ISSN 139-5113 Nonlinear Analysis: Modelling Control, 017, Vol., No. 3, 334 346 https://doi.org/10.15388/na.017.3.4 Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions for the mkdv systems Gangwei Wang a,b, Abdul H. Kara c, Jose Vega-Guzman d, Anjan Biswas e,f a School of Mathematics Statistics, Hebei University of Economics Business, Shijiazhuang, 050061, China pukai111@163.com b School of Mathematics Statistics, Beijing Institute of Technology, Beijing 100081, China c School of Mathematics, Centre for Differential Equations Continuum Mechanics Applications, University of the Witwatersr, Wits 050, Johannesburg, South Africa d Department of Mathematics, Howard University, Washington, DC-0059, USA e Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-77, USA f Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah-1589, Saudi Arabia Received: January 16, 016 / Revised: February 13, 017 / Published online: March 17, 017 Abstract. We study the symmetry groups, conservation laws, solitons, singular solitary waves of some versions of systems of the modified KdV equations. Keywords: systems of modified KdV equation, group analysis, conservation laws, exact solutions. 1 Introduction One of the better known studied partial differential equations PDEs is the Korteweg de Vries KdV equation being, inter alia, a model for shallow water waves. Other versions of the KdV equation are the modified KdV mkdv equation, systems of KdV, the complex KdV equations, amongst others. The literature abounds with these studies, see, e.g., [11 15] [1]. Soliton studies of the equation other equations may be found in [3] [8]. A number of authors have dealt with a 4 4 matrix spectral problem with three potentials. In some cases, a new hierarchy of nonlinear evolution c Vilnius University, 017
Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions 335 equations was derived see [3] [8]. Some typical equations in the hierarchy are a new generalized Hirota Satsuma coupled KdV system. We study the invariances, conservation laws, solitons, singular solitary waves in the study below. Two distinct versions of systems of the modified KdV equations are considered see [6,7] references therein. The equation I The first version of the system of modified KdV equations we consider takes the form F 1 = u t 1 u xxx + 3uu x 3v x w 3w x v = 0, F = v t + v xxx 3v x u = 0, F 3 = w t + w xxx 3w x u = 0. 1.1 Invariance conservation Here, we analyse system 1 using its invariance properties [, 4, 5, 9, 10] conservation laws []. The formal Lagrangian multiplier approaches to the latter are used for the construction of conservation laws..1.1 Adjoint system conditions for self-adjointness Considering system 1, we give the following formal Lagrangian: L = ū u t 1 u xxx + 3uu x 3v x w 3w x v + vv t + v xxx 3v x u + ww t + w xxx 3w x u, where ū, v, w are three new dependent variables. The adjoint system is given by here F 1 = δl δu = 0, F = δl δv = 0, F 3 = δl δw = 0, δl δu = u D t D x + D x D x D x D x D x, u t u x u xx u xxx δl δv = v D t D x + D x D x D x D x D x, v t v x v xx v xxx δl δw = w D t D x + D x D x D x D x D x. w t w x w xx w xxx On the basis of Eq., one can get the adjoint system F 1 = ū t 3uū x 3 vv x 3 ww x + 1 ūxxx = 0, F = v t + 3wū x + 3u x v + 3u v x v xxx = 0, F 3 = w t + 3vū x + 3u x w + 3u w x w xxx = 0. 3 Nonlinear Anal. Model. Control, 3:334 346
