Multisoliton solutions, completely elastic collisions and non-elastic fusion phenomena of two PDEs

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1 Pramana J. Phys. (2017) 88:86 DOI /s Indian Academy of Sciences Multisoliton solutions completely elastic collisions and non-elastic fusion phenomena of two PDEs MST SHEKHA KHATUN 1 MD FAZLUL HOQUE 12 and MD AZIZUR RAHMAN 1 1 Department of Mathematics Pabna University of Science and Technology Pabna 6600 Bangladesh 2 School of Mathematics and Physics University of Queensland Brisbane QLD 4072 Australia Corresponding author. shekhshekhamath88@gmail.com MS received 11 August 2016; revised 8 December 2016; accepted 16 December 2016; published online 31 May 2017 Abstract. A direct rational exponential scheme is introduced and applied to construct exact multisoliton solutions of the clannish random walker s parabolic and the Vakhnenko Parkes equations. We discuss the nature of soliton solutions before and after their interactions and present their fusion (non-elastic) and elastic collisions of the soliton solutions. These soliton solutions of the equations are connected to physical phenomena: weakly non-linear surface internal waves in a rotating ocean and interacting population motions. In addition some three-dimensional and contour plots of the soliton wave solutions are presented to visualize the dynamics of the models. Keywords. Direct rational exponential scheme; the double-subequation method; the clannish random walker s parabolic equation; travelling wave solutions; multisoliton solution. PACS Nos Jr; Wz; Yv; Fg 1. Introduction Due to the broad application of soliton theory exact multisoliton solutions play important roles in applied mathematics physics geophysics biology communications and various issues related to engineering fields. The role of exact solutions of non-linear partial differential equations still has much unexplored open problems that compel scientists and engineers to seek different methods for searching exact single or multisoliton solutions. In recent years internationally there has been great interest to investigate the exact soliton/multisoliton solutions of non-linear equations which can be used to simulate physical phenomena in various areas. Many numerical and analytical methods have been developed to obtain accurate analytic solutions for different problems such as inverse scattering transform method [1] analytical methods [2] the exp-function method [3] the Hirota s bilinear method [4] the Jacobi elliptic function expansion method [5] Backlund transformation method [6] Darboux transformation method [7] the multipleexp-functionmethod [8] the symmetry algebra method [9] the Wronskian technique [10] the exp( (ξ))-expansion method [11] analytical methods [12 14] and so on. Exact solutions Peregrine soliton and Kuznetsov Ma soliton of some non-linear models are discussed in [1516]. They also discussed the spatiotemporal localizations of some models in the non-local non-linear media [17 19]. Recently many researchers are engaged in finding shock waves solitons singular solitons and optical solitons of different models which arise in mathematical physics [20 31]. Some advanced applications of such new analytical methods for practical problems have been found in [32 40]. Studies of completely integrable equations with non-linear phenomena is one successful way to connect with to solitary wave fields and engineering concepts. The case of non-elastic phenomena