Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation

Phase Shift in the Whitham Zone for the Gurevic h-Pitaevskii Sp ecial Solutio n of the Kortew eg-de V ries Equation R. Garifullin a, ∗ , B. Suleimano v a , N. T arkhano v b a Institute of Mathematics, RAS, 45008, 112, Chernushevsko go, Ufa, Russia b Institute of Mathematics, Potsdam Un iversity, A m Neuen Palac e, 14469, Potsdam, Germany Abstract W e get the leading term of the Gur evich - Pitaevskii sp ecial solution of the KdV equation in the oscillation zone without using a v eraging metho ds. Keywor ds: nondissipativ e sho c k w a ve , cusp catastrophe 1. In tro duction The Gurevic h-Pitaevskii ( G P) sp ecial univ ersal solution of the Kortew eg- de V ries (KdV) equation u t + u u x + u xxx = 0 (1.1) w as in tro duced in [1] in connection with the problem of description of colli- sionless sho ck wa v es (Sagdeev sho we d in [2 ] that suc h w av es are of oscillat- ing c haracter). The b ehav iour of the GP sp ecial s olutio n for t → −∞ and x → ±∞ is determined in the main from the cubic canonical equation of the cusp catastrophe x − tu + u 3 = 0 . (1.2) The GP solution to t he KdV equation is one of the most in teresting sp ecial functions o f the mo dern nonlinear mathematical ph ysics. ∗ Corresp o nding author Email addr esses: r ustem @mate m.anrb.ru (R. Garifullin), bisu l@mai l.ru (B. Suleimanov), tarkhanov @math .uni-potsdam.de (N. T a rkhanov) Pr eprint submitte d t o Physics L etters A June 8, 2018 In [1] it is show n that in problems of disp ersion hydrodynamics (in par- ticular, in problems of plasma theory) t he GP sp ecial solution app ears near the p o in ts o f ov erturning of simple w av es . F rom the results of [3 ], [4], [5 ] one actually se es that the same univers al special function app ears near the p oints of o v erturning o f the generic state solutions to dive rse disp ersion p er- turbations of the e quations of one-dimens ional motion of ideal incompressible liquid ρ ′ t + ( v ρ ) ′ x = 0 , v ′ t + v v ′ x + α ( ρ ) ρ ′ x = 0 . Here, ρ is the densit y of the liquid, v t he ve lo cit y and α ( ρ ) = ( c ( ρ )) 2 /ρ , where c ( ρ ) = p p ′ ( ρ ) is the speed of sound and p ( ρ ) the pressure. In particular, this is the case f o r solutions of the shallow water equations h ′ t + ( hA ′ x ) ′ x = ε 2 ( h 3 A ′′ xx ) ′′ xx + O ( ε 4 ) , A ′ t + 1 2 ( A ′ x ) 2 + g h = 1 2 ε 2 ( A ′′′ xxt + A ′ x A ′′′ xxx − ( A ′′ xx ) 2 ) + O ( ε 4 ) , where h is t he free b oundary , A the p otential of b ottom v elo cit y and g the acceleration of grav ity . The rig h t-hand sides can actually b e written as complete series in p o w ers of the pa r a meter ε b y the pro cedure g iv en for instance in [6, Ch. 1 , § 4] (a nd not only as the so-called second appro ximations, as stated in [4]). In t he 199 0s there w ere disco v ered surprising connections of the GP sp e- cial solutions with some problems of quan tum gravit y . In [7] this solution w as sho w ed to sim ultaneously satisfy the fourth o rder ordinary differential equation u xxxx + 5 3 uu xx + 5 6 ( u x ) 2 + 5 18 ( x − tu + u 3 ) = 0 , (1.3) whic h had b een studied for t = 0 in [8 ] and [9 ] in connection with ev aluat- ing nonp erturbative string effects in tw o-dimensional quan tum gra vit y (the equation (1.3) b elongs to a class of massiv e string equations). In [10] the solution of  W 3 − W W X X − 1 2 ( W X ) 2 + 1 10 W X X X X  + 15 32 T  W 2 − 1 3 W X X  = X with a symptotics 3 √ X as X → ±∞ w as treated n umerically in connection with problems of quan tum grav ity . One can sho w tha t this solution U ( t, X ) is also equiv alen t to the G P sp ecial solution of (1 .1) for t ≥ 0 (but not f o r t < 0). 2 Dubro vin sho w ed in [11] and [1 2] directly by means of the theory of ap- pro ximate symm etries [13] that it is the solution o f (1.3) with asymptotics (1.2) that a pp ears near the points o f w a v e o v erturning for the v ery divers e singular disp ersion p erturbations o f the equations of o ne-dimensional h ydro- dynamics. The results o f numerical sim ulations presen ted in [10] demonstrate rather strikingly tha t t he GP sp ecial solution of the K dV equation p ossesses a do- main of undamp ed oscillations for t large enough. The author s of [10] did not conjecture a n y relation of their pap er to the GP sp ecial solution a nd raised the problem of describing this domain of oscillations. Mean while Gurevic h and Pitaevskii [14] had used successfully the self-similar solutions of the av- eraged Whitham equations [15] to solv e t he problem. The self-similar solutions in question w ere constructed in explicit form b y Potemin [1 6 ]. Ho we ve r, the problem on the leading term of asymptotics of t he G P sp ecial solution in the do ma in of Whitham osc illatio ns has b een op en up to no w. One not simple question still unansw ered has b een tha t o n the phase shift. Our purp o se is to sho w how it is p ossible to construct the leading term of the GP sp ecial solution in the zone of oscillations without using an y av eraging metho ds. T o this end we deriv e certain a lgebraic equations for the slowly v arying amplitude and t he leading term of the phase, whic h are actually equiv a lent to those of [16]. Moreo v er, we de termine the phase shift of the solution in the o scillation zone. Our approac h may also b e of use for the study of undamp ed oscillations of other common solutions to integrable partial and ordinary differential equa- tions whic h are of imp or t a nce in ph ysics. In particular, it applies to tw o univ ersal solutions of the KdV equation treated in the recen t article [17]. Almost one problem in the approach is some aw kw ardness o f analytical cal- culations. Ho w ev er, in v oking mo dern programs for sym bol calculations (in this pap er we use Maple) often allows o ne to get rid of suc h problems without particular difficulties. 