1-Multisoliton and other invariant solutions of combined KdV - nKdV equation by using symmetry approach
11-MULTISOLITON AND OTHER INVARIANT SOLUTIONS OFCOMBINED KDV-NKDV EQUATION BY USING LIESYMMETRY APPROACH
SACHIN KUMAR AND DHARMENDRA KUMAR
Abstract.
Lie symmetry method is applied to investigate symmetries of thecombined KdV-nKdV equation, that is a new integrable equation by combin-ing the KdV equation and negative order KdV equation. Symmetries whichare obtained in this article, are further helpful for reducing the combined KdV-nKdV equation into ordinary differential equation. Moreover, a set of eightinvariant solutions for combined KdV-nKdV equation is obtained by usingproposed method. Out of the eight solutions so obtained in which two solu-tions generate progressive wave solutions, five are singular solutions and onemultisoliton solutions which is in terms of WeierstrassZeta function. Introduction
Korteweg and Vries derived KdV equation [6] to model Russell’s phenomenonof solitons like shallow water waves with small but finite amplitudes [26]. Soli-tons are localized waves that propagate without change of it’s shape and velocityproperties and stable against mutual collision [7, 22]. It has also been used to de-scribe a number of important physical phenomena such as magneto hydrodynamicswaves in warm plasma, acoustic waves in an in-harmonic crystal and ion-acousticwaves [2, 8, 17]. A special class of analytical solutions of KdV equation, the so-called traveling waves, for nonlinear evolution equations (NEEs) is of fundamentalimportance because a lot of mathematical-physical models are often described bysuch a wave phenomena. Thus, the investigation of traveling wave solutions is be-coming more and more attractive in nonlinear science nowadays. However, not allequations posed in these fields are solvable. As a result, many new techniques havebeen successfully developed by a diverse group of mathematicians and physicists,such as Rational function method [31, 32], B¨acklund transformation method [14],Hirotas bilinear method [5, 13], Lie symmetry method [1, 9–11, 18, 30], Jacobi el-liptic function method [12], Sine-cosine function method [27], Tanh-coth functionmethod [15], Weierstrass function method [20], Homogeneous balance method [28],Exp-function method [4], ( G (cid:48) /G )-expansion method [21], etc. But, it is extremelydifficult and time consuming to solve nonlinear problems with the well-known tra-ditional methods. This work investigates the combined KdV-nKdV equation u xt + 6 u x u xx + u xxxx + u xxxt + 4 u x u xt + 2 u xx u t = 0 , (1)where u = u ( x, t ). We apply Lie symmetry analysis on combined KdV-nKdV equa-tion, first constructed by Wazwaz [25] using recursion operator [19]. In addition,the combined KdV-nKdV equation (1) possesses the Painlev´e property for complete Key words and phrases.
Combined KdV-nKdV equation; Lie symmetry method; Invariantsolutions; Similarity reduction. a r X i v : . [ m a t h - ph ] M a y ntegrability [3]. In this paper, Lie point symmetry generators of the combinedKdV-nKdV equation were derived. Similarity reductions and number of explicitinvariant solutions for the equation using Lie symmetry method were obtained.All the new invariant solutions of combined KdV-nKdV were analyzed graphically.Also, 1-multisoliton solution obtained in terms of WeierstrassZeta function whichappear in classical mechanics such as, motion in cubic and quartic potentials, de-scription of the movement of a spherical pendulum, and in construction of minimalsurfaces [29]. Some of the outcomes are interesting in physical sciences and arebeautiful in mathematical sciences.The organization of the paper is as follows. In Sec. 2, we discuss the methodologyof Lie symmetry analysis of the general case. In Sec. 3, we obtain infinitesimalgenerators and the Lie point symmetries of the Eq. (1). In Sec. 4, symmetryreductions and exact group invariant solutions for the combined KdV-nKdV eqationwere obtained. In Sec. 5, we discussed all the invariant solutions graphically byFigures 1, 2, 3 and 4. Finally, concluding remarks are summarized in Section 6.2. Method of Lie symmetries