in soliton theory is more complicated and there are a few models in the literature which describe the existence of these phenomena. The interactions between two or more soliton solutions of the integrable models for considering to obtain completely elastic and their amplitude velocity and wave shape which are remained still same after their non-linear collision. In fact some models exist in literature which give soliton solutions that are completely non-elastic and which depend on the conditions between the wave vectors and velocities. Wazwaz [32 34] investigated multiple soliton solutions for such non-elastic phenomena. Burgers equation and Sharma Tasso Olver equation are also such types of models studied in [35]. They obtained non-elastic

2 86 Page 2 of 9 Pramana J. Phys. (2017) 88:86 soliton fusion phenomena for the Burgers equation and non-elastic soliton fission and fusion for phenomena the Sharma Tasso Olver equations. In this article we investigate non-elastic multisoliton solutions for fusion phenomena of the clannish random walker s parabolic equation and completely elastic multisoliton solutions of the Vakhnenko Parkes equation. 2. Multisoliton solutions of two non-linear equations 2.1 The clannish random walker s parabolic equation and its fusion The clannish random walker s parabolic equation is a mathematical model of complex physical problem which arises in many scientific applications such as mathematical biology and physics. The soliton solutions and their nature especially in the case of their non-elastic or elastic interaction is an important subject to understand the mechanism and application of these physical models. In this subsection we consider a direct rational exponential approach to investigate new solutions and explain the nature of these solutions for non-elastic/ elastic interaction. The simplest non-linear clannish random walker s parabolic equation [36]isgivenby u t u xx + α(u 2 ) x αu x = 0 (1) which has both non-linear radiation and diffusion effects. This equation explains the motion of the two interacting populations which tend to be clannish and wish to live near their own kind. For one-soliton solution we first consider trial solution as u(x t) = r(ln f ) x f = a 0 + c 1 exp(k 1 x + w 1 t). (2) Inserting (2)into(1) and then arranging the coefficients of (exp(k 1 x + w 1 t)) i i = to zero yields a system of algebraic equations as follows: c 1 w 1 + c 1 k rαc 1k1 2 αc 1k 1 = 0 w 1 a 0 k1 2 a 0 αk 1 a 0 = 0. Solving this over determined system of algebraic equations for a 0 w 1 r we obtain a 0 = const w 1 = αk 1 + k1 2 and r = 1/α. Hence the solution is u(x t) = 1 α (ln f ) x f = a 0 + c 1 exp(k 1 {x + (α + k 1 )t}) (3) and the corresponding potential function is as follows: v(x t) = u x (x t). (4) To obtain two-soliton solutions we just set u(x t) = r(ln f ) x (5) + a 12 c 1 c 2 exp(ξ 1 + ξ 2 ) ξ 1 = k 1 x + w 1 t ξ 2 = k 2 x + w 2 t. Directly inserting (5) in(1) via commercial software Maple-13 and collecting the coefficients of different power of exponential to zero we attained a system of algebraic equations in terms of r k 1 k 2 w 1 w 2 c 1 c 2 and a 12. Solving this system of algebraic equations for r a 0 w 1 w 2 and a 12 with the software we obtain the following solutions of the unknown parameters. Now according to the cases in the method we have: Set 1: r = 1/α a 0 = const a 12 = 0 w 1 = αk 1 + k1 2 w 2 = αk 2 + k2 2then u(x t) = 1 α (ln f ) x (6) f = a 0 + c 1 exp(k 1 {x + (α + k 1 )t}) + c 2 exp(k 2 {x + (α + k 2 )t}) a 0 c 1 c 2 k 1 k 2 are arbitrary constants. The corresponding potential field reads as v(x t) = u x (x t). (7) In figure 2 the plots of the fusion phenomenon of the two solitons (shock waves) with the parameters for k 1 = 1 k 2 = 1.5 it is shown that two single soliton fusions become one (resonant) soliton after their interaction i.e. after a specific time t = 0. That is when the two populations living nearer to each other after interaction between them with different characteristics (e.g. different sizes different behaviour etc.) and tend to be a single clannish which is completely different from the previous nature (that is different sizes different behaviour etc.). From a careful analysis of (6) and (7) it is concluded that for all the ranges of the two arbitrary parameters k 1 k 2 only fusion occurs. Neither elastic scattering nor fission does exist. Set 2: r = 1/α a 0 = 0 a 12 = const w 1 = αk 1 + k k 1k 2 w 2 = αk 2 + 2k 1 k 2 + k2 2then u(x t) = 1 α (ln f ) x (8) f = c 1 exp(k 1 {x + (α + 2k 2 + k 1 )t}) + c 2 exp(k 2 {x + (α + 2k 1 + k 2 )t})

3 Pramana J. Phys. (2017) 88:86 Page 3 of a 12 c 1 c 2 exp(k 1 {x + (α + 2k 2 + k 1 )t}) exp(k 2 {x + (α + 2k 1 + k 2 )t}) and a 12 c 1 c 2 k 1 k 2 are arbitrary constants. The corresponding potential field reads as v = u x which is also non-elastic scattering and without fission but the fusion phenomenon of the two shock waves exists for all the ranges of arbitrary parameters k 1 k 2 like solutions (6) and(7). Set 3: r = 1/α a 0 = 0 a 12 = 0 w 1 = α(k 1 k 2 ) + k 2 1 k2 2 + w 2 w 2 = const then u(x t) = 1 α (ln f ) x (9) f = c 1 exp(k 1 x +{α(k 1 k 2 ) + k 2 1 k2 2 + w 2)t}) + c 2 exp(k 2 x + w 2 t) and a 12 c 1 c 2 k 1 k 2 are arbitrary constants. The corresponding potential field reads as v = u x. Equation (9) and its corresponding field give the non-elastic soliton-like (8) solution before and after the interaction. Solutions (8) and(9) represent the same characteristics of solution (6) with different dispersion relations. To obtain three-soliton solutions we set u(x t) = r(ln f ) x (10) + c 3 exp(ξ 3 ) + a 12 c 1 c 2 exp(ξ 1 + ξ 2 ) + a 23 c 2 c 3 exp(ξ 2 + ξ 3 ) + a 13 c 1 c 3 exp(ξ 1 + ξ 3 ) + a 123 c 1 c 2 c 3 exp(ξ 1 + ξ 2 + ξ 3 ) with ξ 1 = k 1 x + w 1 t ξ 2 = k 2 x + w 2 t ξ 3 = k 3 x + w 3 t and the corresponding potential field reads as v = u x. Directly inserting eq. (10) ineq.(1) via the commercial software Maple-13 and collecting the coefficients of different power of exponential to zero we obtain a system of algebraic equations and solving the system of algebraic equations via the software we obtain the following set of solutions of the unknown parameters: Set 1: r = 1/α a 0 = const a 12 = 0 a 23 = 0 a 13 = 0 a 123 = 0 w 1 = αk 1 + k 2 1 w 2 = αk 2 + k 2 2 w 3 = αk 3 + k 2 3 then u(x t) = 1 α (ln f ) x (11) f = a 0 + c 1 exp(k 1 {x + (α + k 1 )t}) + c 2 exp(k 2 {x + (α + k 2 )t}) + c 3 exp(k 3 {x + (α + k 3 )t}) Figure 1. (a) Profile of the single-soliton (a shock wave) solution eq. (3) of the clannish random walker parabolic equation and (b) profile of the potential field eq. (4) for k 1 = c 1 = a 0 = α = 1. and a 0 c 1 c 2 c 3 k 1 k 2 k 3 are arbitrary constants. The corresponding potential field reads as v x (x t) = u x (x t). (12) In figure 3 the plots of the fusion phenomenon of the three solitons (shock waves) with the parameters for k 1 = 1 k 2 = 1.5 it is clearly shown that three single soliton fusions present one (resonant) soliton after the interaction i.e. for a specific time t = 0. That is three populations with different characteristics (different sizes different behaviour etc.) live nearer to each other after their interactions tend to be a single clannish which is completely different from the previous nature (different sizes different behaviours etc.). Set 2: r = 1/α a 0 = 0 a 12 = const a 23 = 0 a 13 = 0 a 123 = 0 w 1 = αk 1 + k k 1k 2 w 2 = αk 2 + k k 1k 2 w 3 = αk 3 + k k 1k 2 then

