Asymptotic Waves for a Non Linear System

Transcription

1 Int Jounal of Math Analyss, Vol 3, 9, no 8, Asymptotc Waves fo a Non Lnea System Hamlaou Abdelhamd Dépatement de Mathématques, Faculté des Scences Unvesté Bad Mokhta BP,Annaba, Algea hamdhamlaou@yahoof Boutaba Smal Laboatoe de Physque de Guelma (LPG, Unvesté 8 ma 945 de Guelma, BP 4 Guelma, Algea Boutabba_s_lpg@yahoof Abstact The tem of wave s used n a geneal way to name any soluton of a hypebolc poblem Lu (, and the method used s a genealzaton of a geometcal pocess of optcs (WKB consstng n seekng the soluton n the fom ωϕ u e ( ω g, n tem of a eal paamete ω (fequenc, unknown functons ϕ (phase and g ( attenuatng factos We thus buld such an asymptotc soluton n the cases of smple and multple chaactestcs, povded that the Cauchy data has an asymptotc expanson, and obseve, lke othe authos, that the phase functon s a soluton of the classcal ekonal equaton, and the tems of the sees ae detemned by a ecusve system of dffeental equatons We then deduce a conon fo genune non lneaty of the poblem whch genealzes that of Lax [7] and John [3], and hghlght fom t the sngula behavou of the fst tem of the fomal soluton when the chaactestcs ae dstnct We note that, fo suffcently small fequency, the asymptotc soluton s almost global Keywods : hypebolc, asymptotc sees, non lneaty, blow-up I Intoducton We ae nteested, n ths atcle, wth a system of patal devatve equatons,

2 36 A Hamlaou and S Boutaba hypebolc and nonlnea, and popose the constucton of an asymptotc soluton n the decton gven by GKL [5], GBollat [4], CBuhat [8] The pocess conssts n seekng a soluton n the fom k u ω uk ( x, ωϕ, k n+ ( x a pont of R, ω the fequency s a eal paamete,ϕ the phase s a scala functon and to obseve, afte eplacement n the system, the phase and the tem sees u k Method known as WKB was used by Lax [6], then Ludwg [] and GKL [5] fo the lnea systems, genealzed then wth the nonlnea systems by GBollat[4], YCBuhat [8], and exploted snce pe many authos [],,[9], It s the fom suggested by Lax [6] that we use n ths atcle II Statement of the poblem Let the system : L( u A ( t, x, u u u(, x f ( x summaton n ( n Whee A s a ( k k matx, ( t, x R R, u t, x ( u,, u the unknown functon,, t μ, μ,, n x μ ( k Unde the conons of egulaty of the coeffcents and ntal data wth compact suppot whch ensue the exstence and the uncty of a egula local n soluton u n [, T] R, one poposes to seek n the polycylnde u u ε the ωψ ( x e ω soluton of the poblem ( wth oscllatoy data f ( x f ( x + f ( x ( ωϕ( t, x e n the fom: u( t, x u ( t, x + g ( t, x (3 ( ω n tem of a eal paamete ω (fequenc, of a functon ϕ ( tx, wth eal values (phase whch wll be to detemne as well as the functons g ( tx, By dentfcaton, we have : ϕ(, x ψ ( x, g (, x f ( x, g (, x, (4 Let us note the vecto,, u u k ( ( ( ( ε so that : A u A + A u u + (5 whee ndex zeo s allotted to any functon of u when u u

3 Asymptotc waves fo a non lnea system 36 Whle epotng (3 and (5 n (, one obtans: ( Lu As follows : F ωϕ e ( ϕx F A g whch nvolves,,,, ( ω F (6 (7 ( ϕ x ( ωϕ + e ( A ϕ x g g ( + ϕx ( (, + + F A g A g A g u F A g + G g + H g (9 whee : ( ϕ x + ( ωϕ + ( ϕ x + ( ϕ x G g A g A g u And : e A g g A g g ωϕ ( l ( ωϕ + ( l l H g e A g g l l+ l e A g g (8 III Détemnaton of the phase ϕ Accodng to (7, g s a ght egenvecto coespondng to the egenvalue, fo the appoached chaactestc matx A ( A ϕ x wods, the sets {(, n+ tx R / ϕ ( tx, constante } and d det A ; n othe fom a famly of chaactestc sufaces Let ϕ ( tx, constante to be such a suface, l the dsplacement of a pont of suface, λ ts popagaton velocty ; one has whee N ϕ x, N ( N Dxϕ,, Nn ( λ dl λ and cos (, dx dl N x N dl beng the unt nomal It s shown that : d Dxϕ det A N A One standadzes the poblem by posng A I (unt because A s egula n the neghbouhood of t, u u It s seen that d s a polynomal of k degee n λ The hypebolcty nduces the exstence of two L, espectvely ghts and lefts, of the matx bases of egenvectos { } R and { } A N and eal egenvalues, ( k δ ϕt Dxϕ λ t, x, u λ, wth espectve multplctes m Whle posng + (, one shows that: ( ( AR δ R, ( ( LA δ L and d ( δ m ; (one does not summon compaed to the ndex between backets

