American Journal of Computational Mathematics, 2014, 4, 254270 Published Online June 2014 in SciRes. http://www.scirp.org/journal/ajcm http://dx.doi.org/10.4236/ajcm.2014.43022 How to cite this paper: Miroshnikov, V.A. (2014) Interaction of Two Pulsatory Waves of the Kortewegde Vries Equation in a Zigzag Hyperbolic Structure. American Journal of Computational Mathematics, 4, 254270. http://dx.doi.org/10.4236/ajcm.2014.43022 Interaction of Two Pulsatory Waves of the Kortewegde Vries Equation in a Zigzag Hyperbolic Structure Victor A. Miroshnikov Department of Mathematics, College of Mount Saint Vincent, New York, USA Email: victor.miroshnikov@mou ntsaintvincent .edu Received 5 May 2014; revised 2 June 2014; accepted 8 June 2014 Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract A new exact solution for nonlinear interaction of two pulsatory waves of the Kortewegde Vries (KdV) equation is computed by decomposition in an invariant zigzag hyperbolic tangent (ZHT) structure. A computational algorithm is developed by experimental programming with lists of eq uations and expressions. The structural solution is proved by theoretical programming with sym bolic general terms. Convergence, tolerance, and summation of the ZHT structural approximation are discussed. When a reference level vanishes, the twowave solution is reduced to the twosoli ton solution of the KdV equation. Keywords KdV Equation, Two Pulsatory Waves, ZHT Structure, Experimental, Theoretical Computation 1. Introduction Since the discovery of the Nsoliton solution of the Kortewegde Vries (KdV) equation by experimental compu tation [1], various analytical methods, like the inverse scattering transform, the Bäcklund transform, generalized functions, etc. [2][5], were developed and implemented by theoretical computation [6]. Further developments of the theory of solitons comprise effects of vorticity [7][9] and viscosity [10], while computational methods evolved from asymptotic methods [11][13], to series approximations [14][16], and structural decompositions in invariant structures [17] [18]. In the current paper, an invariant zigzag hyperbolictangent (ZHT) structure is developed to treat nonlinear interaction of two pulsatory waves of the KdV equation. A zigzagging pattern is a ubiquitous phenomenon in fluiddynamic, biological, and chemical systems [19][21]. The structure of this paper is as follows. In section 2, experimental computation with lists of equations and
V. A. Miroshnikov expression is used to develop a computational algorithm in Maple™ and explore the convergence of the ZHT structural approx imation. Theoretical computatio n with symbolic general ter ms is utilized in Section 3 to devel op differentiation and multiplication formulas for the ZHT structure, show its invariance, and consider summa tion, convergence and tolerance of the structural approximation. The twowave solution is visualized and com pared wit h t he twosoliton solution in Section 4, which is followed by a summary of main results in Section 5. 2. Experiment on Interaction of Two Pulsatory Waves in the ZHT Structure 2.1. Formulation of an Experimental Problem in the ZHT Structure Explore experimentally a structural solution of the canonical form of the KdV equa tion (1) for nonlin ear interaction of two pulsatory waves. Construct the stru ctural solution in the ZHT structure of alge braic order , which is used to illustrate a computational algorithm, () () ()() 222324 235 3 0,0 2,01,13,12,24,23,35,3 2,, xthpZZ taZtaZta TbZ taZ taTbZtaZta Tb φµ =++ +++++ + (2) where is a reference level, and are structural coefficients, and are structural functions, and are wave numbers such that , and are propagation variables, and are celerities, and are initial locations of first and second pulsatory waves, and are interaction and complementary parameters such that and , respectively. W hen and the twowave solution is reduced to a onewave solution ( ) ( ) ( )( ) 22 2 , 21 11 6 sec6, 22 x thta h UhhUhxUta φµ =+− =+ −−−− (3) which becomes the onesoliton solution ()( ) 2 1 , sec 22 x thUxUta U φ = −− [3] for . Consider an in stance of (2) with 1 8,3,1,1 3,223,34 h UVqp µ ====== , and . The ZHT structure then is ( ) ( ) () () 23 0,0 2,01,13,1 2423 53 2,24,23,3 5,3 1 ,8 . x tZZtaZtaZtaTb ZtaZtaTbZ taZ taTb φ =+++ +++ + + (4) 2.2. Experimental Differentiation of the ZHT Structure Primarily, symbolic computation of a spatial derivative of differential order of twowave solution (4) yields the ZHT structure of algebraic order () ( ) () () 3 24 1,03,00,12,14,1 3 52 1,2 3,25,2 24 633 54 2,34,3 6,33,4 5,4 , A taAtaAA taA taTb x A taAtaAtaTb A taAtaAtaTbA taA taTb φ ∂= ++++ ∂ ++ + +++ ++ (5) with a general term , where structural coefficients are