4 86 Page 4 of 9 Pramana J. Phys. (2017) 88:86 Figure 2. (a) Profile of two (solitons) shock wave (fusion) solution eq. (6) of the clannish random walker parabolic equation; (b) potential field eq. (7)fork 1 = 1 k 2 = 1.5 c 1 = c 2 = a 0 = 1 α = 1; (c) contour plot of (a) and (d) contour plot of (b). u(x t) = 1 α (ln f ) x (13) u(x t) = γ 1 γ 2 γ 1 = c 1 k 1 exp(ξ 1 ) + c 2 k 2 exp(ξ 2 ) + c 3 k 3 exp(ξ 3 ) + a 12 c 1 c 2 (k 1 + k 2 ) exp(ξ 1 + ξ 2 ) γ 2 = c 1 exp(ξ 1 ) + c 2 exp(ξ 2 ) + c 3 exp(ξ 3 ) + a 12 c 1 c 2 exp(ξ 1 + ξ 2 ) f = c 1 exp(k 1 x +{αk 1 + k k 1k 2 }t) + c 2 exp(k 2 x +{αk 2 + k k 1k 2 }t) + c 3 exp(k 3 x +{αk 3 + k k 1k 2 }t) + a 12 c 1 c 2 exp((k 1 + k 2 )x +{αk 1 + αk 2 + k1 2 + k k 1k 2 }t) and a 0 c 1 c 2 c 3 k 1 k 2 k 3 are arbitrary constants and the corresponding potential field reads v = u x. Set 3: r = 1/α a 0 = 0 a 12 = 0 a 23 = const a 13 = 0 a 123 = 0 w 1 = αk 1 + k k 2k 3 w 2 = αk 2 + k k 2k 3 w 3 = αk 3 + k k 2k 3 then u(x t) = 1 α (ln f ) x (14) f = c 1 exp(k 1 x +{αk 1 + k k 2k 3 }t) + c 2 exp(k 2 x +{αk 2 + k k 2k 3 }t) + c 3 exp(k 3 x +{αk 3 + k k 2k 3 }t) + a 23 c 2 c 3 exp((k 2 + k 3 )x +{αk 2 + αk 3 + k2 2 + k k 2k 3 }t) and a 0 c 1 c 2 c 3 k 1 k 2 k 3 are arbitrary constants and the corresponding potential field reads as v = u x. Set 4: r = 1/α a 0 = 0 a 12 = 0 a 23 = 0 a 13 = const a 123 = 0 w 1 = αk 1 + k k 1k 3 w 2 = αk 2 + k k 1k 3 w 3 = αk 3 + k k 1k 3 then u(x t) = 1 α (ln f ) x (15)

5 Pramana J. Phys. (2017) 88:86 Page 5 of 9 86 Figure 3. (a) Profile of three solitons (fusion) (11) of the clannish random walker parabolic equation; (b) the corresponding potential field (12) fork 1 = 1 k 2 = 1.5 k 3 = 2 c 1 = c 2 = c 3 = a 0 = 1 α = 1; (c) contour plot of (a) and (d) contour plot of (b). f = c 1 exp(k 1 x +{αk 1 + k k 1k 3 }t) + c 2 exp(k 2 x +{αk 2 + k k 1k 3 }t) + c 3 exp(k 3 x +{αk 3 + k k 1k 3 }t) + a 13 c 1 c 3 exp((k 1 + k 3 )x +{αk 1 + αk 3 + k1 2 + k k 1k 3 }t) and a 0 c 1 c 2 c 3 k 1 k 2 k 3 are arbitrary constants and the corresponding potential field reads as v = u x. Set 5: r = 1/α a 0 = 0 a 12 = 0 a 23 = 0 a 13 = 0 a 123 = const w 1 = αk 1 + k1 2 + k 1k 2 + k 1 k 3 + k 2 k 3 w 2 = αk 2 + k2 2 + k 1k 2 + k 1 k 3 + k 2 k 3 w 3 = αk 3 + k3 2 + k 1 k 2 + k 1 k 3 + k 2 k 3 then u(x t) = 1 α (ln f ) x (16) f = c 1 exp(ξ 1 ) + c 2 exp(ξ 2 ) + c 3 exp(ξ 3 ) + a 123 c 1 c 2 c 3 exp(ξ 1 + ξ 2 + ξ 3 ) in which ξ 1 = k 1 x + (αk 1 + k1 2 + k 1k 2 + k 1 k 3 + k 2 k 3 )t ξ 2 = k 2 x + (αk 2 + k2 2 + k 1k 2 + k 1 k 3 + k 2 k 3 )t ξ 3 = k 3 x + (αk 3 + k3 2 + k 1k 2 + k 1 k 3 + k 2 k 3 )t and a 0 c 1 c 2 c 3 k 1 k 2 k 3 are arbitrary constants and the corresponding potential field reads as v = u x. Set 6: r = 1/α a 0 = 0 a 12 = 0 a 23 = 0 a 13 = 0 a 123 = 0 w 1 = const w 2 = w 1 +k2 2 k2 1 +α(k 2 k 1 ) w 3 = w 1 + k3 2 k2 1 + α(k 3 k 1 )then u(x t) = 1 α (ln f ) x (17) f = c 1 exp(k 1 x + w 1 t) + c 2 exp(k 2 x + (w 1 + k2 2 k2 1 + α(k 2 k 1 ))t) + c 3 exp(k 3 x + (w 1 + k3 2 k2 1 + α(k 3 k 1 ))t) and a 0 c 1 c 2 c 3 k 1 k 2 k 3 are arbitrary constants. The corresponding potential field reads as v = u x. Profile

