On exact solutions, conservation laws and invariant analysis of the generalized Rosenau-Hyman equation
OOn exact solutions, conservation laws and invariant analysis of thegeneralized Rosenau-Hyman equation
Pinki Kumari ∗ , R.K. Gupta † , and Sachin Kumar ‡ Department of Mathematics and Statistics, Central University of Punjab, Bathinda,Punjab, India Department of Mathematics, Central University of Haryana, Mahendergarh, Haryana,India
Abstract
In this paper, the nonlinear Rosenau-Hyman equation with time dependent variable coefficients isconsidered for investigating its invariant properties, exact solutions and conservation laws. Using Lieclassical method, we derive symmetries admitted by considered equation. Symmetry reductions areperformed for each components of optimal set. Also nonclassical approach is employed on consideredequation to find some additional supplementary symmetries and corresponding symmetry reductions areperformed. Later three kinds of exact solutions of considered equation are presented graphically fordifferent parameters. In addition, local conservation laws are constructed for considered equation bymultiplier approach.
Keywords:
Classical and Nonclassical symmetries; Rosenau Hyman equation; Conservation laws; Exactsolution
Mathematics Subject Classification:
From past few decades, the theory of nonlinear differential equations has undergone notable achievements.Nonlinear partial differential equations are the mathematical formulations of laws of nature. Generally theyappear in the mathematical analysis of diverse physical phenomena in the area of engineering, applied sciencesand mathematical physics [10]. Most of the problems in physics are nonlinear and often hard to solve inexplicit manner, so each nonlinear problem is studied as an individual problem. These nonlinear phenomenahave several physical and mathematical applications which are usually interpreted by finding their numericalor analytic solutions. In order to obtain numerical solution, numerical, asymptotic and perturbation methodsare usually used with great success; nonetheless, much interest prevails in the direction of finding closedform analytical solutions. Various effective analytical techniques [1, 11–13, 15, 18] have been developed tofind exact solution in literature. Among these techniques, symmetry reduction techniques [5, 6, 9, 14] are themost effective and straightforward, more generally, it is the general theory to construct exact solution interms of solutions of lower dimensional equations and also the most active field of research nowadays.A well known symmetry approach- Lie classical approach, originally proposed by Norwegian mathe-matician Sophus Lie, is completely algorithmic as it does not involve any kind of guesses and have gainedpopularity in obtaining symmetries, similarity transformations and symmetry reductions. Later nonclassicalmethod is developed to find supplementary symmetries, which is not revealed by classical method. As non-classical method involves nonlinear determining equations, which are complicated to solve. So this method isnot exploited much. In this work, we will use both classical and nonclassical approach to explore symmetriesand symmetry reduction for Rosenau-Hyman (RH) equation with time dependent variable coefficients. TheRH equation with variable coefficients is written as u t + α ( t ) uu xxx + δ ( t ) u x u xx + β ( t ) uu x = 0 (1.1) ∗ [email protected] † [email protected] ‡ Corresponding Author: [email protected], [email protected] a r X i v : . [ n li n . S I] J un here u = u ( x, t ), the coefficients α ( t ) , β ( t ) , δ ( t ) are nonzero integrable functions of t and subscripts standfor partial derivatives. For α = β = 1 and δ = 3, this equation illustrate the formation of patterns in liquiddrops [25].The assumption of constant coefficients usually leads to idealization of the physical phenomena in whichnonlinear models appear. That is why, the study of nonlinear models with variable coefficients is gainingmuch attention nowadays. To analyse the sensitivity of physical situations with various significant parametersconstrued by variable coefficients, exact solutions of these models are very important and helpful.One of the popular aspect in the current study on nonlinear partial differential equations is the con-servation laws. Mathematically conservation laws are the differential equation that describes natural laws.Though some conservation laws do not have a physical relevance, but in the context of partial