Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 S69 NEW PERIODIC WAVE SOLUTIONS OF (+)-DIMENSIONAL SOLITON EQUATION by Jiangen LIU a, Pinia WU a, Yufeng ZHANG a*, and Lubin FENG b a School of Mathematics, China University of Mining and Technology, Xuhou, China b School of Mathematics and Information Sciences, Weifang University, Weifang, China Original scientific paper https://doi.org/.98/tsci7s69l In this paper, associating with the Hirota bilinear form, the three-wave method, which is applied to construct some periodic wave solutions of (+)-dimensional soliton equation, is a powerful approach to obtain periodic solutions for many non-linear evolution equations in the integrable systems theory. Key words: periodic wave solutions, three-wave method, Hirota bilinear form, (+)-dimensional soliton equation Introduction It is famous that integrable systems ehibit rich variety of eact solutions, such as soliton, periodic, rational, and compleion solutions for PDE in mathematical physics [- ]. Their eact solutions play an essential role in the proper understanding of qualitative features of the concerned phenomena and processes in different fields of non-linear science, such as non-linear optics, plasma physics and others. They could help us analye the stability of these systems and validate the results of numerical analysis of non-linear PDE. Among them, the periodic solution is the one of the more important solutions to understand some of the natural phenomena. Here periodic solutions could be obtained through the Hirota bilinear form or their generalied counterparts in three-wave method []. The Hirota bilinear form plays an important role in mathematical physics and engineering fields. Once a non-linear PDE is written in bilinear form by a dependent variable s transformation, then multi-soliton solutions, rational solutions, Wronskian and Pfaffan forms of N-soliton solution could be obtained [5-8]. In this paper, we consider the (+)-dimensional soliton equation [7, 8]: Under the transformation: u (ut u uu ) y ( u uy ) + + () u (ln f) () we can change the (+)-dimensional soliton equation into the bilinear form: (DD DD y t DD) y ff () * Corresponding author, e-mail: hangyfcumt@6.com
S7 Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 or, equivalently: ff f f ff yt + f y ft + f f y + fy f f fy ffy () where f f(, yt,,) and the derivatives D,D,D are the Hirota operators [9] defined by: y a b a b t ( ) (,) (,) ' ' ' ', t t t t DD f g f tg t The (+)-dimensional soliton solution to eq. () was studied by many researchers in recent years. For eample, its algebraic-geometrical solutions was eplicitly given in the form of Riemann theta functions by using a non-linearied method of La pair. The N-soliton solution and its Wronskian form of solution have been discussed and derived by using the Hirota method and Wronskian technique [7]. The bilinear Backlund transformation and eplicit solutions have also been obtained that is based on the Hirota bilinear method []. Some periodic wave solutions have been found in [, ]. The aim of this paper is to study the periodic wave solutions of (+)-dimensional soliton eq. (), which is associated with the Hirota bilinear form. Then, with the aid of the corresponding graphic illustration, we would give a better understanding on the evolution of solutions of waves. Describe the three-wave method Here, we briefly show the three-wave method []. We now consider a non-linear PDE of the form: puu (,, ut, uy, u, u,...) (5) where u uyt (,,,), p is a polynomial of u, and its various partial derivatives. Step : By using the bell polynomial theories [, ], eq. (5) can be converted into Hirota bilinear form: Η (D t,d,d y,d,...) ff (6) where f f(, yt,,) and the derivatives D,D y,d are the Hirota operators. Step : Based on the Hitota bilinear of eq. (6), we assume that the solution can be epressed in the form: f a cosh( tk + k + yk + k ) + a cos( tl + l + yl + k ) + + a cosh( tc + c + yc + c ) (7) ( i,, ), kj, l j, and cj ( j,,,) are all real parameters to be determined. Step : Substituting eq. (7) into eq. (6) and collecting all the coefficients about: cosh( tk + k + yk + k), sinh( tk + k + yk + k), cos( tl + l + yl + l), sin( tl + l + yl + l), cosh( tc + c + yc + c), and sinh( tc + c + yc + c) we get the coefficients of these terms to ero and a set of algebraic equations contains parameters. Step : Solving the set of algebraic equations in the Step using computer software, we obtain the values for involving parameters. Step 5: Substituting these parameters into eq. (7) and corresponding transformation, we obtain the eact solution for original equation.
Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 S7 Net, we will show the three-wave method to solve the (+)-dimensional soliton for a PDE. Periodic wave solutions of (+)-dimensional soliton solutions Based on the known Hirota bilinear eq. () and transformation (), we consider: f a cosh( tk + k + yk + k ) + a cos( tl + l + yl + k ) + + a cosh( tc + c + yc + c ) (8) ( i,, ), kj, l j, and cj ( j,,,) are all real parameters to be determined. To get the periodic solutions, substituting (8) into eq. () and collecting all the coefficients about cosh( tk + k + yk + k), sinh( tk + k + yk + k), cos( tl + l + yl + l), sin( tl + l + yl + l), cosh( tc + c + yc + c), sinh( tc + c + yc + c), we can obtain a set of determining equations for a i, k j, l j, and c j: aa( kl + kkl kll kl kl + kl + kl kl) aa( kk kll klk + ll kk + kk-ll + ll) aa( ck + cck + ckk + ck ck + ck + ck ck) aa( cc + ckk + ckc + kk kk + kk + cc cc) aa( cl + ccl cll cl cl + cl + cl cl) aa( cc + cll clc + ll cc + cc ll + ll) a (k k kk + k k ), a (l l + ll l l ) a (c c cc + cc ) Solving this system of equations with the help of symbolic computation, we can present the following solutions of parameters: Case : ll a a a a a c c c c ( + ) l cll ll cl,,,,, c c cll + ll + cl c k k k k k k k k l l l l,,,,,, cl l l, l l, l l, l l, a, k k ( i, ), c, lp ( p,,,), and kj ( j,,) are arbitrary constants. Case : ( ) k kk k ck a a a a a c k c c c c,,, ±,,, k k ( ) k kk k k k, k k, k, k k, l l, l l, l l k
S7 Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 ( ) k kk k k k, k k, k, k k, l l, l l, l l, l l k ( i, ), c, lp ( p,,,), and kj ( j,,) are arbitrary constants. Case : ( ) k ck c a a, a, a a, c ± k, c c, c, c c c ( ) k ck c k k, k, k, k, l l, l l, l l, l l c ( i, ), c, lp ( p,,,), and k are arbitrary constants. Case : ll a a, a a, a, c c, c c, c c, c c, k k ( ) l kll ll + kl kll + ll + kl k k k k,, k kl ( + ) l ll l l l, l l, l, l l l ( i, ), k, cj ( j,,,), and lp ( p,,) are arbitrary constants. Case 5: ll a a a a a a c c c c ( + ) l cll ll cl,,,,, c c cll + ll + cl ll c k k c k ( + ) l cll ll cl,,, cl c c ( + ) cll + ll + cl l ll l k l l l l l l l,,,, cl l ( i, ), c, and lp ( p,,) are arbitrary constants. Thus, we can obtain the solutions of eq. (): where ( f ) u ln (9) ( ) cos( ) + a cosh ( tc + c + yc + c ) f a cosh tk + k + yk + k + a tl + l + yl + l + ai ( i, ), k, l, and c ( j,,,) are given in the Cases to 6. j j j
Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 S7 Now, we present graphic state of some special solutions. Here we mainly analye the Cases and 6. Case : At first, we give one choice of the parameters: a, a, c, l l, l, l Then, we could obtain the following periodic solution: 5 7 6cos + y 9cosh y+ t t u + 5 7 cos + y + cosh y+ t t 5 7 sin + y sinh y+ t t + () 5 7 cos + y + cosh y+ t t Their plots when y and t,, 5 are depicted in fig., respectively, and the wave along different ais is shown in fig.. u Figure. The periodic solution of eq. () with parameters y, t ; (a) perspective view of the wave and (b) D-density plot (for color image see journal web site) (a) 5 5 (b) 5 5 5 5 5 5 u (a) 5 5 5 (b) 5 5 5 Figure. The periodic solution of eq. () with parameters y, t : The wave along the -ais (a) and the wave along the -ais (b) (for color image see journal web site)
S7 Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 Case 6: Second, we give one choice of the parameters: a a, a, a, c, l, l, l Then, we could obtain the following periodic solution: 7 5 cosh y+ 6 cos + y t t u + 7 5 cosh y+ + cos + y t t 7 5 sinh y+ + sin + y t t + 7 5 cosh y+ + cos + y t t Their plots when y and t,, 5 are depicted in figs. and, respectively. () 8 u 7 6 5 (a) (b) Figure. The periodic solution of eq. () with parameters y, t 5; (a) Perspective view of the wave and (b) D-Density plot (for color image see journal web site) u 6 7 5 (a) (b) Figure. The periodic solution of eq. () with parameters y, t 5; (a) Perspective view of the wave and (b) D-Density plot (for color image see journal web site) Conclusion In this paper, with the help of the Hirota bilinear form of the (+)-dimensional soliton equations, we presented the periodic wave solutions through the three-wave method. The
Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 S75 new periodic wave solutions of the (+)-dimensional soliton equation are obtained and some special solutions were illustrated to help us be better to understand the evolution of solutions of waves. The performances of the methods afore mentioned are substantially influential and absolutely reliable for finding new eact solutions of other NPDE. Acknowledgment This work was supported by the Fundamental Research Funds for the Central Universities (No. 7XKZD). Nomenclature a i (i,, ) co-ordinates, [s ] c j (j,,, ) co-ordinates, [s ] D, D y, D the Hirota operators, [K] f(, y,, t) dependent variable, [K] g(, y,, t) dependent variable, [K] k j (j,,, ) co-ordinates, [s ] l j (j,,, ) co-ordinates, [s ] t time co-ordinate, [s] u(, y,, t) dependent variable, [K], y, space co-ordinate, [m] References [] Drain, P. G., Johnson, R. S., Solitons: An Introduction, Cambridge University Press, Cambridge, USA, 989 [] Gardner, C. S., et al., Method for Solving the Korteweg-de Vries Equation, Physical Review Letters, 9 (967), 9, pp. 95-97 [] Xia, B., et al., Darbou Transformation and Multi-Soliton Solutions of the Camassa-Holm Equation and Modified Camassa-Holm Equation, Journal of Mathematical Physics, 57 (6),, pp. 66-66 [] Dong, H., et al., The New Integrable Symplectic Map and the Symmetry of Integrable Nonlinear Lattice Equation, Communications in Nonlinear Science & Numerical Simulation, 6 (6),, pp. 5-65 [5] Dong, H., et al., Generalised (+)-Dimensional Super MKdV Hierarchy for Integrable Systems in Soliton Theory, East Asian Journal on Applied Mathematics, 5 (5),, pp. 56-7 [6] Dong, H. H., et al. Generalied Fractional Supertrace Identity for Hamiltonian Structure of NLS-MKdV Hierarchy with Self-Consistent Sources, Analysis and Mathematical Physics, 6 (6),, pp. 99-9 [7] Yang, X. J., et al., Nonlinear Dynamics for Local Fractional Burgers Equation Arising in Fractal Flow, Nonlinear Dynamics, 8 (5),, pp. -7 [8] Yang, X. J., et al., On Eact Traveling-Wave Solutions for Local Fractional Korteweg-de Vries Equation, Chaos,6 (6), 8, pp. -8 [9] Yang, X. J., et al., Eact Travelling Wave Solutions for the Local Fractional Two-Dimensional Burgers Type Equations, Computers &Mathematics with applications, 7 (7),, pp. - [] Zhang, Y. F., Wang, Y., Generating Integrable Lattice Hierarchies by Some Matri and Operator Lie Algebras, Advances in Difference Equations, 6 (6),, pp. [] Zhang, Y., Ma, W. X., Rational Solutions to a KdV-Like Equation, Applied Mathematics & Computation, 56 (5), C, pp. 5-56 [] Zhang, Y. F., Ma, W. X., A Study on Rational Solutions to a KP-like Eqation, Zeitschrift Fuer Naturf-Orschung, 7 (5), A, pp. 6-68 [] La, P. D., Periodic Solutions of the KdV Equation, Siam Review, 8 (),, pp. 8-6 [] Liu, J., et al., Spatiotemporal Deformation of Multi-Soliton to (+)-Dimensional KdV Equation, Nonlinear Dynamics, 8 (6), -, pp. 55-6 [5] Hirota, R., Reduction of Soliton Equations in Bilinear Form, Physica D Nonlinear Phenomena, 8 (986), s-, pp. 6-7 [6] Wang, J. M., Periodic Wave Solutions to a (+)-Dimensional Soliton Equation, Chinese Physics Letters, 9 (),, pp. -6 [7] Geng, X., Ma, Y., N-Soliton Solution and its Wronskian Form of a (+)-Dimensional Nonlinear Evolution Equation, Physics Letters A, 69 (7),, pp. 85-89 [8] Wu, J. P., Grammian Determinant Solution and Pfaffianiation for a (+)-Dimensional Soliton Equation, Communications in Theoretical Physics, 5 (9),, pp. 79-79 [9] Hirota, R., The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, Mass., USA,
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