6 86 Page 6 of 9 Pramana J. Phys. (2017) 88:86 ocean with non-linear surface and internal wave in the case of seismic sea wave or other non-linear situations. If we consider a transformation φ(x t) = 1 + x u t dx so that φ x = u t and if v x = u v and its derivatives vanish as x Vakhnenko and Parkes [38] reduces to v xxt + v x v t + v x = 0. (19) Then (19) coincides with (18). For single soliton solutions we first consider trial solution as v(x t) = r(ln f ) x f = a 0 + c 1 exp(k 1 x + w 1 t). (20) Inserting (20) into(19) and then arranging the coefficients of (exp(k 1 x+w 1 t)) i i = is zero yields a system of algebraic equations about a 0 c 1 w 1 and k 1 as follows: 2a 0 c 1 + rw 1 c 1 k 1 a 0 4w 1 c 1 k 1 a 0 = 0 w 1 k 1 c1 2 + c2 1 = 0 w 1k 1 a0 2 + a2 0 = 0. Figure 4. (a) Profile of one-soliton (peakon) solution (21) of the Vakhnenko Parkes equation for k 1 = 1 c 1 = a 0 = 1 and (b) contour plot of (a). of the solutions (13) (17) gives similar phenomena like (11). Solutions (13) (17) have the same characteristics like solution (11) with different dispersion relations. 2.2 The Vakhnenko Parkes equation In this section we recall a direct rational exponential approach to explain the completely elastic and nonelastic interaction of the simplest non-linear Vakhnenko Parkes equation [37 40]. Recently a remarkable and important discovery has been made by Vakhnenko and Parkes [37] whohave confirmedanintegrable equations as follows: uu xxt u x u xt + u 2 u t = 0. (18) The equation presents a model for weakly non-linear surface and internal waves in a rotating ocean. The solutions of eq. (18) will describe the sea waves in a rotating By solving we obtain r = 6 w 1 = 1/k 1. Thus onesoliton solution is ( v(x t) = 6(ln f ) x f = a 0 + c 1 exp k 1 x t ) k 1 and u(x t) = 6c 1k1 2 exp(k 1x (t/k 1 )) a 0 +c 1 exp(k 1 x (t/k 1 )) 6c2 1 k2 1 exp(2k 1x (2t/k 1 )) {a 0 + c 1 exp(k 1 x (t/k 1 ))} 2. (21) To obtain two-soliton solutions we set v(x t) = r(ln f ) x (22) + a 12 c 1 c 2 exp(ξ 1 + ξ 2 ) with ξ 1 = k 1 x + w 1 t ξ 2 = k 2 x + w 2 t. Inserting (22)in(19) and applying similar process as above we get u(x t) = (6(ln f ) x ) x (23)

7 Pramana J. Phys. (2017) 88:86 Page 7 of 9 86 Figure 5. (a) Profile of interaction of two-soliton solution (23) of the Vakhnenko Parkes equation for k 1 = 1 k 2 = 2 c 1 = c 2 = a 0 = 1 (b) sameas(a) but different phase and (c) contour plot of (a). + (k 2 k 1 ) 2 (k1 2 k 1k 2 + k2 2) (k 2 + k 1 ) 2 (k1 2 + k 1k 2 + k2 2)c 1c 2 exp(ξ 1 + ξ 2 ) with ξ 1 = k 1 x t k 1 ξ 2 = k 2 x t k 2. Figure 5 shows that the two waves with different amplitudes and velocities collide at t = 0. The waves keeps their shapes and sizes the same before and after collision except for shifting the phase for any parametric values in the solution. Thus we conclude that during the interaction of waves in the presence of weakly nonlinear surface and internal waves in a rotating ocean the two solitons keep their initial shapes and velocities except for shifting the phase. To obtain three-soliton solutions we set v(x t) = r(ln f ) x (24) + c 3 exp(ξ 3 ) + a 12 c 1 c 2 exp(ξ 1 + ξ 2 ) + a 23 c 2 c 3 exp(ξ 2 + ξ 3 ) + a 13 c 1 c 3 exp(ξ 1 + ξ 3 ) + a 123 c 1 c 2 c 3 exp(ξ 1 + ξ 2 + ξ 3 ) with ξ 1 = k 1 x + w 1 t ξ 2 = k 2 x + w 2 t ξ 3 = k 3 x + w 3 t. Inserting (24) in(19) and applying similar process we get u(x t) = (6(ln f ) x ) x (25) ξ 1 = k 1 x t k 1 ξ 2 = k 2 x t k 2 a 12 = (k 2 k 1 ) 2 (k 2 1 k 1k 2 + k 2 2 ) (k 2 + k 1 ) 2 (k k 1k 2 + k 2 2 ) ξ 3 = k 3 x t k 3