differentialequations, they reveals certain qualities like integrability, existence and uniqueness of solutions [4, 19, 21].Besides, exact solutions of partial differential equation can be derived with the help of conserved vectorslinked with Lie symmetries [7, 8, 26]. First of all, Noether theorem [23] has been introduced to constructconservation laws for variational problems. Now several methods like direct construction method, new con-servation method etc. [2, 3, 16, 27] have been developed in which no priori knowledge about Lagrangian isneeded.The paper is arranged in the manner- In sec. 2, Lie point symmetries of eq. (1.1) are obtained andsymmetry reductions are performed. Sec. 3 deals with the nonclassical symmtries of (1.1). In sec. 4, exactsolutions of (1.1) are found. Conservation laws of (1.1) are constructed in sec. 5. Conclusion ends the paperin sec. 6. This section deals with the application of Lie classical method [17, 20] to eq. (1.1). The point symmetriesobtained by this method allow reduction of PDE to ordinary differential equation.First we assume that a continuous group of point transformations ( x ⊕ , t ⊕ , u ⊕ ) in one parameter as t ⊕ = t + (cid:15)τ ( x, t, u ) + O ( (cid:15) ) x ⊕ = x + (cid:15)ξ ( x, t, u ) + O ( (cid:15) ) u ⊕ = u + (cid:15)η ( x, t, u ) + O ( (cid:15) ) (2.1)that leaves (1.1) invariant. Here, (cid:15) is a continuous group parameter and ξ, τ, η, η t , η x , η xx , η xxx are infinites-imals corresponding to x, t, u, u t , u x , u xx , u xxx respectively and extended infinitesimal are computed by thefollowing formulas η t = D t ( η ) − u x D t ( ξ ) − u t D t ( τ ) η x = D x ( η ) − u x D x ( ξ ) − u t D x ( τ ) η xx = D x ( η x ) − u xx D xx ( ξ ) − u tx D xt ( τ ) η xxx = D x ( η xx ) − u xxx D xxx ( ξ ) − u txx D xxt ( τ ) (2.2)Also, we assume that the infinitesimal generator takes the form X ≡ ξ ( x, t, u ) ∂∂x + τ ( x, t, u ) ∂∂t + η ( x, t, u ) ∂∂u (2.3)Invariance of transformation (2.1) on (1.1) leads the following invariance criterion η t + α ( t )[ ηu xxx + α (cid:48) ( t ) uu xxx τ + uη xxx ] + β ( t )[ ηu x + uη x ] + β (cid:48) ( t ) uu x τ + δ ( t )[ η x u x x + u x η xx ] + δ (cid:48) ( t ) u x u xx = 0 (2.4)Putting the values of extended infinitesimals (2.2) in (2.4) and comparing the coefficients of various linearly2ndependent monomials, the following essential system of determining systems is obtained. ξ u = τ x = τ u = 0 α u η uuu + δαuη uu = 03 αuξ x − α t uτ − αη − τ t αu = 0 αβu η x + α u η xxx + αuη t = 03 α u η ux − α u ξ xx + αδuη x = 03 α u η uux + 2 δαuη ux − δαuξ xx = 03 α u η uu − αδη + αδ t uτ − δα t uτ + δαuη u = 02 αβξ x u − αξ t u − α ξ xxx u + 3 η xxu α u + δαuη xx + αβ t u τ − βα t u τ = 0 (2.5)The general solution of (2.5) is written as τ = (cid:82) − c α ( t ) dt + c α ( t ) ξ = c x + c η = ( c + 3 c ) uα = α ( t ) β = c (cid:32)(cid:16) (cid:90) − c α ( t ) dt + c (cid:17) c c (cid:33) α ( t ) δ = c α ( t ) (2.6)where c , c , c , c , c and c are arbitrary constants. If we take c = 0, symmetries obtained in expression(2.6) coincide with the symmetries of RH equation with coefficients α = β = 1 and δ = 3. So it is clear thatsymmetries (2.6) are generalized version. Now we consider c = c to ease our reduction calculation and forthe case, symmetries take the form τ = − c (cid:82) α ( t ) dt + c α ( t ) , ξ = c x + c , η = 4 c uα = α ( t ) β = c (cid:16) (cid:90) − c α ( t ) dt + c (cid:17) α ( t ) δ = c α ( t ) (2.7)The associated Lie algebra spanned by the infinitesimal generators (2.7) is written as V = x ∂∂x − (cid:82) α ( t ) dtα ( t ) ∂∂t + 4 u ∂∂uV = ∂∂x , V = 1 α ( t ) ∂∂t (2.8)We can perform symmetry reduction for any linear combination of aforementioned generators as anylinear combination of generators again gives a generator. There are large number of such combinations. Soto find non equivalent symmetry reductions, we use concept of optimal system [24]. Here, we find optimalset of vector fields with the components V and V + λV Now we tabulate invariants, similarity variables, similarity transformation and variable coefficients for eachcomponent of optimal system in Table 1.3able 1: Similarity ansatzGenerators Invariants Ansatz Variable coefficients α ( t ) = α ( t ) V ( x (cid:82) α ( t ) dt, ( (cid:82) α ( t ) dt ) u ) u = ( (cid:82) α ( t ) dt ) − f ( x (cid:82) α ( t ) dt ) β ( t ) = − c ( (cid:82) α ( t ) dt ) α ( t ) δ ( t ) = c α ( t ) α ( t ) = α ( t ) V + λV ( x − λ (cid:82) α ( t ) dt, u ) u = f ( x − λ (cid:82) α ( t ) dt ) β ( t ) = c λ α ( t ) δ ( t ) = c α ( t ) Theorem 2.1.