V. A. Miroshnikov 1,02,01,1 3,02,03,1 0,11,1 2,11,13,12,2 4,13,14,2 1,21,12,2 3,23,12,24,23,35,24,25,3 2,32,2 3,3 31 31 , , 2 122 12 3 391 , , 4 446 91 33 , , 4642 33 11 3, 3, 42 44 39 , 24 AZZA ZZ AZAZZZ AZZAZZ AZZZZA ZZ AZ ZA =+ =−+ = =−++ =−+ =−+ =−−+ +=−+ =−+ 4,34,2 3,35,3 6,35,3 3,43,3 5,45,3 3915 , 244 15 99 , , . 4 44 ZZZ AZA ZA Z =−−+ =− =−=− (6) Secondly, computation of a temporal derivative of differential order also returns the ZHT structure of algebraic order ( ) () () () 3 24 1,03,00,1 2,14,1 3 52 1,2 3,25,2 2 4 63 2,3 4,3 6,3 3 54 3,4 5,4 T taTtaTT taTtaTb t T taTtaTtaTb T taTtaTtaTb T taTtaTb φ ∂= ++++ ∂ +++ + ++ ++ (7) with a general term and the following structural coeff icients : 1,02,01,1 3,02,03,1 0,11,1 2,11,13,12,2 4,13,14,2 1,21,12,2 3,23,1 2,24,2 3,35,24,2 5,3 2,32,23,3 4, 91 91 , , 2 122 12 99 271 , , 4 446 2713 9 , , 4 642 39 11 9, 9, 42 44 327 , 24 T ZZTZZ TZTZZ Z TZZTZ Z TZZZZTZZ T ZZT =−− =− =− =−− =−=− = +−−=− = − 3 4,23,35,3 6,35,3 3,43,3 5,45,3 32745, 244 45 , 4. 99 , 44 ZZZ TZT ZT Z +− == = = (8) Finally, a spatial derivativ e of order again pr o duc e s the Z HT structure of order ( ) ( ) ( ) ( ) ( ) 335246 1,03,05,00,1 2,14,16,1 3 357 1,2 3,25,27,2 246 83 0,3 2,34,36,38,3 3 574 1,4 3,45,47,4 2 4 65 2,5 4,5 6,5 6 2 3, B taBtaBtaBBtaB taB taTb x B taBtaB taBtaTb BB taBtaBtaBtaTb B taBtaB taBtaTb B taBtaBtaTb B φ ∂=++++++ ∂ ++ + + +++++ ++ + + + ++ + ( ) 3 56 5,6 taB taTb+ (9) with a general term , where structural coefficien ts become
V. A. Miroshnikov 1,02,01,13,1 2,2 3,02,01,13,12,2 4,23,3 5,02,03,1 4,25,3 0,11,13,1 2,2 2,11,13,1 2,24 277 27 1 424 32 16 135961111, 83224 168288 812711, 8168288 9819 , 8 3216 117 837727 32 , 23 38 BZZZZ BZZZ ZZZ BZZZZ B ZZZ BZZZ Z =− −++ = +−−+ =− + −+ = − −+ + = − + + ,2 3,3 4,11,13,12,24,23,35,3 6,13,14,25,3 1,21,13,12,24,23,3 3,21,1 3,1 2,24,2 9, 32 8178327109915 , 3216161232 32 405451593 2438181 , 9, 168323232832 81741153549253 32 328832 Z BZ ZZZZZ BZZZBZZ ZZZ BZZZ Z + =−++−− + =−+−=−−++ =−+ +−− 3,3 5,3 5,23,1 2,24,23,35,3 7,24,25,3 0,31,12,23,3 2,31,1 3,1 2,24,23,35,3 4,3 135 , 16 243 8187381685 16881632 40540581 8181 ,, 83232 1632 812431772431053405 , 32 32883216 243 32 , ZZ BZZZZZ BZZBZ ZZ BZZZ ZZZ BZ + =− −++− =− +=−+ =−+ +−−+ = − 3,12,24,23,35,3 6,34,23,35,3 8,35,3 1,41,12,23,3 3,43,12,24,23,35,3 5,4 4 2436638912295 16816 16 405405 66152835 ,, 8 16 3232 81 2437298124324311791215 , 32 1632321681616 243 8 , , ZZZ Z BZ ZZBZ BZZZB ZZZZZ BZ −++− =−−+ =− =−+−=−−++− = − ,23,35,3 7,45,3 2,52,2 3,3 4,54,23,35,3 6,55,3 3,63,3 5,65,3 729 31233645 , 16 1632 81729 , 8 16 8172912151215 , 8 161616 405 405 , . 1 , 16 6 , Z ZBZ BZZ BZZZBZ B ZB Z −+ =− =−+ =−− +=− =−=− (10) 2.3. Experimental Multiplication of the ZHT Str uctur es Symbolic computation of a product of twowave solution (4) and first spatial derivative (5) once more produces the ZHT structure of algebraic order ( ) () () () () 35246 1,03,05,00,12,1 4,1 6,1 357224 6 83 1,23,2 5,27,22,34,3 6,3 8,3 3579 446810 5 3,45,4 7,4 9,44,56,58,510,5 P taPtaPtaPP taP taP taTb x PtaP taP taPtaTbP taP taP taPtaTb P taP taPtaPtaTbP taP taPtaPtaTb φ φ ∂=++++++ ∂ ++++++++ + +++++++ () () 57911668107 5,67,69,6 11,66,78,7 10,7 P taPtaP taPtaTbPtaP taPtaTb+ ++++++ (11)
V. A. Miroshnikov with a general term and the following structural coeff icients : 1,00,01,03,0 0,03,02,01,0 5,02,0 3,00,10,00,1 2,10,02,12,00,11,1 1,0 4,10,04,12,02,13,1 1,01,13,0 6,13,1 3, 11 , , 88 1 , , 8 1, 8 1, 8 PZAPZAZA PZA PZA PZAZAZA P ZAZAZAZA P ZA =+ =++ == + =++ + =++++ =02,04,11,20,01,21,1 0,1 3,20,03,22,0 1,23,10,11,1 2,12,2 1,0 5,20,05,22,03,23,1 2,11,1 4,14,2 1,02,23,0 7,24,23,03,1 4,12,0 1 , , 8 1, 8 1, 8 Z APZAZA P ZAZAZAZAZA PZAZAZA ZAZA ZA PZAZA ZA +=+ =++ ++ =++++ + + = ++ + + 5,2 2,30,02,31,1 1,22,20,1 4,30,04,32,0 2,33,11,21,13,24,2 0,12,2 2,13,3 1,0 6,30,06,32,04,33,13,21,1 5,24,22,12,24,15,3 1,0 , , 1, 8 1 1 8 8 PZAZAZ A P ZAZAZAZAZAZAZA PZAZAZAZAZA ZA ZA =+++ =++ ++++ =+++ +++ + +3,3 3,0 8,35,33,0 4,24,13,15,2 2,06,3 3,40,03,41,12,32,21,2 3,30,1 5,40,05,42,0 3,43,1 2,31,1 4,34,21,22,2 3,25,3 0,13,3 2,1 7,45,3 2, , 1, 8 1, 8 ZA PZAZA ZAZA P ZAZAZAZA P ZAZAZAZAZAZAZAZA P ZA + =+++ =+++ =++ + +++ ++ + =14,2 3,23,34,13,14,32,2 5,22,0 5,41,16,3 9,45,3 4,14,25,23,1 6,3 4,53,3 1,22,22,31,13,4 6,55,3 1,24,22,33,3 3,23,1 3,42,24,31,1 5,4 8,55,33,24,2 , , , , ZA ZAZAZA ZAZA PZA ZAZA PZAZ AZA PZAZAZAZAZAZA P ZA Z ++ ++ =++ = + =+ ++ = + ++ + ++ 4,33,3 5,23,1 5,42,26,3 10,55,3 5,24,26,35,63,3 2,32,23,4 7,65,3 2,34,2 3,43,34,32,2 5,4 9,65,3 4,34,25,43,3 6,311,65,3 6,36,73,3 3,4 8,75,3 3,43, , , , , ,,, AZAZAZA PZAZA PZAZA PZAZAZAZA PZAZAZA PZA PZA PZA Z ++ =+=+ =+++ =++= = = + + 3 5,410,75,3 5,4 , .A PZA= (12) 2.4. Experimental Solution of an Algebraic KdV Pr o b lem Consider an experimental solution for algebraic order which is a smallest order required to avoid de generation of the subsequent computational algorithm. Substitution of temporal derivative (7), product (11) of the twowave solution and the first spatial derivative, third spatial derivative (9), and collection of structural coefficients reduce differential KdV Equation (1) to an experimental algebraic KdV equation