2. Ev a luation of phase shift Consider the solution of the KdV equation that, for t → −∞ and x → ±∞ , is determined in the main from the cubic equation (1.2). It is kno wn that for this solution for p ositiv e t t here is a domain where dissipationless sho c k w a ve s app ear. 3 W e are aimed at constructing asymptotics of the solution in this domain, when t → ∞ . F ollo wing familiar tec hniques, we change the v ariables b y u = p | t | U ( t, z ) , z = x | t | 3 / 2 . Then equations (1.1) and (1.3) tak e the form tU t + 1 2 ( U − 3 z U z ) + U U z + t − 7 / 2 U z z z = 0 , t − 7 U z z z z + 5 18 t − 7 / 2 (6 U U z z + 3( U z ) 2 ) + 5 18 ( z − U + U 3 ) = 0 . (2.1) W e no w lo ok for a solution U of the sys tem in the form of a symptotic series U = U 0 ( ϕ, z ) + t − 7 / 4 U 1 ( ϕ, z ) + t − 7 / 2 U 2 ( ϕ, z ) + . . . , (2.2) where U 0 , U 1 and U 2 are 2 π -p erio dic in t he fast v ariable ϕ . This latter is assumed to b e of t he form ϕ = t − 7 / 4 f ( z ) + s ( z ) , where b y s ( z ) is mean t precisely the phase shift. F or the unkno wn function U 0 w e get the nonlinear system Q 3 ∂ 3 ϕ U 0 + QR∂ ϕ U 0 + QU 0 ∂ ϕ U 0 = 0 , Q 4 ∂ 4 ϕ U 0 + 5 6 Q 2 (2 U 0 ∂ 2 ϕ U 0 + ( ∂ ϕ U 0 ) 2 ) + 5 18 ( z − U 0 + U 3 0 ) = 0 , while the systems fo r U 1 Q 3 ∂ 3 ϕ U 1 + Q ( R + U 0 ) ∂ ϕ U 1 + Q∂ ϕ U 0 U 1 = F 1 , Q 4 ∂ 4 ϕ U 1 + 5 3 Q 2 ( U 0 ∂ 2 ϕ U 1 + ∂ ϕ U 0 ∂ ϕ U 1 ) + 5 18 (3 U 2 0 + 3 Q 2 ∂ 2 ϕ U 0 − 1) U 1 = F 2 , and for U 2 Q 3 ∂ 3 ϕ U 2 + Q ( R + U 0 ) ∂ ϕ U 2 + Q∂ ϕ U 0 U 2 = G 1 , Q 4 ∂ 4 ϕ U 2 + 5 3 Q 2 ( U 0 ∂ 2 ϕ U 2 + ∂ ϕ U 0 ∂ ϕ U 2 ) + 5 18 (3 U 2 0 + 3 Q 2 ∂ 2 ϕ U 0 − 1) U 2 = G 2 pro v es to b e linear. Here, F 1 and F 2 are explicit functions dep ending o n z and U 0 , and G 1 and G 2 are explicit functions depending on z and U 0 , U 1 , 4 i.e., the righ t-hand sides a r e explicit functions dep ending on z and on the preceding corrections. W e write Q = f ′ , R = 7 4 f f ′ − 3 2 z (2.3) for short. F rom the compatibilit y condition of the equations for U 0 w e o bta in a first order equation Q 2 ( ∂ ϕ U 0 ) 2 + 1 3 U 3 0 + R U 2 0 + 1 3 (18 R 2 − 5) U 0 + 1 3 (15 R − 5 4 R 3 − 5 z ) = 0 . (2.4) F rom the compatibility condition of the equations for U 1 w e derive a nonlinear equation d dz R = 1 9 486 R 4 − 171 R 2 + 9 z R + 5 (54 R 3 − 9 R + z )(2 R + 3 z ) (2.5) for the unkno wn function R = R ( z ). (In Section 4 w e sho w that this equation agrees with results obtained earlier.) When requiring the compatibility of the equations for U 2 , w e deduce that the function U 0 ( ϕ, z ) should satisfy , together with (2 .4), a no nlinear ordinary differen tial equation in the v ariable z of the form ∂ 2 z U 0 − ∂ 2 ϕ U 0 ( ∂ ϕ U 0 ) 2 ( ∂ z U 0 ) 2 + P 3 ( U 0 ) ( ∂ ϕ U 0 ) 2 ∂ z U 0 + ∂ ϕ U 0 ( s ′′ + H s ′ ) + P 4 ( U 0 ) ( ∂ ϕ U 0 ) 2 = 0 . (2.6) Here, P 3 ( U 0 ) and P 4 ( U 0 ) are p olynomials in U 0 of degrees 3 and 4 , respec- tiv ely , with co efficien ts dep ending on z and R . The function H = H ( z , R ) is giv en b y H ( z , R ) = 1 3 N ( z , R ) (54 R 3 − 9 R + z ) 2 (2 R + 3 z ) 2 where N ( z , R ) = 4 5 z 3 + (4860 R 3 − 582 R ) z 2 + (131220 R 6 − 43416 R 4 + 272 1 R 2 − 35) z + 1 39968 R 7 − 59616 R 5 + 604 8 R 3 − 120 R . Note that (2.6) is a Hamiltonian equation with Hamiltonian quadratic relativ e to the impulse, i.e. a ( ∂ z U 0 ) 2 + b∂ z U 0 + c where a , b and c are functions of z and U 0 . 