EXISTENCE OF CONCENTRATED WAVES FOR THE DIFFERENT TYPE OF NONLINEARITY. V. Eremenko and N. Manaenkova

Transcription

1 EXISTENCE OF CONCENTRATED WAVES FOR THE DIFFERENT TYPE OF NONLINEARITY. V. Eremeko ad N. Maaekova Istitte of Terrestrial Magetism, Ioosphere ad Radio Wave Propagatio Rssia Academy of Sciece Abstract. The oliear problem of the wave propagatio is cosidered. I additio to Kerr oliearity the qestio of the existece of cocetrated soltios is aalyzed for the threshold ad satrable oliearity. It is show that both i the case of threshold oliearity, ad i the case of satrable oliearity solitary waves - cocetrated soltios of the correspodig wave eqatios exist. 1. Itrodctio Moder radio commicatios allows so high icreasig of radiated sigal itesity that it geerates modificatio of medim of propagatio ad the problem of determiig the wave field becomes oliear. Theoretical estimates of the effects of ioospheric plasma heatig by powerfl radiatio bega to appear log time ago [1]. Experimetal cofirmatio of the iteractio of powerfl shortwave radiatio with ioospheric plasma at obliqe propagatio [, 3,] made more active the developmet of the theory of this iteractio [4]. As a rle, the theoretical researches of the oliear propagatio of powerfl wave beams are restricted by models of local oliearity. The most commoly sed model is the Kerr oliearity, i which the oliear pertrbatio of the dielectric permittivity is proportioal to the secod degree of the modls of the wave amplitde. This approach allowed s to describe the basic pheomea arisig at the oliear iteractio of radiatio with the eviromet. However sch approach has obvios restrictios. Actally, there is o oliear effects whe the itesity of the wave field is ot eogh powerfl. As soo as the wave amplitde exceeds certai threshold vale, so called "disrptio" of the eviromet takes place ad the oliear depedece of the dielectric permittivity po the wave field amplitde arises. Moreover the oliear depedece becomes more complex with frther growth of the itesity. The pper limit of growth of the 7

2 dielectric permittivity, amely satratio, is observed for most of the real materials. I this case it is reasoable to se a model medim with satrable oliearity. However, this descriptio of the oliear iteractio of radiatio with the medim is local. This approach is possible oly i the case of egligible heat codctivity - whe the size of the wave beam is mch more tha the characteristic scale of the thermal diffsio. Otherwise, yo mst take ito accot the spreadig of the electro desity pertrbatio from the area of the oliear heatig. Noliear effect becomes olocal ad, i this case, it is ecessary to cosider the soltio of the diffsio eqatio for the dielectric permittivity ad the soltio of the wave eqatio joitly, as a system.. Aalysis I order to describe the wave field i this small area we se the Helmholtz eqatio for the wave amplitde V. V k V, where k is the wave mber ad ε is the dielectric permittivity. For the small wave itesity the dielectric permittivity depeds oly o the coordiates. Therefore, i order to determie the distribtio of the wave field i the space, it is eogh to solve the liear problem with the appropriate bodary coditios. Bt, for the high wave itesity, the dielectric permittivity becomes depedet o the wave amplitde, ad it is ecessary to solve the oliear problem for the descriptio of the wave propagatio. Fig.1. The wave beam ad qasi - ray coordiates. We will cosider the propagatio of the arrow shortwave beams. Ths we will costrct the Helmholtz eqatio soltio cocetrated i the small viciity of the ray trajectory. I this viciity, we itrodce the 71

3 orthogoal coordiate system: is the legth of the trajectory arch; is the distace alog the orthogoal directio to the ray (Fig. 1). We represet the complex-valed fctio V i the terms of V expik, where ad - are real fctios. I this approach, the derivatives alog the trajectory are essetially less the the derivatives across the ray directio. Therefore, we ca write i the mai approximatio: d k, d The electric field heats the eviromet ad creates the pertrbatio of the dielectric permittivity. For the slightly ihomogeeos medim we ca write the mai approximatio ad we get the typical problem of oliear wave propagatio. d d d, were is the phase velocity. d This eqatio has the first itegral. d F( ) E, E cost, F( ) ( x) dx d This eqatio admits the existece of cocetrated soltios if E, provided that the eqatio Fx x has soltios x ad x x. Withot loss of geerality, we choose the ceter of the soltio. The x, whe, ad at the ifiity. The amplitde maximm of the wave beam is x write the formal soltio. y x dx F x, ad we ca.1 Kerr oliearity I particlar, if the correctio to the oliear dielectric permittivity has the form, the ad the eqatio has a simple root (Fig. ). F ( ).5 ( ), 7

