q-Analogue of Shock Soliton Solution

q-Analogue of Sho c k Soliton Solution Sengul Nalci and Okta y K . P ashaev Departmen t o f Mathematics, Izmir Institute of T ec hnology Urla-Izmir, 35430, T urk ey No ve mber 12, 2021 Abstract By using Jac kson’s q-exp onentia l function w e introdu ce the generating function, the recursive form ulas an d the second order q -differential equa- tion for the q - Hermite polyn omials. This allo ws us to solv e the q-heat equation in terms of q-Kamp e de F eriet p olyn omials with arbitrary N mo v ing zero es, an d to fi nd op erator solution for the I nitial V alue Problem for t he q -heat equ ation. By the q-analog of the Cole-Hopf transformation w e co n struct the q-Burgers typ e nonlinear heat equation with qu adratic disp ersion and the cubic nonlinearit y . In q → 1 limit it reduces to the standard Burgers equation. Exact solutions for the q-Burgers equation in the form of moving p oles, s in gular and regular q -sho ck soliton solutions are foun d . 1 In tro d uction It is well known that the Bur gers’ equation in one dimension could b e lineariz e d by the Co le-Hopf tr ansformation in ter ms of the linear heat equation. It allows one to solve the initial v alue problem for the Bur g ers eq uation and to get exact solutions in the for m of sho ck solitons and describ e their scattering. In the present pap er we study the q -differential Bur gers type equa tion with quadratic disp e rsion and the c ubic nonlinea rity , and find its linearization in ter ms of the q-heat equation. In terms of the Jackson’s q-exp onential function we introduce the q-Hermite and q-Kampe-de F eriet poly no mials, representing moving po les solution for the q- Burgers equa tion. Then w e deriv e the op erato r s olution o f the initial v alue pr oblem fo r the q-Burg ers equation in terms of the IVP for the q-heat equation. W e find solutio ns of our q-Burg ers t y p e equation in the for m of sing ular and regular q-sho ck s olitons. It tur ns out tha t static q - sho ck soliton solution s hows rema rk able self-similar it y pro per ty in space co ordinate x . 1 2 q-Exp onen tial F unction The q-num b er corr esp onding to the o r dinary num b er n is defined as, [1], [ n ] q = q n − 1 q − 1 , (1) where q is a par ameter, so that n is the limit of [ n ] q as q → 1. A few examples of q-num b ers are given here: [0] q = 0, [1] q = 1, [2] q = 1 + q , [3] q = 1 + q + q 2 . In ter ms of these q-num b ers, the Ja ckson q-exp one ntial function is defined as e q ( x ) = ∞ X n =0 x n [ n ] q ! . (2) F or q > 1 it is e ntire function of x and when q → 1 it reduces to the standar d exp onential function e x . The q -exp onential function ca n also b e expressed in terms o f infinite pro duct e q ( x ) = ∞ Y n =0 1 (1 − (1 − q ) q n x ) = 1 (1 − (1 − q ) x ) ∞ q , (3) when q < 1 and e q ( x ) = ∞ Y n =0  1 + (1 − 1 q ) 1 q n x  =  1 + (1 − 1 q ) x  ∞ 1 /q , (4) when q > 1. Thus, the q-e x po nential function for q < 1 has infinite set of po les at x n = 1 q n (1 − q ) , n = 0 , 1 , .. (5) and for q > 1 the infinite set of zeros at x n = − q n +1 ( q − 1) , n = 0 , 1 , .. (6) The q-der iv ative is defined