V. A. Miroshnikov ()()()() ( () () ) () () ( () () ) 35 2 1,01,0 1,03,03,03,05,05,00,10,1 0,12,12,1 2,1 46 3 4,14,14,16,16,11,21,2 1,23,23,23,2 5 72 5,25,2 5,27,27,20,3 6666 6 6 666 66 BP TtaBPTtaBPtaBP TBP Tta BPT taBPtaTbBPT taBPTta BPTtaBPta TbBB ++ ++++++++ + + +++ +++ + + ++ +++ +++ +++ ()() ( () () ) ()() ( () () ) () () ( 24 2,32,3 2,34,34,3 4,3 6 8335 6,36,36,38,38,31,43,43,43,45,45,45,4 794246 7,47,4 7,49,49,42,54,54,5 4,56,56,5 6,5 66 66 66 66 66 PT taBPT ta BPTtaBPtaTbBtaBPTtaBPTta BPTta BPtaTb Bta BPTta BPTta ++ +++ +++ ++++++++ ++++++++ +++ + + + ( ) ) ()() ( ) ( ) ( ) ( ) 81053579116 8,5 8,510,53,65,65,67,67,69,611,6 46 8101277 911 4,76,7 6,78,710,712,77,89,811,8 13 88 13,88,9 10, 6 66666 666 6666 6 66 BPtataPTbBtaBPtaBPtaP taPtaTb BtaBPtaPtaPtaPtaTbPtaPtaPta PtaTbP taP +++++++ + ++++++ ++ +++ + + ( ) 10129 9 12,9 6 0.taP a Tbt=+ (13) To vanish structural coefficients of , construct binomial systems by vanishing structural coefficients of of odd binomial orders with The binomial systems have two equations, four equa tions, and five equations for and , respectively. The degenerate d bi nomial sy stems for and are, respectively, 1,01,0 1,00,10,10,1 60, 60,T PBTPB + +=+ += (14) 3,03,0 3,02,12,1 2,11,21,2 1,20,3 60, 60, 60, 0.TPBT PBT PBB ++=+ +=+ +== (15) In agreement with onewave solution (3), structural coefficients of the twowave solution are initia lized by (16) For the experimental solution, (17) Solving Equations (14) and (15) with respect to , and yields 1,1 3,12,24,23,3 2,16 9,2,8 3,2. ZZZ ZZ==− ==−= (18) For the binomial system has five equations 5,0 5,04,14,14,13,23,2 3,22,32,3 2,31,4 60,60, 60, 60,0. PBTPBT PBT PBB=++=+ +=+ +==+ (19) Solving first and fifth equations of (19) with respect to and , respectively, gives (20) Substitution of (20) in second, th ird , and fourth equations of system (19) reduc es th em to identities. For , the binomial system also has five equati ons 6,16,15,25,2 5,24,34,3 4,33,43,4 3,42,5 60,6 0,6 0,6 0,0.PBT PBT PBT PBB=+ +=+ +=++==+ (21) Solving first equation of system (21) respect to returns (22) Substitution of computed structural coefficients (17), (18), (20), and (22) in (4) yields an experimental two wave solution in the ZHT structure of algebraic order ( ) 23242353464 9 81683240 ,2 222 89 9399 xttata taTbta taTbtataTbtataTb φ =−+− +−+−+− (23) Substitution of the computed structural coefficients in the lefthandside part of (13) returns an experimental remainder of the ZHT structural approximation
V. A. Miroshnikov ( ) 5 724683 3 5794 246810 5 3 57 5525 1055910 ,722787281 2251765 13405 14230 8 83681 4055805430719081 17536 8 82927 1215 1611 319 82 e rx ttataTbtatataTb tatatataTb tatatatataTb tata ta =−++− +− +− +−+ +− +−+− −+ + 911 6 4681012 7 791113 8 12108 9 3416 61760 3 81 405 60160 189 190904 4 81 3920 1600 54 3293 3200 320 72 9. tata Tb tatatatata Tb tatatata Tb tatataTb +− +−+− +− +− ++− +−+ − (2 4) Rate of convergence of the ZHT structural approximation is examined in Table 1, where a tolerance ( ) () () , , max , e tx r xt ε ∈−∞∞ −∞∞ = (25) and a CPU time are given versus algebraic order for various reference levels . Table 1 was co m puted on a workstatio n in Maple 17.02 by using a sixcore A MD6300 processor with frequency 3.50 GH z and RAM 12.0 GB. The CPU time depends mainly on order of approximation . Tolerance significantly depends upon and through interaction para mete r . For propagatio n ce lerities and , for 1 8,0,180.125,0,0.125,h= −=− respectively. Surface plots of show a uniform convergence of the twowave solution in the ZHT structure. The experimental solutions of Section 2 were computed by experimental programming with lists of equations and expressions in the virtual environment of a global variable with 4 procedur es of 185 code lines in to tal. 3. Theory of Interaction of Two Pulsatory Waves in the ZHT Structure 3.1. Formulation of the Theoretical Problem in the ZHT Structure Compute theoretically a stru ctural solu tion of (1) for nonlinear interac tion of two pulsatory w aves. Constru ct the structural solution in the ZHT structure of algebraic order and diffe rential or de r ( ) 2 2, 0 1 0 ,, MN mmN k m Nkm m N K sxthSTbCta +−+ −+ + == = +∑∑ (26) where is a scale, is a structural coefficient, and are symbolic limits of sum Table 1. Convergence of the experimental solutions in the ZHT structure. 10 20 40 80 160 0.156 0.407 1.782 11.672 99.343 0.125 0.00112 0.312 0.609 2.250 13.312 111.718 1.03 0.056 0.250 0.578 2.328 14.282 116.812
