taken into account . one can formulate the following simple argument that explains the enhanced fluxon transmission across the obstacle of the width @xmath19 in the q1d case . the obstacle can be described as a localized ( both in the @xmath1 and @xmath2 directions ) potential barrier . only the central part ( @xmath129 ) of the initially homogeneous in the @xmath2 direction fluxon takes part in the interaction process , while the marginal areas @xmath130 do not . thus , if the energy in the tails is sufficient enough to overcome the barrier , the fluxon will pass . if @xmath113 , the energy in the non - interacting part of the fluxon tends to infinity , and , consequently , it will overcome any localized obstacle . finally , we would like to outline the future research directions . in our opinion they lie beyond the rigid rod approximation used in this letter . in this case the fluxon center of mass depends also on the transverse coordinate @xmath2 and the effective equations of motion are pde s and not ode s , like eq . ( [ 7 ] ) . accounting for the dependence on @xmath2 becomes important in the following cases : ( i ) the junction width is too large and the flexural oscillations along the fluxon line appear ; ( ii ) the dissipation is rather small and/or the junction length is @xmath131 , thus the spatial distortions of the fluxon line do not have enough time to die out . these cases are also interesting in connection with the arc - like fluxons found in the infinitely wide junction @xcite and the variety of the excitations that travel along the fluxon crest in the dissipationless 2d junctions @xcite . the combined effect of the dissipation and the finite junction width makes the transverse - invariant fluxon front the energetically most favourable solution . however , other attractors of the system which are non - transverse - invariant may exist as suggested by figs . [ 1]j - l , especially when dissipation is small . finding them and investigating how they will manifest themselves on the junction current - voltage dependence is an important problem . authors are indebted to dffd ( project f35/544 - 2011 ) for financial support . s. sakai , h. akoh and h. hayakawa , japan . ( 1985 ) l771 ; h. akoh , s. sakai , a. yagi and h. hayakawa , ieee trans . ( 1985 ) 737 ; i.l . serpuchenko and a.v . ustinov , pisma zh . 46 ( 1987 ) 435 [ sov jetp lett . 46 ( 1987 ) 549 ] .
fluxon transmission through impurities of different shape in a quasi - one - dimensional long josephson junction is investigated . 
 the junction width is significantly less than its length but , at the same time , is of the order of the josephson penetration length @xmath0 or exceeds it . 
 the retrapping current on the impurities of the point , line and rectangular shape is computed as a function of the junction width both numerically and analytically . 
 good agreement between the analytic formulae and the numerical simulation results for the intermediate ( several @xmath0 ) junction width is observed .    solitons , josephson junctions , fluxons , impurities