The similarity variable ζ = x (cid:82) α ( t ) dt and similarity transformation u = ( (cid:82) α ( t ) dt ) − f ( ζ ) for vector field V reduces eq. (1.1) into the following nonlinear ordinary differential equation. − f + zf (cid:48) + f f (cid:48)(cid:48)(cid:48) − c f f (cid:48) + c f (cid:48) f (cid:48)(cid:48) = 0 (2.9) Here, (cid:48) denotes the first order derivative w.r.t. ζ . Theorem 2.2.
For the vector field V + λV , the similarity transformation u = f ( ζ ) with similarity variable ζ = x − λ (cid:82) α ( t ) dt reduces eq. (1.1) into the following ODE − λ f (cid:48) + f f (cid:48)(cid:48)(cid:48) + c λ f f (cid:48) + c f (cid:48) f (cid:48)(cid:48) = 0 (2.10) This section presents nonclassical symmetries of eq. (1.1) via compatibility conditions explained in Refs.[28, 29]. First we consider nonclassical symmetry of the form u t = ξ ( x, t, u ) u x + η ( x, t, u ) (3.1)In order to find compatibility condition, substitute (3.1) into (1.1) and expression reads as ξu x + η + α ( t ) uu xxx + β ( t ) uu x + δ ( t ) u x u xx = 0 (3.2)Next, equality of u tt between (1.1) and (3.1) yields the following equation (3.3) ξ t u x + ξ u u t u x + ξu xt + η t + η u u t + α t uu xxx + αu t u xxx + αuu xxxt + β t uu x + βu t u x + βuu xt + δ t u x u xx δu xt u xx + δu x u xxt = 0Further, by eliminating differential consequences of u t and highest order term u xxx from eq. (3.3) with (3.1)and (3.2), we obtain the following set of nonlinear determining equations u xx : δη x − αuξ xx + 3 αuη xu = 00 : η t + βuη x − η α t α + αuη xxx + 3 ηξ x − η u = 0 u x u xx : δ t − αuξ xu + 3 αη uu u − ηδu + η u δ − δ α t α = 0 u x : − βu α t α + β t u + 3 ηξ u − ξξ x + δη xx − ξ t u + ξ α t α + ηξu + 2 uβξ x + 3 αη xxu − αuξ xxx = 0 u x u xx : − αuξ uu = 0 u x : − ξξ u + 2 δη xu − δξ xx + 3 uβξ u + 3 αuη xuu − αuξ xxx = 0 u x : δη uu − δξ xu + αuη uuu − αuξ xuu = 0 u xx : − αuξ u = 0 u x : − δξ uu − αuξ uuu = 0 (3.4)4nd the solution of system (3.4) yields ξ, η and variable coefficients as ξ = f ( t )( x + c ) η = c f ( t ) uα ( t ) = c f ( t ) exp( − (cid:90) ( c − f ( t ) dt ) β ( t ) = c f ( t ) exp( − (cid:90) ( c − f ( t ) dt ) δ ( t ) = c f ( t ) exp( − (cid:90) ( c − f ( t ) dt ) (3.5)Here, c , c , c , c , c are arbitrary constants and f ( t ) is arbitrary integrable function. Thus, nonclassicalsymmetries of variable coefficients RH equation (1.1) is written as u t + f ( t )( x + c ) u x − c f ( t ) u ≡ X ≡ f ( t )( x + c ) ∂∂x + ∂∂t + c uf ( t ) ∂∂u (3.7)Now we will obtained similarity transformation for eq. (1.1) by solving the following characteristic equation dxf ( t )( x + c ) = dt duc f ( t ) u (3.8)For the nonclassical symmetries (3.7), eq. (3.8) produces similarity variable as z = ( x + c ) exp ( − (cid:82) f ( t ) dt )and similarity transformation as u = exp ( c (cid:82) f ( t ) dt ) g ( z ), which reduce the eq. (1.1) into the following ODE − zg (cid:48) + c g + c gg (cid:48)(cid:48)(cid:48) + c gg (cid:48) + c g (cid:48) g (cid:48)(cid:48) = 0 (3.9) In this section, we aim to find the exact solutions of eq. (1.1) by seeking the solutions for reduced equations(2.9), (2.10) and (3.9). Also graphical representation will be shown by taking several different values ofparameters. (2.9)