as D x q f ( x ) = f ( q x ) − f ( x ) ( q − 1) x , (7) and when q → 1 it r educes to the standard deriv ative D x q f ( x ) → f ′ ( x ). Using the definition of the q -deriv ative one can eas ily see that D x q ( ax n ) = a [ n ] q x n − 1 , (8) D x q e q ( ax ) = ae q ( ax ) . (9) 2 3 q-Hermite P olynomials W e define the q-Her mite p olynomials acco r ding to the ge ner ating function e q ( − t 2 ) e q ([2] q tx ) = ∞ X n =0 H n ( x ; q ) t n [ n ] q ! . (10) F rom this g enerating function we hav e the sp ecia l v alues H 2 n (0; q ) = ( − 1 ) n [2 n ] q ! [ n ] q ! , (11) H 2 n +1 (0; q ) = 0 , (12) where [ n ] q ! = [1 ] q [2] q ... [ n ] q , and the parity r elation H n ( − x ; q ) = ( − 1) n H n ( x ; q ) . (13) By q-differentiating the gener ating function (1 0) acco r ding to x and t we have the r ecurrence r elations cor resp ondingly D x H n ( x ; q ) = [2] q [ n ] q H n − 1 ( x ; q ) , (14) H n +1 ( x ; q ) = [2] q xH n ( x ; q ) − [ n ] q H n − 1 ( q x ; q ) − [ n ] q q n +1 2 H n − 1 ( √ q x ; q ) . (15) Using op erator M q = q x d dx , (16) so that M q f ( x ) = f ( q x ) , (17) relation (15 ) ca n b e rewritten as H n +1 ( x ; q ) = [2] q xH n ( x ; q ) − [ n ] q ( M q + q n +1 2 M √ q ) H n − 1 ( x ; q ) . (18) Substituting (14 ) to (18) we get H n +1 ( x ; q ) = [2] q x − M q + q n +1 2 M √ q [2] q D x ! H n ( x ; q ) (19) By the recursion, starting fro m n = 0 and H 0 ( x ) = 1 we ha ve next representation for the q-Her mite p olynomials H n ( x ; q ) = n Y k =1 [2] q x − M q + q k 2 M √ q [2] q D x ! · 1 (20) W e notice that the genera ting function and the fo rm of our q-Her mite p olyno- mials a r e different from the known ones in the literature, [2 ], [4], [3], [5]. In the ab ov e expressio n the o pe rator M q + q n 2 M √ q = 2 q n 4 q 3 4 x d dx cosh[(ln q 1 4 )( x d dx − n )] (21) 3 is expr essible in terms of the q-spher ical mea ns as cosh[(ln q ) x d dx ] f ( x ) = 1 2 ( f ( q x ) + f ( 1 q x )) . (22) Using definition, [1], ( x − a ) n q = ( x − a )( x − qa ) · · · ( x − q n − 1 a ) , n = 1 , 2 , .. which now we a pply for op erators , we should distinguish the direction of multi- plication. W e consider tw o ca ses ( x − a ) n q < = ( x − a )( x − qa ) · · · ( x − q n − 1 a ) , (23) and ( x − a ) n q > = ( x − q n − 1 a ) · · · ( x − qa )( x − a ) . (24) Then, w e can rewrite (2 0) shor tly as H n ( x ; q ) =  ([2] q x − M q D x [2] q ) − q 1 2 M √ q D x [2] q  n √ q > · 1 First few p olynomia ls are H 0 ( x ; q ) = 1 , H 1 ( x ; q ) = [2 ] q x, H 2 ( x ; q ) = [2] 2 q x 2 − [2] q , H 3 ( x ; q ) = [2] 3 q x 3 − [2] 2 q [3] q x, H 4 ( x ; q ) = [2 ] 4 q x 4 − [2] 2 q [3] q [4] q x 2 + [2] q [3] q [2] q 2 . When q → 1 these p olynomials reduce to the standard Hermite po lynomials. 3.1 q-Differen tial Equation Applying D x to bo th sides of (1 9) and using recurrence form ula (14) w e get q-differential equation for q- Her mite p olynomia ls 1 [2] q D x ( M q + q n +1 2 M √ q ) D x H n ( x ; q ) − [2] q q xD x H n ( x ; q ) + [2] q [ n ] q q H n ( x ; q ) = 0 . 