336 G. Wang et al. System 1 is said to be nonlinearly self-adjoint on the condition of the adjoint system 3 if it satisfies the following formulae: F 1 ū=p x,t,u,v,w, v=qx,t,u,v,w, w=rx,t,u,v,w = λ 11 F 1 + λ 1 F + λ 13 F 3 = 0, F ū=p x,t,u,v,w, v=qx,t,u,v,w, w=rx,t,u,v,w = λ 1 F 1 + λ F + λ 3 F 3 = 0, F 3 ū=p x,t,u,v,w, v=qx,t,u,v,w, w=rx,t,u,v,w = λ 31 F 1 + λ 3 F + λ 33 F 3 = 0 with ū = P x, t, u, v, w, v = Qx, t, u, v, w, w = Rx, t, u, v, w not equal to zero, the coefficients λ ij i, j = 1,, 3 are to be fixed later. From the coefficients of u t, v t, w t one can get λ 11 = P u, λ 1 = P v, λ 13 = P w, λ 1 = Q u, λ = Q v, λ 3 = Q w, λ 31 = R u, λ 3 = R v, λ 33 = R w, then putting these into the system, one can derive P = c 1 tu 1 3 c 1x + c u + c 3, Q = c 1 t + c w, R = c 1 t + c v. 4 We have the following theorem. Theorem 1. System 1 is said to be nonlinear self-adjoint if P, Q, R are given by 4..1. Conservation laws Invariance analysis It can be shown by a well established procedure that 1 admits the algebra of Lie point symmetries generated by X 1 = x, X = t, X 3 = 3t t + x x u u 4v v, X 4 = v v w w. These may be associated with conservation laws of the system. In fact, in the case of a variational system with Noether symmetries, the Noether symmetries are associated with conservation laws via Noether s theorem. This procedure is not employed here. A conservation law of 1 is the divergence equation D t Φ t + D x Φ x = 0 satisfied along the solutions of the equations in which T = Φ x, Φ t is the conserved vector/flow. In order to derive the conservation laws, we employ the following theorem. Theorem. See [5]. For any Lie point, Lie Bäcklund nonlocal symmetry X = ξ i x i + η α u α provides a conservation law for a system its adjoint system. Then the https://www.mii.vu.lt/na
Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions 337 conservation vector T is defined by the expression where W α = η α ξ j u α j. Φ i = ξ i L + W α [ u α i [ + D j W α u α ij D j u α ij D k u α ijk + D j D k + u α ijk Since system 1 is nonlinearly self-adjoint, we can get the conserved vectors using Theorem. Also, via the symmetry operator the formal Lagragian, they become Φ t = ξ 1 L + W 1 + W u t v t Φ x = ξ L + W 1 + Dx u x + W 3 + Dx w x w xxx + D x W D x + Dx W 1 u xxx v xxx + W 3 w t + W, ], + D x u xxx v x + D x W 1 D x u xxx + D x W 3 D x w xxx + Dx W + Dx v xxx v xxx ] W 3 w xxx where W 1 = η 1 ξ 1 u t ξ u x, W = η ξ 1 v t ξ v x, W 3 = η 3 ξ 1 w t ξ w x. For system 1, we have the following set of conserved vectors:, T 1 = T = [ 1 3u wv t + v 6w + w t u xx w x v xx + v x w xx, u + 1 ] wv x vw x, [ 1 xu + tu 3 u x 6 + 1 4 tu x + tv x w x + 1 6 xu xx 1 tuu xx 5 twv xx + vxw tw xx, 1 6 xu + 3tu 6tvw ], T 3 = [u 3 + u x 4 + v xw x 1 uu xx wv xx vw xx, 1 u vw ]. One can show that these correspond, respectively, with the multipliers Q 1 = 1, w x, v x, Q = 13 x + tu, tw, tv, Q 3 = u, w, v. Nonlinear Anal. Model. Control, 3:334 346