4 36 A Hamlaou and S Boutaba The equaton d s thus equvalent to PDE fo the functon ϕ : ϕt + Dxϕ λ ( t, x, u ϕ(, ψ ( ( whch admt each one a sngle local soluton (the taonal ekonal equaton Thee ae thus chaactestc famles of sufaces ϕ, and each one of them defnes an asymptotc wave fo u Whle seekng then u n the fom: ωϕ e +, that one epots n ( u u g ( ω F Lu, one obtans fnally:, whee, whle by vanshng ωϕ Lu ( e F fo each and, ( ω one fnds the equatons (7, (8, (9 whee ϕ s eplaced by ϕ and g by g VI Calculaton of the attenuatng factos Let us wte the system bcaactestc elatng to the poblem (: dτ ( dx ϕxμ μ ( λ, x (,,, μ yμ μ n dτ Dxϕ dp ( ( Dxϕ λ, dτ p( p( p' λ (, y, f ( dpμ ( ( Dxϕ μλ, pμ ( ψx (,( p, p' Dϕ μ dτ The detemnaton of the factos g etuns to the esoluton of the systems (7, (8, (9 along the bcaactestcs soluton of ( The equaton (7 shows that g belongs to subspace geneated by the ght egenvectos assocated to the null egenvalue δ Lemma : If L and R ae espectvely the egenvectos left and ght of A assocated wth the same egenvalue δ, then, along the bcaactestcs soluton of (, one has the elaton: dx μ μ LA R LR, μ,, n Lemma : Standadzaton : L R δ ( symbol of Konecke g ( ( Smple chaactestcs: n ths case, A has k eal egenvalues dstnct k δ,,δ and two bases of egenvectos { L } K and { R } k Let us fx the null

5 Asymptotc waves fo a non lnea system 363 egenvalue δ, wth whch ae assocated the egenvectos L and R Fom (7, one deduces that: g R (3 n+ whee ( t, x s a functon wth eal values on R whch t s necessay to calculate Whle epotng (3 n LF, (8, usng lemmas, and whle placng oneself on a chaactestc ( t, x( t,, one obtans the dffeental equaton of d Benoull: + Γ + Θ( (4 whee (, ( ωϕ ( t, e L( A ϕx R R Γ t y LA R+ L A R u Θ (5 Accodng to (4 and the expesson of ( R (,, one has: (, y L(, f ( (6 The poblem (4, (6 defnes n a sngle way ( t, By epotng ts value n the system (8, ths one becomes an algebac lnea system compaed to g and thus has a patcula soluton h modulo R, and so g h + R whee s a scala functon whch one wll detemne by the elaton Let us eason by nducton on Let us suppose known the factos g,, g such as : g h + R The equalty (9 : F A g + G( g + H ( g s a lnea system n f n the base { } g whch the esoluton gves us a patcula soluton h modulo R, thus g h + R whee the unknown functon wll be then soluton of the d equaton LF, whch s: + LG( R LG( h LH ( g It s a lnea dffeental equaton fo and the ntal value (, L(, h (, s obtaned by expessng the vecto h n the base { R } Thus all the factos can be calculated successvely Multple chaactestcs: we teat the case of a multplcty m wth an egen subspace of dmenson m Let the bases of the egenvectos { R } and { L ε },, ε,,m, espectvely on the ght and on the left of the matx A, coespondng to the null egenvalue δ Fom F A g one deduces that g s a lnea combnaton of the vectos R, so : R (7 g that one epots n L ε F, by usng lemma, to obtan:

6 364 A Hamlaou and S Boutaba ε d γ γ ( L R + Γε + Θε (8, wth, ε, γ vayng fom to m ; and (, ( x ( ε ε ε Γ LA R + L A R u γ ωϕ ε γ ωϕ ε γ ε e L A R R e D ϕ AλR L R Θ Let us obseve that (8 s a system of m dffeental equatons to the m m unknown factos,, whose ntal values ae solutons of the system of Came: f ( g (, (, R (, (9 whee one ponts out that ndcates the numbe of oots of the chaactestc polynomal, each one of them of multplcty m, m k, so that the system { } R s complete Thus, (8, (9 detemne n a sngle way the m functons m,, and consequently the facto g The elaton F consdeed as a lnea system n g gves us a patcula soluton h modulo an egenvecto R, and so : g h + R Let us eason by nducton on Let us suppose known the factos g,, g such as: g h + R ; then the lnea F A g G( g H( g g : system n + + have a patcula soluton h modulo R, then g h + R, that one epots n L ε F, by usng lemma, to lead fnally to the system of m dffeental equatons to the m unknown ( d ε ( ( ( ε ε ε LR + LG R LG h LH g factos : Intal values (, can be gven by the system (, R (, h (, vansh Thus the pocess s complete whose detemnant [ R ] does not V PRINCIPAL RESULTS V Genune non lneaty In the lnea case, A does not depend on u, Θ, and the equaton (4 s educed to that obtaned by Lax [6] and Ludwg [] It s obvous that fo a tuly nonlnea case, Θ does not vansh and t s not enough whch A depends on u The conon of nonlneaty equed by Lax [7] when n, k, and by F John [3] when n, k unspecfed, s : ( λ R

7 Asymptotc waves fo a non lnea system 365 Let us show that ths conon s the same one hee: ωϕ ωϕ Θ e L A ϕ R R e L A R R ( x ( ( LA R R ( L R A R + L( A R R L( A R R e ωϕ Θ Because AR In adon: ( LA R R ( δ L R R ( δ R LR+ δ ( L R R ( λ R ωϕ ωϕ Because δ and LR ; and so Θ e ( δ R e Dxϕ ( δ R and consequently Θ ( λ R V Blow up of the soluton It s known that, n the case of ths system, the soluton exsts only n the neghbouhood of the ntal data Let us show that one obseves the same phenomenon hee The fst tem of the asymptotc development of u beng g R whee d checks : (4 +Γ ( t, ( ( +Θ t, y ; (6 ( (, y L y f ( along the chaactestcs ( txty, (, whee L( L, y, f (, Dψ ( t ( h+ One poses h( t, Γ( s, dsand z( t, e ωϕ ( t, Let us epot the value of dawn fom ( n (4, (6, the poblem becomes: dz h e Dxϕ ( λ R z a ( t, z ( ωψ ( z(, e L( f ( whee a( t, y does not vansh snce ( λ R Lke y suppf B compact n n R, a constant M such as M a( t,, ( t, [ O, T] B Let us consde the chaactestcs C N and C N + wth espectve ntal phases Nπ ( N + π ψ ( and ψ (, N Z on whch z( t, y s eal and one have: (, z t y ω ω L( f ( t ( ( ( L y f y a s, y ds on C N, and L( f ( (, on C t N + + L( f ( a( s, ds Let B f and let us consde the two pats z t y supp

8 366 A Hamlaou and S Boutaba + { ( / ( } { ( / ( } B y B L y f y m B y B L y f y m Let us notce that the contnuous functons z ( t, and m z ( t, fo mmt t m M ( If ( λ R, then a( t m mmt t, cannot exst espectvely fo t ( m M and, M On the chaactestcs C N ssued fom (, y whee y B +, a contnues soluton z( t, y of ( cannot exst fo t ( m M, because z( t, z ( t, On C N + ssued fom (, y whee y B, one have z( t, z ( t,, then T p ( m M If ( λ R f, then a( t, M p T T On C N ssued fom y B, one have : z( t, z ( t,, then p ( m M On C N + ssued fom y B +, one have : z( t, z ( t,, then p ( m M In all the cases, a egula soluton of ( exsts only fo p t T ( max mm Wth m mn ( m, m Note: fo a suffcently small fequency ω, y n a compact set, ψ ( may not take the values Nπ ω, N ntege, the soluton (, z t y of ( wth complex values, emans lmted, and consequently ths last poposal does not exclude the exstence of global solutons Refeences [] A Mada, Nonlnea geometc optcs, IMA vol, Math, Appl,, Spnge New Yok (986 [] D Ludwg, Exact and asymptotque soluton of, C P A M, 3 (96 [3] F John, Fomaton of sngulates, C P A M, 7 (974 [4] G Bollat, La popagaton des ondes, Gauthe-Vllas, Pas (965

9 Asymptotc waves fo a non lnea system 367 [5] Leay-Kotake-Gadng, Unfomsaton et soluton du poblème de Cauchy, Bulletn de la Socété de Mathématque, tome 9, pp [6] P Lax, Asymptotc solutons of oscllatoy ntal value poblems, Duke MathJ4 (957 [7] P Lax, The fomaton and decay of shock waves, A M M 79 (97 [8] Y Choquet-Buhat, Ondes asymptotques, J M P A 48, (969 [9] YCBuhat, Sémnaes et congés9,socété Mathématque de Fance, (4 Receved : Decembe, 8