Consider solution of eq. (2.9) has series form as f ( ζ ) = a + bζ + cζ (4.1)where a, b, and c are unknown constants that need to be determined.On putting (4.1) into (2.9), we obtain a = 0 , b = − c , c = 0 (4.2)So by reverting back the original variables, the exact solution of (1.1) can be written as u ( x, t ) = − c x (cid:16) (cid:90) α ( t ) dt (cid:17) − (4.3)3D and contour plots of the solution (4.3) is shown in figures (a)-(i) with some different parameters.5a) α ( t ) = e − t , c = 1 (b) α ( t ) = e − t , c = 1 (c) α ( t ) = e − t , c = 300(d) α ( t ) = sinh ( t ), c = 1 (e) α ( t ) = sinh ( t ), c = 1 (f) α ( t ) = sinh ( t ), c = 300(g) α ( t ) = ln ( t ), c = 1 (h) α ( t ) = ln ( t ), c = 1 (i) α ( t ) = ln ( t ), c = 300 (2.10) For arbitrary constant c , the work to find exact solution is in progress. Presently, we seek the solution for c = 1. Now eq. (2.10) is written as − λ f (cid:48) + f f (cid:48)(cid:48)(cid:48) + c λ f f (cid:48) + f (cid:48) f (cid:48)(cid:48) = 0 (4.4)Eq. (4.4), on integrating w.r.t. ζ , can be expressed as − λ f + f f (cid:48)(cid:48) + c λ f = 0 (4.5)With the aid of Maple software, the general solution of eq. (4.5) is obtained as f ( z ) = c sin ( 1 √ λ √ c z ) + c cos ( 1 √ λ √ c z ) + 2 c λ (4.6)Consequently, the general solution admitted by (1.1) can be written as (4.7) u ( x, t ) = c sin (cid:16) √ λ √ c ( x − λ (cid:90) α ( t ) dt ) (cid:17) + c cos (cid:16) √ λ √ c ( x − λ (cid:90) α ( t ) dt ) (cid:17) + 2 c λ
6D profile of solution (4.7) is displayed in figures (j)-(o) and contour plots are shown in figures (j.1)-(l.1)with some parametric values.(j) α (t)=1, c =1, c =1, (k) α (t)=1, c =1, c =1, (l) α (t)=1, c =3, c =1, c =1, λ =1 c =1, λ =3 c =1, λ =3(j.1) α (t)=1, c =1, c =1, (k.1) α (t)=1, c =1, c =1, (l.1) α (t)=1, c =3, c =1, c =1, λ =1 c =1, λ =3 c =1, λ =3(m) α (t)=t, c =1, c =1, (n) α (t)=t, c =1, c =1, (o) α (t)=t, c =3, c =1, c =1, λ =1 c =1, λ =3 c =1, λ =3 (3.9) We assume that the eq. (3.9) takes the series solution as follows g ( z ) = a + bz + cz (4.8)where a, b, and c are unknown constants need to be determined. By substituting (4.8) into (3.9), we get a = 0 , b = 1 − c c , c = 0 (4.9)Thus by reverting back the original variables, the exact solution of (1.1) is found as u ( x, t ) = 1 − c c ( x + c ) exp (cid:16) (cid:90) ( c − f ( t ) dt (cid:17) (4.10)7D graph and contour plots of (4.10) are represented in figures (p)-(u) by appointing various functionparameters.(p) f ( t ) = exp ( − t ), c =1, (q) f ( t ) = sin ( t ), c =1, (r) f ( t ) = tan ( t ), c =1, c =2, c =1 c =2, c =1 c =2, c =1(s) f ( t ) = exp ( − t ), c =1, (t) f ( t ) = sin ( t ), c =1, (u) f ( t ) = tan ( t ), c =1, c =2, c =1 c =2, c =1 c =2, c =1 Here we intend to construct local conservation laws of RH equation (1.1) by multiplier method [22, 27]. Inthis method, no priori knowledge about Lagrangian of equation of motion is necessary. The method is quitewell known, effective and