Let us consider a system of partial differential equations as follows:Λ ν ( x, u ( n ) ) = 0 , ν = 1 , , ..., l, (2)where u = ( u , u , ..., u q ) , x = ( x , x , ...x p ) , u ( n ) denotes all the derivatives of u ofall orders from 0 to n . The one-parameter Lie group of infinitesimal transformationsfor Eq. (2) is given by˜ x i = x i + (cid:15) ξ i ( x, u ) + O ( (cid:15) ); i = 1 , , ...p, ˜ u j = u j + (cid:15) φ j ( x, u ) + O ( (cid:15) ); j = 1 , , ...q, where (cid:15) is the group parameter, and the Lie algebra of Eq. (1) is spanned by vectorfield of the form V = p (cid:88) i =1 ξ i ( x, u ) ∂∂x i + q (cid:88) j =1 η j ( x, u ) ∂∂u j (3)A symmetry of a partial differential equation is a transformation which keeps thesolution invariant in the transformed space. The system of nonlinear PDEs leadsto the following invariance condition under the infinitesimal transformations P r ( n ) V [Λ ν ( x, u ( n ) )] = 0 , ν = 1 , , ...l along with Λ( x, u ( n ) ) = 0In the above condition, P r ( n ) is termed as n th -order prolongation [16] of the infin-itesimal generator V which is given by P r ( n ) V = V + q (cid:88) j =1 (cid:88) J η Jj ( x, u ( n ) ) ∂∂u jJ (4)the second summation being over all (unordered) multi-indices J = ( i , ..., i k ),1 ≤ i k ≤ p, ≤ k ≤ n . The coefficient functions η Jj of P r ( n ) V are given by thefollowing expression η Jj ( x, u ( n ) ) = D J (cid:18) η j − p (cid:88) i =1 ξ i u ji (cid:19) + p (cid:88) i =1 ξ i u jJ,i (5) here u ji = ∂u j ∂x i , u jJ,i = ∂u jJ ∂x i and D J denotes total derivative.3. Lie symmetry analysis for the combined KdV-nKdV equation
In this section, authors explained briefly all the steps of the STM method to keepthis work self-confined. The Lie symmetries for the Eq. (1) have generated andthen its similarity solutions are found. Therefore, one can consider the followingone-parameter ( (cid:15) ) Lie group of infinitesimal transformations˜ x = x + (cid:15) ξ ( x, t, u ) + O ( (cid:15) ) , ˜ t = t + (cid:15) τ ( x, t, u ) + O ( (cid:15) ) , ˜ u = u + (cid:15) η ( x, t, u ) + O ( (cid:15) ) , (6)where ξ, τ and η are infinitesimals for the variables x, t and u respectively, and u ( x, t ) is the solution of Eq. (1). Therefore, the associated vector field is V = ξ ( x, t, u ) ∂∂x + τ ( x, t, u ) ∂∂t + η ( x, t, u ) ∂∂u . (7)Lie symmetry of Eq. (1) will be generated by Eq. (7). Use fourth prolongation P r (4) V gives rise to the symmetry condition for Eq. (1) as follows: η xt + 6 η x u xx + 6 η xx u x + η xxxx + η xxxt + 4 η x u xt +4 η xt u x + 2 η t u xx + 2 η xx u t = 0 , (8)where η x , η t , η xt , η xx , η xxxx , and η xxxt are the coefficient of P r (4) V , values are givenin many references [1, 16]. Incorporating all the expressions into Eq. (8), and thenequating the various differential coefficients of u to zero, we derive following systemof Eight determining equation ξ u = ξ xx = 0 , ξ x + ξ t = τ t , τ x = τ u = 0 ,ξ x + 2 η x = ξ x + η u = 0 ,ξ x + 12 τ t = η t . (9)Solving the above system of equations, we obtain following infinitesimals for (1)using software Maple , ξ = ( x − t ) a + a + f ( t ) , τ = f ( t ) , η = ( t − x − u ) a + a + 12 f ( t ) , (10)where a , a and a are arbitrary constants whereas f ( t ) is an arbitrary function.The symmetries under which Eq. (1) is invariant can be spanned by the followingfour infinitesimal generators if we assume f ( t ) = c , a constant Then all of theinfinitesimal generators of Eq. (1) can be expressed as V = a V + a V + a V + cV . here V = 2( x − t ) ∂∂x + ( t − x − u ) ∂∂u ,V = ∂∂x ,V = ∂∂u ,V = ∂∂x + ∂∂t + ∂∂u . (11) Table 1.