4 3.2 Op erator Represen tation Prop ositi on 1 We have next identity e q  − 1 [2] q D 2 x  e q ([2] q xt ) = e q ( − t 2 ) e q ([2] q xt ) . (25) Pro of 1 By q- differ entiating the q-exp onential function in x D n x e q ([2] q xt ) = ([2] t ) n e q ([2] q xt ) , (26) and c ombining then to the sum ∞ X n =0 a n [ n ] q ! D 2 n x e q ([2] q xt ) = ∞ X n =0 [2] n q a n t 2 n [ n ] q ! e q ([2] q xt ) , (27) we have r elation e q ( aD 2 x ) e q ([2] q xt ) = e q ([2] q at 2 ) e q ([2] q xt ) . (28) By cho osing a = − 1 / [2] q we get e q  − 1 [2] q D 2 x  e q ([2] q xt ) = e q ( − t 2 ) e q ([2] q xt ) . (29) Prop ositi on 2 The next identity is valid H n ( x ; q ) = [2 ] n e q  − 1 [2] q D 2 x  x n . (30) Pro of 2 The rig ht hand side of ( 25 ) is the gener ating function for q-Hermite p olynomials. Henc e exp anding b oth sides in t we get the r esult. Prop ositi on 3 e q  − D 2 x [2] q  x n +1 = 1 [2] q [2] q x − ( M q + q n +1 2 M √ q ) D x [2] q ! e q  − D 2 x [2] q  x n . (31) Pro of 3 we use (30 ) and r elation (19) . Corrollary 1 If fun ction f ( x ) is analytic and exp andable to p ower series f ( x ) = P ∞ n =0 a n x n then we have next q-Hermite series e q  − 1 [2] q D 2 x  f ( x ) = ∞ X n =0 a n H n ( x ; q ) [2] n q . (32) 5 4 q- Kamp e-de F eriet P olynomials W e define q -Kamp e-de F er iet p o ly nomials as H n ( x, t ; q ) = ( − ν t ) n 2 H n  x [2] q √ − ν t  , (33) and from (19) we hav e the next recursion fo rmula H n +1 ( x, t ; q ) =  x + ( M q + q n +1 2 M √ q ) ν tD x  H n ( x, t ; q ) . By re cursion it gives H n ( x, t ; q ) = n Y k =1  x + ( M q + q k 2 M √ q ) ν tD x  · 1 (34) or by notatio n (24) H n ( x, t ; q ) =  ( x + M q ν t D x ) + q 1 2 M √ q ν tD x  n √ q > · 1 First few p olynomia ls are H 0 ( x, t ; q ) = 1 , H 1 ( x, t ; q ) = x, H 2 ( x, t ; q ) = x 2 + [2] q ν t, H 3 ( x, t ; q ) = x 3 + [2] q [3] q ν t x, H 4 ( x, t ; q ) = x 4 + [3] q [4] q ν t x 2 + [2] q [3] q [2] q 2 ν 2 t 2 . 5 q-Heat Equation W e consider the q-heat equation ( D t − ν D 2 x ) φ ( x, t ) = 0 . (35) Solution of this equation expanded in ter ms of parameter k φ ( x, t ) = e q ( ν k 2 t ) e q ( k x ) = ∞ X n =0 k n [ n ]! H n ( x, t ; q ) , (36) 6 gives the s e t of q-Kamp e-de F eriet p olynomial solutions for the equation. Then we find time evolution of zero es for these solutions in terms of zero es z k ( n, q ) o f q-Hermite p oly nomials, H n ( z k ( n, q ) , q ) = 0 , (37) so that x k ( t ) = [2] z k ( n, q ) √ − ν t. (38) F or n=2 we hav e t wo zer o s determined by q-num b e rs, x 1 ( t ) = q [2] q √ − ν t, (39) x 2 ( t ) = − q [2] q √ − ν t, (40) and moving in opp osite directions according to (38). F or n=3 we have zeros determined b y q-num b er s , x 1 ( t ) = − q [3] q ! √ − ν t, (41) x 2 ( t ) = 0 , (42) x 3 ( t ) = q [3] q ! √ − ν t, (43) t wo of which ar e moving in opp osite dire c tion accor ding to (38) and one is in the r est. 6 Ev olution Op erator F ollowing similar c a lculations as in Prop os ition I w e hav e nex t re la tion e q  ν tD 2 x  e q ( k x ) = e q ( ν tk 2 ) e q ( k x ) . (44) The righ t hand side of this expres sion is the pla ne w av e type solution of the q-heat e q uation ( D t − ν D 2 x ) φ ( x, t ) = 0 . (45) Expanding b oth sides in p ower series in k we get q-Kamp e de F er ie p olynomial