338 G. Wang et al.. Soliton solutions This section will retrieve solitary wave, shock waves, singular solitary wave solutions to the model given by system 1. The method of undetermined coefficients will be employed to obtain these solutions. The derivation of results is distributed in the following few subsections...1 Solitary waves To study solitary waves of system 1, we first assume a solution of the form ux, t = sech p1 τ, 6 vx, t = A sech p τ, 7 wx, t = A 3 sech p3 τ, 8 where τ is given by τ = Bx ct 9 with B representing the inverse width, while c is the corresponding speed of such nonlinear wave. The parameter A j for j = 1,, 3 are the amplitudes of the three solitary wave components. After substituting the trial functions 6 8 into system 1, one get, in a simplified form, p 1 c + p 1 B sech p1 τ 6p 1 A 1 sech p1 τ p 1 1 + p 1 + p 1 B sech +p1 τ + 6p + p 3 A A 3 sech p+p3 τ = 0, 10 p B sech p τ 3 sech p1+p τ 1 + p + p B sech +p τ = 0, 11 p 3B sech p3 τ 3 sech p1+p3 τ 1 + p 3 + p 3 B sech +p3 τ = 0 1 for each equation in 1, respectively. The subtle balance between nonlinearity dispersion allows one to equate + p 1 = p 1, 13 + p 1 = p + p 3, 14 p 1 + p = + p, 15 p 1 + p 3 = + p 3 16 from which one can obtain p 1 =, p =, p 3 =. Substituting this values of p i into 10 1 leads to c + B sech τ + 3 A A 3 A 1 B sech 4 τ = 0, 17 4B sech τ + 3 + 4B sech 4 τ = 0. https://www.mii.vu.lt/na
Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions 339 After adding both equations, one get from the coefficients of the linearly independent functions sech j τ for j =, 4 c = + B 18 the relation 4B + 1 B + A A 3 = 0. 19 Also, from Eq. 17, setting the coefficients of the linearly independent functions to zero, one get the relations B = c Equating the values of B from 0 1, we also get 0 B = A 1 A A 3. 1 c = A A 3 A 1. Thus, the solitary waves solution of system 1 takes the form ux, t = sech [ Bx ct ], vx, t = A sech [ Bx ct ], wx, t = A 3 sech [ Bx ct ], where the speed of the waves is given by 18 or, the amplitudes are related in 19, the soliton width is provided in 0 or 1 provided.. Conserved quantites 0. From the densities given by system 5 the conserved quantities are as follows: I 1 = I = [ u + wvx vw x ] dx = 4 B, xu 3tu + 6tvw dx = 4t A A 3 A 1. B For I to be a conserved quantity, it is necessary to have di /dt = 0. Thus, this conserved quantity exists provided A 1 A A 3 =, I 3 = u vw dx = 4 A 3B 1 A A 3, where the densities are computed using the solitary wave solutions. Nonlinear Anal. Model. Control, 3:334 346
340 G. Wang et al...3 Shock waves We assume shock wave solutions of the form ux, t = tanh p1 τ, 3 vx, t = A tanh p τ, 4 wx, t = A 3 tanh p3 τ, 5 where τ is defined as in 9. In this case, the parameters A j for j = 1,, 3 B are free parameters. Then, by substituting ansatz 3 5 into system 1 simplifying, one get 6p 1 A 1 tanh p1 1 τ 6p 1 A 1 tanh p1+1 τ 6p + p 3 A A 3 tanh p+p3 1 τ + 6p + p 3 A A 3 tanh p+p3+1 τ + p 1 p 1 + 1p 1 + B tanh p1+3 τ p 1 p 1 p 1 1 B tanh p1 3 τ + p 1 [ c B + 3p 1 p 1 1B ] tanh p1 1 τ p 1 [ c B + 3p 1 p 1 + 1B ] tanh p1+1 τ = 0, 6 [ + 3p 1 + p ] B tanh p+1 τ [ + 3p p 1 ] B tanh p 1 τ p + 1p + B tanh p+3 τ + p 1p B tanh p 3 τ + 3 tanh p1+p+1 τ 3 tanh p1+p 1 τ = 0, 7 [ + 3p3 1 + p 3 ] B tanh p3+1 τ [ + 3p 3 p 3 1 ] B tanh p3 1 τ p 3 + 1p 3 + B tanh p3+3 τ + p 3 1p 3 B tanh p3 3 τ + 3 tanh p1+p3+1 τ 3 tanh p1+p3 1 τ = 0. 8 As for the case in the previous section, balancing principle leads to p 1 =, p =, p 3 =. This values of p i reduce 6 8 into c 4B tanh τ + [ c 10B + 3A 1 6A A 3 ] tanh 3 τ + 3 [ B + A A 3 A 1] tanh 5 τ = 0, 9 8B tanh τ + 0B 3 tanh 3 τ + 3 4B tanh 5 τ = 0. In this case, adding both equations setting the coefficients of the resulting linearly independent functions tanh j τ for j = 1, 3, 5 allow one to obtain two different expressions for the speed: c = 4 B, 30 c = 3 + 10B 3 + 6A A 3 0B 31 https://www.mii.vu.lt/na
Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions 341 the identity B + A A 3 + 4B 3A 1 = 0. 3 After coupling the 30 31, it follows that = 1 1 + B + 1 + 4B 3B + 8A A 3. 33 Notice also that by setting to zero the coefficients of the linearly independent functions in equation 9 we obtain c = 4B 34 whenever B = A 1 A A 3, 35 A 1 A A 3 A1 > 0. 36 Then we have that the shock waves solution of system 1 is in the form ux, t = tanh [ Bx ct ], vx, t = A tanh [ Bx ct ], wx, t = A 3 tanh [ Bx ct ], where is provided in 33, while its relation with the amplitudes A A 3 is given in identity 3. Two expressions for the soliton speed 30 31 allow to obtain an expression for, while the speed provided in 34 leads to the formula of the inverse soliton width showed in 35 subject to the constraint 36..3 Singular solitary waves type I To obtain the first type of singular solitary waves, we adopt the solutions of the form ux, t = csch p1 τ, 37 vx, t = A csch p τ, 38 wx, t = A 3 csch p3 τ 39 with τ as defined in 9. In order to calculate the unknown parameters p i for i = 1,, 3, we make a substitution of 37 39 into system 1 simplify, it takes the form p 1 c + p 1 B csch p1 τ 6p 1 A 1 csch p1 τ + p 1 1 + p 1 + p 1 B csch +p1 τ + 6p + p 3 A A 3 csch p+p3 τ = 0, 40 p B csch p τ 3 csch p1+p τ + 1 + p + p B csch p+ τ = 0, 41 Nonlinear Anal. Model. Control, 3:334 346
34 G. Wang et al. p 3B csch p3 τ 3 csch p1+p3 τ + 1 + p 3 + p 3 B csch p3+ τ = 0. 4 In this case, we have + p 1 = p 1, + p 1 = p + p 3, + p = p 1 + p, which yields p 1 = p = p 3 =. This values of p i reduce 40 4 into c + B csch τ + 3 B + A A 3 A 1 csch 4 τ = 0 43 4B csch τ 3 4B csch 4 τ = 0. By adding the last two equations one get, after equating to zero the coefficients of the linearly independent functions csch j τ for j =, 4, the identity c = B 44 4B 1 + B + A A 3 = 0. 45 Notice also thet from Eq. 43 it is possible to retrieve 0. Finally, we have that the type-i singular solitary waves solution of system 1 are in the form ux, t = csch [ Bx ct ], vx, t = A csch [ Bx ct ], wx, t = A 3 csch [ Bx ct ], where the corresponding speed is given in 44, relation between soliton width, amplitudes A A 3 is provided in 45. Also, another possible expression for the speed is given in 18 along with two possible expressions for the solitary wave width 0 1 depending on the situation at h..4 Singular solitary waves type II For the case of type-ii singular solitary waves, we adopt a solution of the form ux, t = coth p1 τ, 46 vx, t = A coth p τ, 47 wx, t = A 3 coth p3 τ 48 https://www.mii.vu.lt/na
Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions 343 with τ as defined in 9. The substitution of ansatz 46 48 into system 1 leads to [ p 1 c B + 3p 1 p 1 + 1B ] coth p1+1 τ [ p 1 c B + 3p 1 p 1 1B ] coth p1 1 τ p 1 p 1 + 1p 1 + B coth p1+3 τ + p 1 p 1 1p 1 B coth p1 3 τ 6p + p 3 A A 3 coth p+p3+1 τ + 6p + p 3 A A 3 coth p+p3 1 τ + 6p 1 A 1 coth p1+1 τ 6p 1 A 1 coth p1 1 τ = 0, 49 3 coth p1+p+1 τ 3 coth p1+p 1 τ p + 1p + B coth p+3 τ + p 1p B coth p 3 τ + + 3p p + 1 B coth p+1 τ + 3p p 1 B coth p 1 τ = 0, 50 3 coth p1+p3+1 τ 3 coth p1+p3 1 τ p 3 + 1p 3 + B coth p3+3 τ + p 3 1p 3 B coth p3 3 τ + + 3p 3 p 3 + 1 B coth p3+1 τ + 3p 3 p 3 1 B coth p3 1 τ = 0. 51 From the balancing principle one get the parameter values p 1 = p = p 3 =. Then 49 51 reduce to c 4B coth τ + [ 6A A 3 c 10B 3A ] 1 coth 3 τ 3 B + A A 3 A 1 coth 5 τ = 0 5 8B coth τ + 0B 3 coth 3 τ + 3 1B coth 5 τ = 0. In this case, one get two different expressions for the speed the identity c = 4 + B, 53 c = 10B 3 3 + 0B + 6A A 3 54 1 + A1 B 4B A A 3 = 0. 55 From the two expressions of the speed 53 54 it is possible to obtain = 1 B 1 + 1 + 4B + 3B + 8A A 3. 56 Nonlinear Anal. Model. Control, 3:334 346
344 G. Wang et al. Additionally, from Eq. 5, setting the coefficients of the linearly independent functions, one get the relations B = c 4 57 B = c + 3 6A A 3 10. 58 Equating both expressions of B leads to another expression for the speed c = A 1 A A 3. 59 Therefore, the type-ii singular solitary waves solution of system 1 is ux, t = coth [ Bx ct ], vx, t = A coth [ Bx ct ], wx, t = A 3 coth [ Bx ct ], where the speed of the waves is given by 53 or 54, the amplitude is given in 56, its relation with A, A 3, the soliton width is provided in 55. The soliton width may be retrieve either from 57 or 58, another expression for the speed given in 59 comes from this two expressions of B. 3 The equation II Another version of the mkdv system is given by u t 1 u xxx + 3u u x 3 vw x,x 3uvw x = 0, v t + v xxx + 3vu x x + 3v w x 6uvu x 3u v x = 0, w t + w xxx + 3wu x x + 3w v x 6uwu x 3u w x = 0. 3.1 Adjoint system conditions for self-adjointness Here, we get the following formal Lagrangian: L = ū u t 1 u xxx + 3u u x 3 vw x,x 3uvw x + v v t + v xxx + 3vu x x + 3v w x 6uvu x 3u v x + w w t + w xxx + 3wu x x + 3w v x 6uwu x 3u w x, https://www.mii.vu.lt/na
Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions 345 where ū, v w are three new dependent variables. As previous step, one can derive the adjoint system F 1 = ū t + 1 ūxxx 3u ū x + 3ū x vw + 3 v xx v + 3 v x v x 3v v x 6vv x v + 6uvv x + 3w w xx + 3 w x w x + 6uw w x = 0, F = v t v xxx 3 ūxxw + 3ūu x w 3u x v x + 6 vvu x + 3u v x 6ww x w 3w w x = 0, F 3 = w t w xxx 3 ūxxv + 3uū x v 3 w x u x + 6w wv x + 3u w x = 0. Also, we get the conditions of nonlinear self-adjointness to be P = c 1, Q = 0, R = 0. 3. Lie symmetry analysis conservation laws The generators of the algebra of Lie point symmetries are X 1 = x, X = t, X 3 = 3t t x x + u u + v v, X 4 = 3t t x x + u u + w w, the system admits the conserved vector [ 1 T = 4ku 3 + 3u 4 18u vw + 3v w + u x 6kwv x 4 v x w x ku xx u 3wv x + 3vkw + w x + u xx + wv xx + v6wu x 3kw x + w xx, 1 ku + u + vw ] with associated multiplier Q = u + k, 1 w, 1 v, where k is a constant. Acknowledgment. The authors really appreciate Editors Reviewers helpful comments valuable suggestions that improved this paper. References 1. M. Antonova, A. Biswas, Adiabatic parameter dynamics of perturbed solitary waves, Commun. Nonlinear Sci. Numer. Simul., 143:734 748, 009. Nonlinear Anal. Model. Control, 3:334 346
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