The commutator table of the vector fields (11) * V V V V V − V + V V V V V − V V − V V − V i, j )thentry in Table 1 is the Lie bracket [ V i , V j ] = V i · V j − V j · V i . Table 1 is skew -symmetric with zero diagonal elements. Table 1 shows that the generators V , V , V and V are linearly independent. Thus, to obtain the similarity solutions of Eq. (1),the corresponding associated Lagrange system is dxξ ( x, t, u ) = dtτ ( x, t, u ) = duη ( x, t, u ) . (12)4. Invariant solutions of the combined KdV-nKdV equation
To proceed further, selection of f ( t ) and by assigning the particular values to a i ’s (1 ≤ i ≤ f ( t ): Case 1:
For f ( t ) = a t + b t + c, a (cid:54) = 0 , b (cid:54) = 0 , c (cid:54) = 0, then Eqs. (10) and (12) gives dx ( x − t ) a + a + f ( t ) = dtf ( t ) = du ( t − x − u ) a + a + f ( t ) . (13)The similarity form suggested by Eq. (13) is given by u = α + 14 (3 t − x ) + e − a β tan − ( βT ) U ( X ) , (14)with similarity variable X = ( x − t + A ) e − a β tan − ( βT ) , (15)where α = a + 4 a a , β = 1 √ ac − b , A = a a , T = 2 at + b. (16) a) Progressive wave shown by Eq. (19) (b)
Singularity on 1 + x − t = 0 in Eq. (20) Figure 1.
Invariant solution profiles for Eq. (19) and Eq. (20)
Inserting the value of u from Eq. (14) into Eq. (1), we get the following fourthorder ordinary differential equation X U
XXXX + 4 U XXX + 6
X U X U XX + 2 U U XX + 8 U X = 0 , (17)where X is given by Eq. (15) and U X = dUdX , U XX = d UdX , etc.Any how, we could not find the general solution of Eq. (17) still two particularsolutions are found as below U ( X ) = c and U ( X ) = c X , (18)where c and c are arbitrary constants. Thus, from Eqs. (14) and (18), we gettwo invariant solutions of Eq. (1) given below u ( x, t ) = α + 14 (3 t − x ) + c e − a β tan − ( βT ) , (19) u ( x, t ) = α + 14 (3 t − x ) + c A + x − t , (20)where α , β, T and A are given by Eq. (16). Case 2:
For f ( t ) = c, , c (cid:54) = 0; a = 0 , a (cid:54) = 0 , a (cid:54) = 0, Eq. (12) are of the form dxc + a = dtc = du c + a . (21)The similarity solution to Eq. (21) can be written as u = (cid:18)
12 + a c (cid:19) t + U ( X ) , (22)where X = x − (1 + a c ) t is a similarity variable.Inserting the value of u from Eq. (22) in Eq. (1), we get the fourth order ordinarydifferential equation in Ua U XXXX + ( a − a + 6 a U X ) U XX = 0 . (23)Assume a = 2 a . Without loss of generality, we can assume a (cid:54) = 0, then Eq. (34)reduces to U XXXX + 6 U X U XX = 0 . (24) he general solution of Eq. (24) in terms of WeiestrassZeta function as U ( X ) = c + ( − W eierstrassZeta [ α ( x, t ) , α ] , (25)where α ( x, t ) = ( − ) (cid:18) x − (1 + a c ) t + c (cid:19) , α = { − c , c } and c , c and c are arbitrary constants. From Eq. (25) with Eq. (22), we have another invariantsolution of Eq. (1) u ( x, t ) = c + (cid:18)