V. A. Miroshnikov mation. Other variables and parameters are the same as in (2), but instead of experimental instances of section 2 they receive symbolic values to compute theoretically a general term of the structural solution. In agreement with (26) and (2), a twowave solution is constructed with algebra ic order and differential order using a generalized Einstein notation for summation, which is extended for exponents, as follows: ( ) ( ) 22 2 , 2, ,2 , mm m mmmm x thptaZtaZTb φµ + + =++ (27) where When and initialization condition (16) is invoked, the two wave solution is reduced to onewave s o lution (3). 3.2. Theoretical Differentiation of the Invariant ZHT Structure Primarily, computation a fir s t spa tial derivative of a binomial term yields ( ) 1 111 , 11,1,, 1 , nm nmnn mnm nmn mn mnm ta Tbdata TbdataataTbdata Tb xd µ − −++ − −++ + ∂ =++ ∂ (28) ( )( ) ( )( ) 2 , 1, 11,1, 1,1,, 1, 1 ,, , ,, ,,, nmnmn mnm n mn mnmnm da aqda an d dnm mdnm d nmaandamna dm −− −− ++++ = = = = =−=− = = (29) where structural coefficients (29) of binomial derivative (28), called differential binomial coefficients, are ma trix functions . Here, indices and , which are equal to powers of and in the binomial derivative, d efine names of the matrix fun ctions. Indices and which are equal to powe rs of and in the binomial term , determine definition s of the matrix functions. Similarly, computation of a firs t temporal derivative of the binomial term gives ( ) 1 111 , 11,1,, 1 , nm nmnn mnm nmn mn mnm ta TbdttaTbdttattaTbdtta Tb td µ − −++ − −++ + ∂ =++ ∂ (30) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 222 , 1, 11,1, 2 22 1,1,, 1, 1 22,, ,, 3,223 , 223 ,223. nmnmn mn m n mn mnmnm dtdtqqh mdtth nnm d dtth ndt nm d nmdtq hmnm µµ µµ −− −− ++ ++ = = = ==−+=−+ =+=+ (31) Thus, spatial and temporal derivatives (28) and (30) have the same structure but vary in differential binomial coefficients (29) and (31), respectively. Increase in order of differentiation produces a twodimensional (2d) differentiation cascade, which is shown in Figure 1. The onedimensional (1d) diff erentiation cascade o f invariant hyp erbolicsecant structures is asym metrical as the cascade spreads only towards higher powers [18]. To the contrary of the 1d differentiation cas cade, the 2d differentiation cascade spreads in symmetrical square waves, which resemble circular waves on the water surface generated by a point source. Similar to the 1d differentiatio n of the invariant trigonometric, hyper bolic, and elliptic stru ctures [18], the 2d differentiation of an even order preserves structure of binomial deriva tives and the 2d differentiation of an odd order converts structure of binomial derivatives to complementary ones. Finally, computation a third spatial deriv ative of the binomial term returns a binomial derivative of the following structure: () ( ) () 3331122 ,31,21,22,1, 1 3 2131 13 2, 13,1,1,3, 2 2, 1,12, nm nmnn mnn nmn mnmnmnm nmnnnnm n mnmnmnmnm nn n mnmn ta TbdbtaTbdbtadbta Tbdbtadbta x dbtaTbdb tadbtadbtadbta Tb dbtadab tdb µ −− +−− −−−+−− −− +−−−++ +−−− + + − −++ + ∂ = ++++ ∂ ++ +++ +++ ( ) ( ) 21 1 1 12 1, 21,2 3 ,3 nm m n nm nm nm nm nm ta Tb dbtadbta Tb dbta Tb ++ + − ++ −+ ++ + + + + + (32) with differential binomial coefficients
V. A. Miroshnikov (a) (b) (c) (d) Figure 1. Differentiation cascade of the binomial structure: (a) N = 1; (b) N = 3; (c) N = 2; (d) N = 4. ( )()()( )() ( )()( )() ( )()() 64 , 3,31, 21, 2 42 1,21,22, 12, 1 2 2222 ,1 ,12,12,1 12, 31, 31, 31, 2 16 ,, ,, ,3, 3, nmnmn mn m nmnmn mn m nmnmn mn m dbdbqm mmdbdbqmmn dbbq mmndbdbq mnn db nm nm d nmnm d nmmnmbq mqqmndbdbq −−−− −− +−+−− −−− −−+− +− ==−−=− =−−= = = = − = −=−+ == + ( ) ( )()()( )() ( )()( )()() ( )()( ) 22 3,3,1, 1, 22 1, 1,3,3, 2 2,12,1,1,1 1, 12, 2631, 2631 , , 12, 31, , 2 ,, ,, nm nmnm nm nm nmnm nm n mn mnmnm dnmd nm d nm mn n dbbn nndbbqmn nn dbbqmn nndbbn nn dbbmnn d nm db dbdnmnmmq −− −− ++ ++ −+ −+++ + = −−=−++− =+++= = = == −+ + −==−= = ( ) ( )()( )() ( )()()()() 22 2, 12, 11, 21,2 1, 21, 2, 3, 3 ,, 316 , 31, , 3 1, ,31, 12. n mn mnmnm n mn mnmnm d nmnm d qmm n dbbmn ndbdbmmn dbb nm m mndbdbmmnmm + ++ +−+−+ ++ ++++ ++ =−+ =+ =− +=−+ + = = + = = (33) Construct now a sequence of terms of the zigzag structure ( ) 2 , 2, mkmkmk mkmmkm Z taZtaTb +++ + + ++ + with , derivatives of which make a contribution to the general term of a zigzag derivative through binomial derivatives (28), (30), and (32). Substitute then the binomial derivatives of 2 , mk mkmkmk ta TbtaTb + +++ + and collect like terms of a structural coefficient of the general term proportional to . In agreement with (5), a first spatial derivative of the twowave solution is the invariant ZHT structure of algebraic order and diffe rential or de r ( ) 32 113 1, 1,3, 2, m mmm mm mm mm p AtaAtaAtaTb x φµ −++ −++ + ∂= + ∂ (34)