it directly uses the definition of local conservation laws. First we assume simplemultiplier of the form Λ( x, t, u ). The multiplier Λ for (1.1) have the propertyΛ( u t + α ( t ) uu xxx + β ( t ) uu x + δ ( t ) u x u xx ) = D t C t + D x C x (5.1)for all solutions of u ( x, t ). Here, D t and D x are total derivative operator defined by D t = ∂∂t + u t ∂∂u + u tt ∂∂u t + u xt ∂∂u x + · · · D x = ∂∂x + u x ∂∂u + u tx ∂∂u t + u xx ∂∂u x + · · · (5.2)The determining equation for the multiplier Λ follows by (5.3) δδu (cid:104) Λ( u t + α ( t ) uu xxx + β ( t ) uu x + δ ( t ) u x u xx = 0 (cid:105) (1.1) = 0where Euler operator, δδu is expressed as (5.4) δδu = ∂∂u − D t ∂∂u t − D x ∂∂u x + D xx ∂∂u xx + D tt ∂∂u tt + D xt ∂∂u xt + · · ·
8n expansion of (5.3), the following set of determining equations is obtained.Λ x δ − x α − ux αu = 02Λ ux δ − ux α − uux αu = 0Λ xx δ − xx α − xux αu = 0Λ uu δ − uu α − Λ uuu αu = 0 , − Λ x β − xxx αu − Λ t = 03Λ u δ − u α − uu αu = 0 (5.5)Now we split the general solutions of eq. (5.5) into cases with different pivots in order to get maximumnumber of conserved vectors. The conserved vectors are constructed for each case explicitly. Case (a) If α (cid:54) = 0 , δ ( δ − α ) (cid:54) = 0 , δ t α − δα t (cid:54) = 0, the multiplier takes the formΛ = c (5.6)For this case, the conserved components of (1.1) are written as C t = c uC x = αc uu xx − αc u x + 12 δc u x + 12 βc u (5.7) Case (b)
For α (cid:54) = 0 , δ ( δ − α ) (cid:54) = 0 , δ (cid:54) = 0 , δ t α − δα t = 0, two subcases arise. Subcase (b.1)
For δ = c α, c ( c − (cid:54) = 0, the multiplier isΛ = c + c u c − (5.8)So the conserved vectors associated with multiplier (5.8) are expressed as C t = c c u c + c uC x = αc uu xx + αc uu xx + 12 βc u + 12 αc c u x − c αc u x + βc c + 1 u c +1 (5.9) Subcase (b.2)
The multiplier for δ = α takes the formΛ = c ln ( u ) + c (5.10)The expression for conserved vectors with (5.10) are as follows C t = u ( c ln ( u ) + c − c ) C x = αc ln ( u ) uu xx + αc uu xx − αc u x + βc (cid:16) u ln ( u ) − u (cid:17) + 12 c u (5.11) Case (c)
For α (cid:54) = 0 , δ ( δ − α ) = 0 , δ (cid:54) = 0 , δ t β − δβ t (cid:54) = 0, we obtain the following multiplierΛ = c u + c (5.12)So the conserved fluxes for multiplier (5.12) are expressed as C t = 13 c u + c uC x = αc u u xx + 14 βc u + αc u x + 12 βc u + αc uu xx (5.13) Case (d)
For α (cid:54) = 0 , δ ( δ − α ) = 0 , δ (cid:54) = 0 , δ t β − δβ t = 0, two subcases are encountered. Subcase (d.1) If α = δ, β = c δ, c >
0, the multiplier is obtained asΛ = c u + c + c sin ( √ c x ) + c cos ( √ c x ) (5.14)9he conserved vectors for multiplier (5.14) are computed as C t = 13 c u + c u + c usin ( √ c x ) + c cos ( √ c x ) uC x = 112 δ (4 c u x + c c u + 2 c c u + 4 c u x sin ( √ c x ) + 4 c u x cos ( √ c x ))+4 c u u xx + 4 c uu xx + 4 c √ c usin ( √ c x ) u x − c √ c ucos ( √ c x ) u x + 4 c √ c ucos ( √ c x ) u xx + 4 c √ c usin ( √ c x ) u xx (5.15) Subcase (d.2) If α = δ, β = − c δ, c >