8 86 Page 8 of 9 Pramana J. Phys. (2017) 88:86 Remark: All the solutions available in this paper have been checked with the help of Maple-13 and we observe that they satisfy the corresponding original equation. 3. Soliton propagation and collisions 3.1 Collisions of multisoliton solutions of the clannish random walker s parabolic equation It can be seen that the amplitudes and velocities of the solitons depend on both k and α α is a parameter related to non-linearity. Figure 1 shows the single-soliton solutions (3) and its potential field (4). Overtaking collisions between the two solitons for two different amplitude modes are illustrated in figure 2 of the solution (6) and its potential field (7). After the collision the two solitons fuss to one soliton with similar shape but amplitude increases after the phase shifts. We also provided contour plot to visualize the real density of the waves and their directions. It is concluded that the overtaking collisions between the two solitons for two modes are completely non-elastic. Similarly overtaking collisions among the three solitons for different amplitudes are illustrated in figure 3 of solution (11)and(12) and we see similar non-elastic fusion phenomena. 3.2 Collisions of multisoliton solutions of the Vakhnenko and Parkes equation Figure 6. (a) Profile of interaction of three-soliton solution (25) of the Vakhnenko Parkes equation for k 1 = 1 k 2 = 2 k 3 = 3.5 c 1 = c 2 = c 3 = a 0 = 1 and (b) contour plot of (a). a 13 = (k 3 k 1 ) 2 (k 2 1 k 1k 3 + k 2 3 ) (k 3 + k 1 ) 2 (k k 1k 3 + k 2 3 ) a 23 = (k 3 k 2 ) 2 (k2 2 k 2k 3 + k3 2) (k 3 + k 2 ) 2 (k2 2 + k 2k 3 + k3 2) and a 123 = a 12 a 13 a 23. Figure 6 shows that three waves with different amplitudes and velocities collide at t = 0. The shapes and sizes of the waves remain the same before and after collision except at the phase shift for any parametric values involved in the solution. Thus we conclude that during the interaction of waves in the presence of weakly nonlinear surface and internal waves in a rotating ocean the three solitons keep their initial shapes and velocities except for the phase shifts. It can be seen that the amplitudes and velocities of the solitons depend on k only. Figure 4 shows the one soliton (peakon) via solution (21). Overtaking collisions between the two solitons for different amplitude modes are illustrated in figure 5 of the solution (23). After the collision the two solitons keep their initial shapes and velocities except for the phase shifts. We also provided contour plot to visualize the real density of the waves and their directions. We observe that before and after collision the amplitude shape and density of the same wave remain the same with same direction. We can conclude that the overtaking collisions between the two solitons for two modes are completely elastic. Similarly overtaking collisions among three solitons for different amplitudes are illustrated in figure 6 of the solution (25) and we see similar elastic collision. 4. Conclusion One of the main results of this paper presented the nature of the soliton solutions of the clannish random walker s parabolic equation and the Vakhnenko Parkes equation. We obtain new multisoliton solutions in numerical form and explicitly express the exact three-soliton solutions

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