V. A. Miroshnikov where and structural coefficients are following: () 1,, 11, 11,, 1,, 11, 11,,1,2,, 11,1 3,1,2,, 13, 1 ,, ,, , , ,. , mmnmmmnmmm mmnmmmnmmmnmmm nmmm mmnmmmnm mm nmi jnmi j ADa ZDa Z ADaZDa ZDa ZDa Z ADa ZDa Z d ijDa ZaZ −+ −−− +++−+−+− ++ +++− ++ = + =++ + =+ = (3 5) Equations (34)(35) are complemented by truncation conditions of the twowave solution (36) which are set by (27). Conditions (36) result in truncation conditions for the first spatial derivative: ,3,1, 1, 0 for 0,1,3; 0 for ; 0,0 for 1. nmmmmmmm AnnmnmAmMAAmM + +− =<<−>+= >== >+ (37) Similarly, a first temporal derivative of the two wave solution again yields the invariant ZHT structure of al gebraic order and diffe rential or de r ( ) 32 113 1, 1,3, 2 , m mmm mm mm mm p TtaTtaTtaTb t φµ −++ −++ ∂= ++ ∂ (38) where structural coefficients are ( ) 1,, 11,11,,1,, 11,11,,1,2, , 11, 13,1,2,, 13, 1,,,, , ,,,, mmnmmmnm mmmmnmmmnm mmnm mm nmmmmmnmmmnmmmnmi jnmi j TDt ZDt ZTDt ZDt ZDt Z DtZTDtZDtZDt ZtZd ij −+−− −+++− +−+ −++++ +−++ =+=+ += + += (39) and truncation conditions are given by (36).Truncation conditions for the first temporal derivative become ,3,1, 1, 0 for 0,1,3; 0 for ; 0,0 for 1. nmmmm mm m TnnmnmTmMTTmM + +− =<<−>+= >== >+ (40) Finally, computation of a third spatial derivative of the twowave solution produces the invariant ZHT struc ture of algebraic order and differential order ( ) 352 31135 3,1,1,3,5, 3 ,2 mm mmmm mmmmmm mmmm p BtaBtaBtaBtaBtaTb x φµ −−++ + − −++ + ∂=+++ + ∂ (4 1) where structural coefficients are 3,,33, 31,22,22, 11,13,, 1,,31, 31,22, 21, 2,2,11,1 2,11, 11,,3,2, , mmnmmmnmmmnm mmnmmm mmnmmmnmm mnmmmnmmm nmm mnmmmnmm BDbZDb ZDb ZDbZ BDbZDbZDbZDb Z DbZDb ZDbZ −+− −−+− −− +−−− −+−−++− −−+−+−− − ++−−−+ =+++ = + + ++ ++ 2, 11, 1 1,1,2,22,11, 1,11, 11,, 1,2,, 11, 12, 13, 11,22,2 3,2,11, 13 , , mn mmm m mn mmmnmmmnmm mn mmm nmm mnmmmn mmmnmmm mmnmmmn Db Z BDbZDbZDb ZDb Z Db ZDb ZDbZDbZ BDb ZDb − −++ +++−++−−+ +−+ −+−++−−++−− ++ ++++−+ ++ + = + ++ = ++ + ,,1,2,2, 11,1 ,13,11, 22,21,24, 2,33, 3 5,3,2,2, 13, 11,24,2, 35,3 , , , mmm nmmm nmmm nmmmnmmmnmm mnmmm mmnmmmnm mmnmmmnmmm nm i ZDb ZDbZ DbZDbZ DbZDbZ BDb ZDbZDbZDb Z Db Z +++ −++ −+++− ++−− ++−++ +++ +−+++−++−++ + ++ + = + + ++ + ( ) ,, , ,, jnmi j i Zjdb= (42) and truncation conditions are set by (36). Truncation conditions for the third spatial derivative are ,5, 3, 1,1,3, 0 for 0,3,5; 0 for ;0 for 1; 0 for 2; 0,0 for 3. nmm mmm mmmm mm Bnnm nmBmMBmM BmM BBmM ++ + −− =<<−>+=>=> + = >+==>+ (4 3) Equations (34), (38), and (41) show that the ZHT structure is invariant with respect to differentiation of orders and , which only modify algebraic orders and structural coefficients. The zigzag structure of two wave solution (27) and its derivatives (34) and (38) together with product of (27) and (34) are shown in Figure 2 in a virtual space of computational indices and of structural coefficients , ,
V. A. Miroshnikov (a) (b) (c) (d) Figure 2. The invariant ZHT structure in the virtual space (n, m): (a)—(27), (b)—(34), (c)—(38), (d)—(53). and , which coincides with a virtual space of computational powers and of . Differentia tion increases the width of the invariant ZHT structure and the effect of multip lication is similar to that of diffe rentiation. 3.3. Theoretical Multiplication of the Invariant ZHT Structures Continuation of spatial derivatives (34) and (38) in the invariant ZHT structure to any differential order in the generalized Einstein notatio n gives 22 2 2, ,2 NmNkm m Nkm N N pta Tb xR φµ + −+ −+ ∂= ∂ (44)
V. A. Miroshnikov where and The spatial derivative of order then becomes 12 2 1 1 21 1 , 2 . NmNkm mN k m N N pta Tb xQ µ φ + −+ − − + −+ + ∂= ∂ (45) Differentiation of (45) with respect to and reduction of all terms to a general term by substitutions and in a term proportional to , in a term proportional to in a term proportional to yields ()( ) ()() 22 21,2 ,1 2 2 1,1 22, 211 2 2 11 , NmN k mmN km mN km mN k mmN km N N mN kQmQ mN kQm x tTq aQ p b µ φ + −+ −−+− −+ −+ +−++ − −+−−− + −+ −+ + ∂= ∂ (46) where struc tural coefficients of (44) a re c on ne c ted with st ructura l coeffici e nt s of (46) by a recurrent re lation ()() ()( ) 2,2 1,2,1 22, 1 2 1, 21 1 211 . mN kmmN k mmN km mN k mmNkm RmN kQmQ m NkQmqQ −+−+ −−+− −+ +−++ − −+ −− − +−+ +++ = (47) Thus, the invariance of the ZHT structure with respect to spatial differentiation of any order is proved by in duction. Set up two spatial derivatives of the twowave solution in the invariant ZHT structures of algebraic order and differential orders and with structural coefficients and as 12 1 112 2 12 2 222 2 22 2, 2, 2 ,2, NmN