0, the multiplier isΛ = c u + c + c e √ c x + c e −√ c x (5.16)The conserved vectors for multiplier (5.16) are expressed as C t = 13 c u + c u + c ue √ c x + c e −√ c x uC x = 112 δ (cid:16) c u x − c c u − c c u + 4 c u x e √ c x + 4 c u x e −√ c x + 4 c u u xx +4 c uu xx + 4 c √ c ue −√ c x u x − c √ c ue √ c x u x +4 c √ c ue −√ c x u xx + 4 c √ c ue √ c x u xx (cid:17) (5.17) Remark 1
It is verified that conservation laws of the original RH equation with α = β = 1 and δ = 3 aresame as explained in sub-case (d.1). Remark 2
In case (a), we found only one conserved vector. Two conserved vectors are encountered in case(b) and case (c). In both sub-cases of case (d), we get four conserved fluxes.
The work investigates the effectiveness of Lie classical method on Rosenau-Hyman equation with time depen-dent variable coefficients. Also, generalized supplementary symmetries, which was not revealed by classicaltheory, are obtained by nonclassical approach and symmetry reductions for both type of symmetries areperformed successfully. Exact solutions of RH equation for each component of optimal set, generated bythree vector fields, as well as for nonclassical symmetries are presented. The extracted solutions are discussedgraphically by taking different values of parameters and may be significant in dynamic study of RH model.Moreover we have tried to find maximum number of conserved vectors for RH model (1.1) by multiplierapproach. The obtained conserved vectors can be used to find the solution of eq. (1.1) and will be subjectto future work.
Acknowledgement
The autor, Pinki Kumari expresses her gratitude to the University Grants Commission for financial assistance(Grant no. 19/06/2016(i)EU-V).
References [1] S Abbasbandy and A Shirzadi. The first integral method for modified Benjamin–Bona–Mahony equa-tion.
Communications in Nonlinear Science and Numerical Simulation , 15(7):1759–1764, 2010.[2] Stephen C Anco and George Bluman. Direct construction method for conservation laws of partialdifferential equations part i: Examples of conservation law classifications.
European Journal of AppliedMathematics , 13(5):545–566, 2002.[3] Stephen C Anco and George Bluman. Direct construction method for conservation laws of partialdifferential equations part ii: General treatment.
European Journal of Applied Mathematics , 13(5):567–585, 2002. 104] Thomas Brooke Benjamin. The stability of solitary waves.
Proceedings of the Royal Society of London.A. Mathematical and Physical Sciences , 328(1573):153–183, 1972.[5] George Bluman and Stephen Anco.
Symmetry and Integration Methods for Differential Equations ,volume 154. Springer Science & Business Media, 2008.[6] George W Bluman and Julian D Cole.
Similarity Methods for Differential Equations , volume 13. SpringerScience & Business Media, 2012.[7] Ashfaque H Bokhari, Ahmad Y Al-Dweik, AH Kara, FM Mahomed, and FD Zaman. Double reductionof a nonlinear (2+ 1) wave equation via conservation laws.
Communications in Nonlinear Science andNumerical Simulation , 16(3):1244–1253, 2011.[8] GL Caraffini and M Galvani. Symmetries and exact solutions via conservation laws for some partialdifferential equations of mathematical physics.