4 Fig.. The Ker oliearity. Ths, soltio has the form of the simple solito with the phase velocity.5. It is a well-kow soltio. This approach allowed s to describe basic pheomea arisig at the oliear iteractio of radiatio with the eviromet.. Threshold oliearity However as metioed above, sch approach has obvios restrictios. Actally, there is o oliear effects whe the itesity of the wave field is ot eogh powerfl. As soo as the wave amplitde exceeds certai threshold vale, there is a "disrptio" of the eviromet ad the oliear depedece of the dielectric permittivity po the wave field amplitde arises. I this case, we ca write the model of the threshold oliearity. The oliear pertrbatio of the dielectric permittivity for this sitatio ca be represet by the formla. ) ( A is the threshold vale., x - is the Heaviside theta fctio, A 4 I this case F( ) t dt A ( A ) 4, ad the expressio (1) reslts i the elliptic itegral. Nevertheless it is obvios (Fig. 3) that i this case, we have also the cocetrated soltio 4 with the phase velocity.5 A. / The cocetrated soltio with the threshold oliearity is very close to the sal solito, bt it is somewhat arrower i the ceter of the 73

5 beam ad it has a "log" tails. The iteractio betwee these beams differs from the iteractio of the solito collisios. Fig.3. The threshold oliearity.3 Satrable oliearity The oliear depedece becomes more complex with magificatio of the wave power. The pper limit, of the dielectric permittivity is observed for most of real materials. I this case it is reasoable to se a model medim with satrable oliearity, i which the depedece of the dielectric permittivity o the wave itesity is described by fractioal-liear fctio [5, 6]. The pertrbatio of the dielectric permittivity. 1 I this case F( ) t dt l(1 ) N Fig. 4. The satrable oliearity We ca see from Fig. 4 that it is also possible the existece of the cocetrated soltios. The phase velocity has the limit /, 74

6 whe the wave itesity teds to the ifiity. I this case, the eviromet becomes liear. However, at ay fiite amplitde, the cocetrated soltio exists..4 Nolocal oliearity The previos descriptios of the oliear problems were local. That approaches are possible oly i the case of egligible heat codctivity, whe the size of the wave beam is mch more tha the characteristic scale of the thermal diffsio. Otherwise, it is ecessary to take ito accot the spreadig of the electro desity pertrbatio from the area of the oliear heatig. Noliear effect becomes olocal [1]. I this case for the, we have to write the diffsio eqatio. The comptatioal soltio allow s to assert that the cocetrated soltio exist for ay vale of the diffsio scale. The cocetrated soltio has ot ay siglarities for ay vale of the scale of the diffsio process ad it looks like the solito bt more wide. Natrally, the diffsio process elarges the solito. 5. Refereces [1] Gizbrg V ad Grevich A 196 Noliear pheomea i plasma occrrig i a alteratig electromagetic field Sccesses of physical scieces 7 pp [] Bochkarev G et al 1979 Iteractio decameter radio waves at freqecies ear the F MUF o obliqe propagatio Geomagetism ad Aeroomy 19 pp [3] Bochkarev G Eremeko V et al 198 Noliear Iteractio of Decameter Radio Waves at close Freqecies o obliqe propagatio Joral of Atmos. ad Terr.Phys 44 N 1 [4] Bochkarev G Eremeko V et al 198 Modelig the impact of the powerfl waves o the ioosphere at obliqe icidece Geomagetism ad Aeroomy pp [5] Gotz Sad ad Herrma J 1991 Solito propagatio i materials with starable oliearity J.Opt.Soc. Am. B 8 N 11 [6] Eremeko V ad Molotkov I 1998 Featres of behavior of wave beams i media with satrable oliearity Radiotechology ad eletroika 43 N 7 [7] Eremeko V ad Cherkashi Y 1999 Nolocal oliear iteractio of high-freqecy waves with the ioosphere Math. Colleges ad iversities. Radiophysics 4 pp