5 W e pro ceed to study equations (2.4)- (2.6). Equation (2.4) is autonomo us in the fast v ariable ϕ , hence the arbitrary constant of the g eneral solution is con tained in the phase shift whic h w e tak e in to account in the v a riable s ( z ). The general solution of this equation is sought in the form U 0 = A dn 2  B Q ϕ ; k  + C , where dn is the elliptic function of Jacobi and A , B , C and k are t o b e de- fined. On substituting U 0 in to equation (2.4) and equating the co efficien t s of differen t p ow ers of the Jacobi function to zero w e get the system o f algebraic equations A 2 ( A − 12 B 2 ) = 0 , A 2 ((8 − 2 k 2 ) B 2 + C + R ) = 0 , A ((12 k 2 − 12) AB 2 + 18 A 2 + 6 RC − 5) = 0 , C 3 + 3 RC 2 + (18 R 2 − 5) C − 54 R 3 + 15 R − 5 z = 0 . (2.7) F rom the assumption o n the 2 π -p erio dicit y of U 0 it follows that B Q = K ( k ) π , (2.8) where K ( k ) is the complete elliptic integral of first kind. The system o f equations (2.3), (2.5 ) , (2.7) and (2.8) obtained in this w a y is o v erdetermined. It consists of 6 a lg ebraic equations and 2 differential equations for the unknow ns f , Q , R , A , B , C and k . Our next concern will b e to sho w that one can find all slo wly v arying unkno wns without solving the differen tial equations. The unkno wns f , Q , R , B and C can b e determined imme diately from this system through A , k and z . More precisely , B = r A 12 , C = ( k 2 − 2)(4 k 4 − 5 k 2 + 5) A 3 − 10( K ( k ) − 2) A − 4 5 z 14( k 4 − k 2 + 1) A 2 − 30 , R = ( k 2 − 2) A 3 − C , Q = π B K ( k ) , f = Q (4 R + 6 z ) 7 . (2.9) 6 On substituting these expressions into (2.7) we arriv e at one algebraic equa- tion (8 k 12 − 24 k 10 + 43 k 8 − 46 k 6 − 43 k 4 + 24 k 2 − 8) A 6 − 140( k 4 − k 2 + 1) 2 A 4 + 50 z ( k 2 − 2)(2 k 2 − 1)( k 2 + 1) A 3 + 500 ( k 4 − k 2 + 1) A 2 + 337 5 z 2 = 500 (2.10) for A and k . Differen tiating this equalit y in z and substituting the result- ing ex pression along with (2.9) in to (2.3) and (2.5), w e obtain 3 equations con taining A ′ ( z ) and k ′ ( z ). On eliminating these deriv ativ es, we get a no ther algebraic equation for A and k whic h con tains the quotient of tw o complete elliptic inte gra ls q = E ( k ) K ( k ) . Using (2 .10) to eliminate the highest p o w er of z from the latter equation we bring it to the form 21 k 4 ( k 2 − 1) 2 A 3 + 10 ((2 q − 1 ) k 6 − (3 q + 1) k 4 − (3 q − 4) k 2 + (2 q − 2) ) A + 315 z ((2 q − 1) k 4 − (2 q − 3) k 2 + (2 q − 2) ) = 0 . (2.11) When eliminating the v ariable A from (2.10) a nd (2.11), one finds k as an im- plicit f unction of z . The other functions can b e expressed explicitly t hrough k . This metho d allo ws one to get t he explicit formu las of [16] without using a v eraging pro cedure. The do main o f z in whic h o scillations ar e p ossible is determined from the condition that k ∈ [0 , 1]. T o the p oint k = 0 there corresp onds leading w av e fron t set z l = − √ 2 and to the p oint k = 1 there corresp onds t r a iling w av e fron t set z t = √ 10 / 27. The second o r dina r y differen tial equation of (2.1) enables us a lso to de- termine the phase shift s ( z ) o f the solution. F