Applied Mathematics and Computation , 219(4):1474–1484, 2012.[9] Peter A Clarkson and Elizabeth L Mansfield. Algorithms for the nonclassical method of symmetryreductions.
SIAM Journal on Applied Mathematics , 54(6):1693–1719, 1994.[10] Lokenath Debnath.
Nonlinear partial differential equations for scientists and engineers . Springer Science& Business Media, 2011.[11] Engui Fan and Jian Zhang. Applications of the Jacobi elliptic function method to special-type nonlinearequations.
Physics Letters A , 305(6):383–392, 2002.[12] Zhaosheng Feng. The first-integral method to study the Burgers–Korteweg–de Vries equation.
Journalof Physics A: Mathematical and General , 35(2):343, 2002.[13] Zuntao Fu, Shikuo Liu, Shida Liu, and Qiang Zhao. New Jacobi elliptic function expansion and newperiodic solutions of nonlinear wave equations.
Physics Letters A , 290(1-2):72–76, 2001.[14] Maria Luz Gandarias and MS Bruzon. Classical and nonclassical symmetries of a generalized Boussinesqequation.
Journal of Nonlinear Mathematical Physics , 5(1):8–12, 1998.[15] Hossein Jafari, M Zabihi, and M Saidy. Application of homotopy perturbation method for solving gasdynamics equation.
Appl. Math. Sci , 2(48):2393–2396, 2008.[16] AH Kara and FM Mahomed. Relationship between symmetries and conservation laws.
InternationalJournal of Theoretical Physics , 39(1):23–40, 2000.[17] Lakhveer Kaur and RK Gupta. Kawahara equation and modified Kawahara equation with time depen-dent coefficients: symmetry analysis and generalized-expansion method.
Mathematical Methods in theApplied Sciences , 36(5):584–600, 2013.[18] Kamruzzaman Khan and M Ali Akbar. Exact and solitary wave solutions for the Tzitzeica–Dodd–Bullough and the modified Kdv–Zakharov–Kuznetsov equations using the modified simple equationmethod.
Ain Shams Engineering Journal , 4(4):903–909, 2013.[19] RJ Knops and Charles Alexander Stuart. Quasiconvexity and uniqueness of equilibrium solutions innonlinear elasticity.
Archive for rational mechanics and analysis , 86(3):233–249, 1984.[20] Sachin Kumar, K Singh, and RK Gupta. Coupled Higgs field equation and Hamiltonian amplitudeequation: Lie classical approach and (G’/G)-expansion method.
Pramana , 79(1):41–60, 2012.[21] Peter D Lax. Integrals of nonlinear equations of evolution and solitary waves.
Communications on pureand applied mathematics , 21(5):467–490, 1968.[22] Rehana Naz. Conservation laws for some systems of nonlinear partial differential equations via multiplierapproach.
Journal of Applied Mathematics , 2012:1–13, 2012.[23] E Noether. Invariante Variationsprobleme. Nachr. d. K¨onig. Gesellsch. d. Wiss. zu G¨ottingen, math-phys. klasse, 235-257 (1918). Translated by MA Travel.
Transport Theory and Statistical Physics ,1(3):183–207, 1971.[24] Peter J Olver.
Applications of Lie groups to Differential Equations , volume 107. Springer Science &Business Media, 2000.[25] Philip Rosenau and James M Hyman. Compactons: solitons with finite wavelength.
Physical ReviewLetters , 70(5):564–567, 1993.[26] A Sj¨oberg. Double reduction of pdes from the association of symmetries with conservation laws withapplications.
Applied Mathematics and Computation , 184(2):608–616, 2007.[27] Heinz Steudel. ¨Uber die zuordnung zwischen lnvarianzeigenschaften und erhaltungss¨atzen.
Zeitschriftf¨ur Naturforschung A , 17(2):129–132, 1962.[28] Zhi-lian Yan and Xi-qiang Liu. Symmetry and similarity solutions of variable coefficients generalizedzakharov–kuznetsov equation.
Applied mathematics and computation , 180(1):288–294, 2006.1129] Zhi-lian Yan, Jian-ping Zhou, and Xi-qiang Liu. Symmetry reductions and similarity solutions of the(3+ 1)-dimensional breaking soliton equation.