solutions of this equation H n ( x, t ; q ) = e q  ν tD 2 x  x n . (46) Consider an ar bitrary a nalytic function f ( x ) = P ∞ n =0 a n x n , then function f ( x, t ) = e q  ν tD 2 x  f ( x ) = ∞ X n =0 a n e q  ν tD 2 x  x n (47) = ∞ X n =0 a n H n ( x, t ; q ) , (48) 7 is a time dep endent solution of the q-heat equation (4 5). According to this we hav e the evolution op erato r for the q- heat equation as U ( t ) = e q  ν tD 2 x  . (49) It a llows us to solve the initial v alue problem ( D t − ν D 2 x ) φ ( x, t ) = 0 , (50 ) φ ( x, 0 + ) = f ( x ) , (51) in the for m φ ( x, t ) = e q  ν tD 2 x  φ ( x, 0 + ) = e q  ν tD 2 x  f ( x ) . (52) 7 q-Burgers’ T yp e Equation Let us consider the q-Cole-Ho pf tra nsformation u ( x, t ) = − 2 ν D x φ ( x, t ) φ ( x, t ) , (53) then u ( x, t ) satisfies the q- Burgers ’ type Equa tion with quadratic disp ersio n and cubic nonlinearity D t u ( x, t ) − D 2 x u ( x, t ) = 1 2  ( u ( x, q t ) − u ( x, t ) M x q ) D x u ( x, t )  − 1 2 [ D x ( u ( q x, t ) u ( x, t ))] + 1 4  u ( q 2 x, t ) − u ( x, q t )  u ( q x, t ) u ( x, t ) . When q → 1 it reduces to the standar s Burger s’ Equation. 7.1 I.V.P . for q-Burgers’ T yp e Equation Substituting the op erator s olution (52) to (53) we find op er ator solution for the q-Burger s type eq ua tion in the form u ( x, t ) = − 2 ν e q  ν tD 2 x  D x f ( x ) e q ( ν tD 2 x ) f ( x ) . (54) This so lutio n corr esp onds to the initial function u ( x, 0 + ) = − 2 ν D x f ( x ) f ( x ) . (55) Thu s, for ar bitrary initial v a lue problem for the q-Burgers equation with u ( x, 0 + ) = F ( x ) we need to so lve the initial v alue problem for the q- heat equation with ini- tial function f ( x ) satisfying the first or der q-differential equation ( D x + 1 2 ν F ( x )) f ( x ) = 0 . (56) 8 8 q-Sho c k soliton solution As a solutio n of q- heat equa tion we choose first φ ( x, t ) = e q  k 2 t  e q ( k x ) , (57) then we find solution of the q- B urgers equation as a cons tant u ( x, t ) = − 2 ν k . (58) If we choos e φ ( x, t ) = e q  k 2 1 t  e q ( k 1 x ) + e q  k 2 2 t  e q ( k 2 x ) , (59) then we hav e the q- Sho ck s oliton so lution u ( x, t ) = − 2 ν k 1 e q  k 2 1 t  e q ( k 1 x ) + k 2 e q  k 2 2 t  e q ( k 2 x ) e q ( k 2 1 t ) e q ( k 1 x ) + e q ( k 2 2 t ) e q ( k 2 x ) . (60) Due to z ero es of the q-exp o nential function this expression admits singularities for so me v alues of pa rameters k 1 and k 2 . In Fig.1 we plo t the singula r q- sho ck soliton fo r k 1 = 1 a nd k 2 = 10 at time t = 0. Out[2]= - 4 - 2 2 4 - 20 - 10 10 20 Figure 1 : Singular q-Sho ck Soliton How ever for so me sp ecific v a lues o f the parameters w e found the regular q-sho ck s oliton so lution. W e in tr o duce the q - hyperb olic function cosh q ( x ) = e q ( x ) + e q ( − x ) 2 , (61) 9 or cosh q ( x ) = 1 2 e q ( x ) + 1 e 1 q ( x ) ! , (62) then by using infinite pro duct representation for q-exp onential function we have cosh q ( x ) = 1 2  1 + (1 − 1 q ) x  ∞ 1 /q +  1 − (1 − 1 q ) x  ∞ q ! . F rom (5 ),(6) we find that zer o es of the fir s t pro duct are lo cated on negative axis x , while for the second pro duct on the p ositive axis x . Therefo r e the function has no zeros for r eal x and cosh q (0) = 1. If k 1 = 1, a nd k 2 = − 1, the time dep endent facto r s in nominator and the denominator of (60) ca ncel each other and we hav e the stationary sho ck soliton u ( x, t ) = − 2 ν e q ( x ) − e q ( − x ) e q ( x ) + e q ( − x ) ≡ − 2 ν tanh q ( x ) . (63) Due to ab ove consider ation this function has no s ingularity on re a l a xis and we hav e reg ular q-sho ck soliton. In Fig.2, Fig.3 and Fig.4 we plot the reg ular q -sho ck soliton for k 1 = 1 and k 2 = − 1 at different r a nges of x . It is rema rk able fact that the structure of o ur sho ck s oliton shows self-s imilarity pr op erty in spa ce co ordinate x . Indeed at the ranges of para meter x = 5 0 , 500 0 , 5000 00 the s tructure of sho ck lo oks a lmo st the s ame. F or the se t o f arbitra ry num b ers k 1 , ..., k N φ ( x, t ) = N X n =1 e q  k 2 n t  e q ( k n x ) , (64) we have multi-shock solutio n in the form u ( x, t ) = − 2 ν P N n =1 k n e q  k 2 n t  e q ( k n x ) P N n =1 e q ( k 2 n t ) e q ( k n x ) . (65) In ge ne r al this solutio n admits several singularities. T o hav e regula r multi- sho ck so lution we can consider the even num b er of ter ms N = 2 k with o ppo site wa ve n umber s. When N = 4 a nd k 1 = 1, k 2 = − 1, k 3 = 10 , k 4 = − 10 we hav e q-multi-shock soliton so lution, u ( x, t ) = − 2 ν e q ( t ) s inh q ( x ) + 10 e q (100 t ) sinh q (10 x ) e q ( t ) c o sh q ( x ) + e q (100 t ) cosh q (10 x ) . (66) In Fig. 5 we plot N = 4 case with v alues of the w ave num b ers k 1 = 1, k 2 = − 1, k 3 = 10 , k 4 = − 1 0 at t = 0. T o have reg ular solution for any time t a nd given bas e q , we should cho ose prop er n um b er s k i which a r e not in the form of power o f q . This question is under the study now. 10 Out[3]= - 40 - 20 20 40 - 1.5 - 1.0 - 0.5 0.5 1.0 1.5 Figure 2 : Regular q-Sho ck Solito n F or k 1 = 1, k 2 = − 1 , at rang e (-50, 50) Ac kno wledgmen ts One of the authors (SN) was partially supp orted by National Scholarship of the Scientific and T echnological Research Council of T urkey (TUBIT AK). This work was supp orted partia lly by Izmir Institute of T echnology , T urk ey . References [1] V. Kac and P . Cheung , Q ua nt um Ca lculus, Spring er, New Y ork, 2002 . [2] H. Exton, q-Hyp ergeo metric F unctions and Applications, John Wiley and Sons, 1 9 83. [3] P . Ra j kovic a nd S. Ma rinko vic, On Q-analo gies of gener alized Hermite’s po lynomials, Filomat 15, 277 , 200 1 . [4] J . Cigler and J. Zeng , Two curious q -Analogues of Hermite Polynomials arXiv:090 5.0228 , 2 009. [5] J . Negro , The F a c to rization Metho d and Hiera rchies of q-Os cillator Hamil- tonians Cen tr e de Rec herches Ma thema tiques CRM Pro ceedings and Lec- ture Notes, V olume 9, 239, 1996. 11 Out[3]= - 40 - 20 20 40 - 1.5 - 1.0 - 0.5 0.5 1.0 1.5 Figure 3: The regula r q - sho ck solito n for k 1 = 1, k 2 = − 1 at r ange (-50 00, 5 000) Out[3]= - 40 - 20 20 40 - 1.5 - 1.0 - 0.5 0.5 1.0 1.5 Figure 4: The regular q -sho ck so liton for k 1 = 1 , k 2 = − 1 at rang e (-500 000, 50000 0) 12 Out[3]= - 40 - 20 20 40 - 1.5 - 1.0 - 0.5 0.5 1.0 1.5 Figure 5 : q-sho ck regular 13