kNmNk mm mN kmmN NN NN km pta TbptaTb x QR x φφ µµ +−++− + −+− + ∂∂ = = ∂∂ (4849) where and for the first structure, and for the second structure. Product of (48) with a binomial substitution and (49) with a binomial substitution returns 12 1 12 12 12 2 42 2,2 2, 4 ,4 N NmN Nkm nNln mnNkl NN mn NN Qpta Tb xx R φφ µ + +−− + − +−−+ −− ∂∂ = ∂∂ (50) where , , and . Thus, the ZHT struc ture is also invariant with respect to multiplication since a general term of the product is the ZHT structure of algebraic or der and differentiation order 12 12 1 2 2 1 12 4 42 2 4, N NmNNkm m NN NN NN k ptaTbP xx φµ φ + +−− + −−+ ∂∂ = ∂∂ (51) where the structural coefficients are 12 12 22,22, . mNNknNlnm n Nklm n P QR −− +−+−− +−− = (52) Summation of a general term of the product of twowave solution (27) and first spatial derivative (34) by (51)(52) with yields ( ) 511 35 1, 1,3 4, 5, ,4 m mmmm mm mm mmmm p PtaPtaPtaPtaTb x φ φµ −++ + −++ + ∂=++ + ∂ (53) where structural coefficients are obtained for and by constructing a list of sums of general terms for with truncation conditions which follows from (27), 1,1,,1,1,,1,2, 3,3,,1,2,5,5,,3, 2, , , , . mmmnmn nnmmmnmn nnmnmn nn mmmnmnnnmnmn nnmmmnmn nnmnmn nn PAZPAZAZ PAZAZ PAZAZ −−− −+−+−−− −+ +−+−−+−++−+−−+− + ==+ =+=+ (54) Structure of the twowave solution is reduced to that of a spatial der ivative b y subs titution (55) Truncation conditions for product of the twowave solution and the first spatial derivative become
V. A. Miroshnikov , 5, 3, 1, 1,3, 0 for 0,1,5; 0 for 2; 0,0,0,0 for 21. nmm m mmmmmmmm Pnnm nmPm M PP PPmM + ++−− =<<− >+=> ==== >+ (56) 3.4. Theoretical Solution of the Algebraic KdV Problem Substitution of temporal derivative (38), product (53) of the twowave solution and the first order spatial deriva tive, third order spatial d erivative ( 41), and collection of structural coefficients reduce differential KdV Equation (1) to a theoretical algebr aic KdV equatio n ( ) () () ( ) 523 3254521 3,1,1, 1, 3254521 3254523 1,1, 1,3,3,3, 54 525 5, 5, 22 24 2 2 2422242 24 20, mm mmmmmmmm mm mmmm mmmmmm mm mm mm mm pB tapTpPpBta pTpPpB tapTpPpBta pPpBta Tb µµµµ µ µµµµµ µµ −− −− −− ++ + +++++ + ++ ++ + ++ ++++ ++= (57) which is complemented by truncation conditions (40), (56), and (43). For this equation to be satisfied exactly for all para meters and variables, all structural coefficients of should vanish. Therefore, five structural coefficients of (57), which are supposed to be vanished, are 222 222 2 3, 1,1,1, 1,1,1, 22 222 2 3,3, 3,5,5, 0, 120, 120, 12 0,12 0. mmmmmmmmmmmm mm mmmm mmmm mm BTpPB TpPB TpP BpP B µµµ µµ µµ µµ −−− −++ + + ++++ =+ +=+ += + +=+= (58) These equations constitute a polynomial system of equations with respect to leading structural coefficients of increasing orders. To combine equations with respect to of same binomial orders , construct binomial systems as structural coe ff icie nts of , where for . The binomial syst ems of orde rs and have two and four equations, respectively, 22 222 2 1,01,0 1,00,10,1 0,1 120, 120,TpP BTpP B µµ µµ + +=+ += (59) 22222 222 22 3,03,03,02,12,12,1 1,21,21,20,3 120, 120,120,0.TpPBTpP B TpPBB µµµµµµµ ++=++=++== (60) Substituting structural coefficients through by (39)(40), (54)(56), (42)(43), using initialization (16), and solving (59)(60) with respect to and gives 22 1,1 3,12,24,23,3 2,,2,2 ,2.ZZpZZpZ==−==− = (61) For binomial orders where for the binomial system has five equations 5, 22 2222 5,4, 14, 14, 1 222 3, 23, 23,2 22 22 2, 32,32,31,4 120, 120, 12 0, 120, 0. ll llllllll llll ll llll llll pPBTpPB TpP B TpP BB µµµ µ µµ µµ µ + +++++ ++ ++++ ++ ++++ ++++ += ++= + += ++= = (62) Solving first and fifth equations of (62) with respect to and respectively, yields 22 2 5,34,46,45,51, 1, 4 ,2,5 ,2,,2, M MMM ZpZZpZZMpZ +− =−==−==− = (6 3) where last two terms are obtained by induction. In agreement with the experimental solution of section 2.4, second, third, and fourth equations of system (62) are satisf ied identically. For binomial orders , where for , the binomial system also has five equations 22222 2 2, 32, 31, 21, 21, 2 22 222 22 , 1, 1, 11,1,1,2, 1 120, 120, 12 0,12 0,0. M MMMMMMMMM MMMMMMM MM MMMMM pPB TpPB TpP BTpP BB µµµµ µµ µµµ + −+ −+−+−+− − −−− −−−+ +=+ += + +=+ +== (64) Solving first equation of (64) with respect to returns