or t his purpo se w e mak e use of equation (2.6). Note that U 0 is an eve n function of ϕ , hence ∂ ϕ U 0 is o dd and ∂ z U 0 , ∂ 2 z U 0 are ev en in ϕ . Multiplying (2.6) b y ( ∂ ϕ U 0 ) 3 , in tegrating in ϕ o v er the whole p erio d 2 π and taking into accoun t that t he mean v alue o f an o dd p erio dic function ov er the whole p erio d v anishes, we get t he equation s ′′ + H ( z , R ) s ′ = 0 . (2.12) In order to c ho ose a concrete solution to (2.12 ), one has to use the asymp- totics of the solution for z → z l . Suc h an asymptotics is giv en in [3] and it 7 sho ws in particular that the solution do es not con tain log ( z − z l ). In our case w e get k = 2 7 / 8 √ 5 ( z − z l ) 1 / 4  1 − 2 3 / 4 10 ( z − z l ) 1 / 2 + 131 1280 √ 2 ( z − z l ) + O (( z − z l ) 3 / 2 )  , a = − √ 2 6 + 1 40 ( z − z l ) + 7 2560 √ 2 ( z − z l ) 2 + O (( z − z l ) 3 ) , H = 1 z − z l − 543 1600 √ 2 + O ( z − z l ) . F rom t he asymptotics of H it follo ws that a solution of (2.12) con ta ins terms of the form log( z + √ 2) whic h are not permitted by [3]. Hence , this solution en ters into the linear com bination of solutions with co efficien t 0 , a nd so s ( z ) = s 0 is constan t. The h yp othesis on the constancy of the phase shift s ( z ) has b een formu- lated in [3]. Ho wev er, in [3] it was based solely o n the asymptotics given there. It is clear that a priori one migh t not exclude the situatio n where the phase shift fails to b e constan t but tends exp onen tially fast to a constan t as z → z l . F rom (2.12) and the asymptotics of H ( z , R ) for z → z l w e see that suc h is not the case, and so s ( z ) prov es to b e constan t. T o ev aluate t he constan t s 0 w e inv ok e a nu merical sim ula t io n. Namely , w e compare a n umerical solution with the solution constructed b y using asymptotic formulas. 3. Numerical sim ulations T o this end w e ha ve written a sp ecial prog r am. The results of numeric al sim ulations a r e presen ted in Figures 1 – 3. In Figures 1-2 one can observ e n umerical solutio ns f or function U ( t, z ) for negativ e and p ositive v a lue t . It can b e show n that function U ( t, z ) for negativ e v alue t practically councide with the ro o t of the equation Λ 3 + Λ = − z . In F igure 3 the difference b et w een these solutions is sho wn. Both figures corresp ond to t = 2 0. The constan t s 0 has pro v ed to b e equal to 3 . 1254 ≈ π . The difference b etw een tw o solutions is a m ultiple of t − 1 . 77 ≈ t − 7 / 4 whic h agrees with the order o f the first correction in form ula ( 2.2). 8 20 15 10 5 0 -5 -10 -15 -20 2 1 0 -1 -2 Figure 1: The numerical simulation for the function U ( t, z ) corres po nding to t = − 7 . W e b eliev e that the constan t s 0 just amoun ts to π and the small difference is caused b y a computation error . Figure 3 mak es it eviden t that the error increases for z close to − √ 2 and √ 10 / 27. This manifests the nonuniform character of the constructed asymp- totic form ula in the en tire domain of the v ariable z . In neigh b ourho o ds o f the leading and trailing w a ve front sets one should construct other asymptotic form ulas whic h can b e made consisten t with asymptotic expansion (2.2), a s it is described in [18]. 