12 + a c (cid:19) t + ( − W eierstrassZeta [ α ( x, t ) , α ] . (26) Case 2A: f ( t ) = c ; a = 0 , a (cid:54) = 0 , a (cid:54) = 0, Eq. (13) becomes dx ( x − t ) a + c = dtc = du ( t − x − u ) a + a + c . (27)In this case, we get u = a a + 14 (3 t − x ) + e − a c t U ( X ) , (28)where X = ( x − t ) e − a c t . Substituting the value of u in Eq. (1), again we obtainthe same fourth order ordinary differential equation (17). Some particular solutionsare given below U ( X ) = c , U ( X ) = c X , (29)where c and c are arbitrary constants. Therefore, using Eq. (29) in Eq. (28), weobtain the following two exact solutions for Eq. (1) u ( x, t ) = a a + 14 (3 t − x ) + c e − a c t , (30) u ( x, t ) = a a + 14 (3 t − x ) + c x − t . (31) Case 3:
For f ( t ) = 0; a (cid:54) = 0 , a (cid:54) = 0 , a (cid:54) = 0, Eq. (13) modified as dx ( x − t ) a + a = dt du ( t − x − u ) a + a . (32)The group invariant solution is given as u = a x ( x − t ) − a x + 4 U ( T )4 a ( t − x ) − a , (33)with T = t . Substituting the value of u from Eq. (33) into Eq. (1), we get thefollowing reduced ordinary differential equation (cid:2) a (cid:0) t − (cid:1) − a ( a t − a t + 2 U ( t )) − a ( a + 4 a ) (cid:3) (cid:2) a t − a + 2 a − U (cid:48) ( t ) (cid:3) = 0 . (34) Hence, from Eq.(34) we found two values of U given as U ( T ) = 3 a (cid:0) t − (cid:1) − a − a ) a t − a ( a + 4 a )4 a , (35) U ( T ) = 14 (3 a t − a t + 4 a t + 4 c ) , (36) a) (b)(c) (d)Figure 2. Evolution profiles of 1-Multisoliton solution for Eq. (26). where c is an arbitrary constant of integration. Here, Eq. (35) gives same solutionas Eq. (20) with c = . Also, using Eq. (36) in Eq. (33) gives new invariantsolution of Eq. (1) as u ( x, t ) = 4 ( a ( t − x ) + c ) + a (cid:0) t − tx + x (cid:1) − a t a ( t − x ) − a , (37)where α and A are given by Eq. (16). Case 3A:
For f ( t ) = 0; a = 0 , a (cid:54) = 0 , a (cid:54) = 0, from Eq. (32), we have dx ( x − t ) a = dt du ( t − x − u ) a + a . (38)Therefore, the similarity transformation method gives u = 4 xa − x ( x − t ) a − U ( T )4( a + ( x − t ) a ) , (39)where T = t and U ( T ) is the similarity function. Substituting the value of u in Eq.(1) we get reduced ordinary differential equation given as (cid:2) a (cid:0) a (cid:0) t − (cid:1) − U ( t ) (cid:1) + 2 a a t − a (cid:3) [3 a t + a − U (cid:48) ( t )] = 0 . (40)Again it gives two values of U as U ( T ) = 3 a ( t −
1) + 2 a a t − a a , (41) U ( T ) = 14 (2 a t + 3 a t + 4 c ) , (42) a) Progressive wave shown by solution Eq.(30) (b)
Singularity shown by solution Eq. (31) (c)
Singularity on 1 + x − t = 0 in Eq. (37) (d) Singularity in the solution Eq. (43)
Figure 3.