V. A. Miroshnikov ( ) 22 2 6,4 7,52, 5,6,1 , MM ZpZp ZMp + =− =−=−+ (6 5) where a last term is also generated by induction, the proof of which will be completed in Section 4. Finally, a general solution fo r th e ZHT structural coefficients may be written as ( ) 2 , 2, 2,1 , mmmm Z Zmp + = =−+ (66) and general term (27) of the twowave solution in the ZHT structure becomes ( )() 222 2 , 221 m mm x thptamptaTb φµ + =+−+ (67) for .Theoretical formula for the ZHT structural sum of the twowave solution is ( )() 222222 1 , 2121. Mm mm m xthpptatamp taTb φµ + = =+− +−+ ∑ (68) Substitution of initialized (16) and computed (66) stru ctural coefficients in the left handside part of theoreti cal algebraic KdV Equation (57) yields a theoretical remaind er of the invariant ZHT structure ( ) () ( ) () ( ) 3213 22 1,23,2,1 2,1 411135 4, 11,1,3,5, 311 35 3,1,1, 3,5, ,2 MMMMM tMMMMMMMM MMM MMMM M MMMMMM MMM mmmm m mmmm mmmmmm rxtpCtaCtaTbCtaC ta Cta TbCtaCtaCtaCtaTb CtaC taCtaCtaCtaT µ + +−+ +−+−−+ − +−− +++ +−−+ ++ −−+++ − −++ + =+++ ++++ + ++++ + 21 1 , Mm mM b + = + ∑ (69) where a structural coefficient 22 2 ,,, , 12 nm nmnmnm C TpPB µµ =++ (70) and truncation conditions (40), (56), and (43) are invoked. The theoretical solutions of Section 3 in the invariant ZHT structure were computed using theoretical pro gramming methods with symbolic general terms by the generalized Einstein notation in the virtual environment of a global variable with 16 procedures of 923 code lines in total. The theore tical formulas for twowave solution (68), first spatial derivative (34)(36), first temporal derivative (38)(40), third spatial derivative (41)(43), product of the twowave solution and the first spatial derivative (53)(56), and structural remainder (69)(70) were justified by the correspondent experimental solutions for algebraic order . 4. Discussion and Visualization Through an expansion variable , where as and , summation of the ZHT structural approximation yields ( ) ( ) ( ) 222222 2 21 ,, 1 pptata Tb xt htaTb µ φ −− = +− (71) since a partial sum of the Taylor series expansion of (71) in of order returns the same expression as the ZHT structural sum (68) with the same general term as general term (67) of the invariant ZHT struc tur e. A functional form of (71) expressed through two regular hyperbolic functions and , five parameters , and two propagation variables and becomes ( )()()() ()( ) 2 2222 22 2 tanhanh tanh t 21t a ,. 1 nhtanh XXp Y XY pq xt hq µ φµ µν µν −− = +− (7 2) Differentiation and substitution of Equation (72) into differential KdV Equation (1) completes the proof by induction of (66)(68) and returns another verification of (71) as Equation (1) is satisf ied identically. Conversion of (72) through two singular hyperbolic functions and , two regular hyperbolic functions and , three parameters and two variables and gives
V. A. Miroshnikov ( ) ( ) ()() ()( ) 222 2 2 22 csch sech coth 2 ,tanh .xt hXY XY µν ν µ µν µν µ φν − = + + − (73) When , the twowave solution reduces to the conventional form [6] of the twosoliton solution in two singular hyperbol i c f u nc t ions and , two regul ar hyperbolic funct ions and , two parameters and , and two propagation variables and () ( ) ()( ) ( )( ) 22 2 csch sec 22 co h ,th2tan2 . 2 h UVU xt UX VVY UUX VVY φ − = + − (74) A remainder of the Taylor series approximation of twowave solution (71) ( )() 2222 1 ,22 1mm smM rxtpmptataTb µ ∞ = + = −+ ∑ (75) converges slower than the ZHT structural remainder (69) because of the infinite limit of summation and trunca tion condition s (40), (56), and (43). So, the method of decomposition in the invariant ZHT structure is more ro bust than the method of expansion in the Taylor series. Interaction of two pulsatory waves is visualized by spatiotemporal plots in Figure 3 for positive and negative reference levels . Negative values of considerably increase amplitudes and decrease dispersions of pul satory waves compared with those of solitons with , because pulsatory waves propagate on a more shal low water in this case. The effect of positive values of is opposite and results in decrease of amplitudes and increase of dispersions. Similar to interaction of two solitons, interaction of two pulsatory waves is also con servative and preserves onewave solutions before and after a nonlinear interaction at the moment of merging. Animations of the twowave solution show that the merging process may be considered as a flow of a faster flu id of the first pulsatory wave into the second pulsatory wave with a slower fluid. 5. Conclusions The analytical methods of solving PDEs by undetermined coefficients and series expansions are generalized by the computational method of solving nonlinear PDEs by decomposition in the invariant ZHT structure. The computational algorithm is developed by experimental computing using lists of equations and expressions im plemented in four procedures of 185 code lines in total. Afterwards, the computational method is proved by theoretical computing with symbolic gene ral terms implemented in 16 procedures with 923 code lines in to tal. Figure 3. Spatiotemporal plots of the twowave solution for U = 1. 8, V = 1, a = 8. 4, b = 18, h = 1/8 (left) and h = −1/8 (right).