4. Reduction of equations to st andard form T o start with, w e bring the equations obtained in [16] to a simpler for m. These are z − Y j + Z j = 0 for j = 1 , 2 , 3, where Y 1 = 1 3 S 1 + 2 3 l 1 − l 2 1 − q , Y 2 = 1 3 S 1 − 2 3 ( l 1 − l 2 )(1 − k 2 ) 1 − q − k 2 , Y 3 = 1 3 S 1 + 2 3 l 3 − l 2 q , S 1 = l 1 + l 2 + l 3 , S 2 = l 1 l 2 + l 2 l 3 + l 3 l 2 , S 3 = l 1 l 2 l 3 and Z j = 1 35  V + (3 Y j − S j ) V ′ l j  with V = 5 S 3 1 − 12 S 1 S 2 + 8 S 3 . 9 0 -1 -2 2 1 0 -1 Figure 2: The numerical simu la tio n fo r the function U ( t, z ) corres p o nding to t = 20 and ro ot of equatio n z − Λ + Λ 3 = 0. Ob viously , one can eliminate tw o v ariables from three equations. On eliminating z and q w e get an equation for l 1 , l 2 and l 3 , whic h do es not con tain z and q . Namely , 3 ( l 2 1 + l 2 2 + l 2 3 ) + 2 ( l 1 l 2 + l 2 l 3 + l 3 l 1 ) − 5 = 0 . (4.1) W e now eliminate either of the v a riables z and q in the equations and use (4.1) to eliminate all p o w ers of l 1 greater than the first o ne. Then w e get t w o more equations z = 2 45  l 1 (8 l 2 2 + 4 l 2 l 3 + 8 l 2 3 − 1 5 ) − ( l 2 + l 3 )(24 l 2 2 − 8 l 2 l 3 + 24 l 2 3 − 2 5 )  , q = 1 2 ( l 2 − l 3 )(3 l 2 l 3 + 3 l 3 l 1 + 9 l 2 3 − 5 ) l 1 (2 l 2 2 + l 2 l 3 + 2 l 2 3 − 5) − ( l 2 + l 3 )(6 l 2 2 − 2 l 2 l 3 + 6 l 2 3 − 5) . (4.2) The system of equations (2.3), (2.7) and (2.8) is equiv alen t to the sys- tem of a v eraged equations (4.1 ) and (4.2) whic h w as obtained in [16] by Whitham’s metho d. T o pro v e this, w e pass in the system of the equation for C in (2.9) and equations (2.10), (2.11) to Whitham’s v ar iables l 1 ( z ), l 2 ( z ) and l 3 ( z ) b y A = 2 ( l 3 − l 1 ) , C = l 1 + l 2 − l 3 , k 2 = l 2 − l 1 l 3 − l 1 . 10 0 -1 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 Figure 3: The difference b etw een the numerical solutio ns and a s ymptotic for U ( t, z ) for t = 20 . On eliminating z and q from the obtained equations w e arriv e precise ly at (4.1). On eliminating either of z and q and all p o w ers of l 1 greater than the first one, w e get (4.2), as desired. 5. Conclusion It should b e noted that the v alue of the phase shift s 0 deriv ed from the nu- merical sim ulation contradicts [3, 4 ]. F rom the results of the pa p er it follow s that the function s ( z ) tends to π / 2, as z → z l . But there is an arithmetical error in f orm ula (17)[3] and in form ula (21)[4]. The righ t calculation with using mono dromic da te from [19] sho we d t ha t s ( z ) tends to π , and so s ( z ) ≡ π . Ac knowled gemen ts The authors a r e greatly indepted t o V. Adler fo r first n umerical exp eriments in this problem. The deriv ation of equation (2.5) without using the av eraging method is due to our colleague V. Kudashev 1 . The researc h of the first author w as supp orted b y the DFG gran t T A 28 9 /4-1 and b y the Prog r am for Supp orting Y oung Scien tists, gran t MK-28 12.2010.1 . 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