Evolution profiles of various invariant solutions. where c is constant of integration. Here, Eq. (41) gives same solution as Eq. (20).Also, using (42) with (39) gives another new invariant solution of (1) as u ( x, t ) = 2 a (2 x − t ) − a (cid:0) t − tx + x (cid:1) − c a ( x − t ) + a ) . (43) Case 4:
For f ( t ) = t ; a (cid:54) = 0 , a (cid:54) = 0 , a (cid:54) = 0, from Eq. (32) we get similarity formsuggested by Eq.(13) is given by u = α + 14 (3 t − x ) + e a t U ( X ) , (44)with similarity variable X = ( x − t + A ) e a t and A is given by (16). In this case,we get same ordinary differential equation as Eq. (17), one particular solution is asfollows U ( X ) = c , (45)where c is arbitrary constant. Thus, from Eqs. (44) and (45) we get another newinvariant solution of Eq. (1) given below u ( x, t ) = α + 14 (3 t − x ) + c e a t . (46) . Discussion
The results of the combined KdV-nKdV equation presented in this paper havericher physical structure then earliar outcomes in the literature [25]. The recordedresults are significant in the context of nonlinear dynamics, physical science, mathe-matical physics etc. The invariant solutions obtained can illustrate various dynamicbehaviour due to existence of arbitrary constants. The nonlinear behaviour of theresults are analyzed in the following manner:
Figure 1 : Fig 1(a) shows progressive wave soluion in Eq. (19) for a = 1 . , a =0 . , a = 1 . , a = 0 . , b = 0 . c = 0 .
9. Fig 1(b) shows presence of singular-ity in the plane 1 + x − t = 0 in Eq. (20) with values a = 2 . , a = 2 . , a =3 . c = 0 . Figure 2 : The evolution profiles of 1-Multisoliton solution is given by Eq. (26)as shown in this Figure. We have recorded the physical nature with variation inparameters. Fig 2(a) For a = 1 . , a = 1 . , c = 9 . , c = 1 . c = 0 . a = 5 , c =1 , a = 1 , c = 1 and c = 1 only few solitons are shown; Fig 2(c) Front orthographicprojection is shown for a = 1 . , a = 1 . , c = 9 . , c = 5 . c = 4 . a = 1 . , a =1 . , c = 9 . , c = 1 . c = 0 . Figure 3 : Fig 3(a) For a = 0 . , a = 0 . c = 1, Eq. (30) exhibits progressivewave; Fig 3(b), shows singularity in planes for Eqs. (31) for a = 0 . , a = 2 , a =0 . c = c = c = 1; Fig 3(c, d) shows singularity wave profile for Eq. (37)and Eq. (43) which explain the transition of nonlinear behaviour in the form ofopposite rotatory folded sheets. Figure 4 : u ( x, t ) exhibits singularity near t = 0 but asymptotic structures is ob-served near t = 0 for parameters a = 1 , a = 1 , a = 1 and c = 1.6. Conclusion
In this paper, the similarity reductions and invariant solutions for the combinedKdV-nKdV are presented. This paper obtained the 1-multisoliton and other invari-ant solution of the equation. The method that was used to obtain the exact groupinvariant solutions is the Lie symmetry analysis approach. All the solutions aredifferent from earlier work which have been obtained by Wazwaz [25]. Eventually,the stucture of combined KdV-nKdV equation is an non-trivial one, which can be
Figure 4.
Asymptotic structures at t = 0 for Eq. (46). learly seen from the graphically results of invariant solutions. The Lie symmetryanalysis method extracts the new forms of analytic solutions which are of physicalimportance such as condensed matter physics and plasma physics. Acknowledgment
The second author sincerely and genuinely thanks Department of Mathematics,SGTB Khalsa College, University of Delhi for financial support.
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Z. Natur-forsch. (4)a (2015) 263–268. Department of Mathematics, University of Delhi, Delhi -110007, India
E-mail address : [email protected] Department of Mathematics, SGTB Khalsa College, University of Delhi-110007, In-dia
E-mail address : [email protected]@sgtbkhalsa.du.ac.in