V. A. Miroshnikov The invariance of the ZHT structure with respect to differentiation and multiplication is shown by using 2d differentiation cas cade of binomial stru ctures and mathematical indu ction. Contrary to the asymmetric differen tiation cascade in one dimension [18], the 2d differentiation cascade spreads in symmetric square wav es. Com pared with the 2d series expansion, the ZHT structure considerably saves computational resources and simplifies results since it implies a loworder polynomial in one dimension and a series expansion in another dimension. The invariance of the ZHT structure enables other computational applications in nonlinear PDEs with solutions approaching a constantor vanishing at infinity. The ZHT structural approximation and remainder are computed theoretically to any algebraic order. Summa tion of the twowave solution in the invariant ZHT structures is implemented and presented both through regular and singular hyperbolic functions. When a reference level vanishes, the twowave solution is reduced to the two soliton solution. Negative reference levels considerably increase amplitudes and decrease dispersions of pulsa tory waves compared with those for solitons with a vanishing reference level. The effects of positive reference levels are opposite, i.e. amplitudes of pulsatory waves are decreased and dispersions are increased. Acknowledgements The author thanks I. Tari for the stimulating discussion at the 2013 SIAM Annual Meeting. Support of the Col lege of Mount Saint Vincent and CAAM is gratefully acknowledged. References [1] Zabusky, N.J. and Kruskal, M.D. (1965) Interaction of “Solitons” in a Collisionless Plasma and the Recurrence of Ini tial States. Ph ysical Review Letters , 15, 240243. http://dx.doi.org/10.1103/PhysRevLett.15.240 [2] Hirota, R. (1971) Exact Solutions of the Kortewegde Vries Equation for Multiple Solitons. Physical Review Letters, 27, 11921194. http://dx.doi.org/10.1103/PhysRevLett.27.1192 [3] Drazin, P.G. (1983) Solitons. In: Reid, M., Ed., London Mathematical Society Lecture Note Series, No. 85, Cambridge University Press, Cambridge, 136. [4] Izrar, B., Lusseyran, F. and Miroshnikov, V. (1995) TwoLevel Solitary Waves as Generalized Solutions of the KdV Equation. Physics of Fluids, 7, 10561062. http://dx.doi.org/10.1063/1.868548 [5] Varley, E. and Seymour, B.R. (1998) A Simple Derivation of the NSoliton Solutions to the Kortewegde Vries Equa tion. SIAM Journal on Applied Mathematics, 58, 904911. http://dx.doi.org/10.1137/S0036139996303270 [6] Vvedensky, D.D. (1992) Partial Differential Equations with Mathematica. AddisonWesley Publishing Company, Wo kingham. [7] Miroshnikov, V.A. (1995) Solitary Wave on the Surface of a Shear Stream in Crossed Electric and Magnetic Fields: The Formation of a Single Vortex. Magnetohydrodynamics, 31, 149165. [8] Miroshnikov, V.A. (1996) The FiniteAmplitude Solitary Wave on a Stream with Linear Vorticity. European Journal of Mechanics, B/Fluids, 15, 395411. [9] Miroshnikov, V.A. (2002) The BoussinesqRayleigh Approximation for Rotational Solitary Waves on Shallow Water with Uniform Vorticity. Journal of Fluid Mechanics, 456, 132. http://dx.doi.org/10.1017/S0022112001007352 [10] Miroshnikovs, V. (1996) Coupled Solitary Waves in Viscous MHD a nd Geop hysical Flows. Comptes Rendu Académie des Sciences Paris, 323, 2330. [11] Keller, J.B. (1948) The Solitary Wave and Periodic Waves in Shallow Water. Communications in Pure and Applied Mathematics, 1, 323339. [12] Laitone, E.V. (1960) The Second Approximation to Cnoidal and Solitary Waves. Journal of Fluid Mechanics, 9, 430 444. http://dx.doi.org/10.1017/S0022112060001201 [13] Grimshaw, R. (1971) The Solitary Wave in Water of Variable Depth. Part 2. Journal of Fluid Mechanics, 46, 611622. http://dx.doi.org/10.1017/S0022112071000739 [14] LonguetHiggins, M.S. and Fenton, J.D. (1974) On the Mass, Momentum, Energy, and Circulation of a Solitary Wave II. Proceedings of the Royal Society A, 340, 471493. http://dx.doi.org/10.1098/rspa.1974.0166 [15] Pennell, S.A. and Su, C.H. (1984) A SeventeenthOrder Series Expansion for the Solitary Wave. Journal of Fluid Me chanics, 149, 431443. http://dx.doi.org/10.1017/S0022112084002731 [16] Pennell, S.A. (1987) On a Series Expansion for the Solitary Wave. Journal of Fluid Mechanics, 179, 557561. http://dx.doi.org/10.1017/S0022112087001666
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