Group Classification of a Higher-Order Boussinesq Equation
aa r X i v : . [ n li n . S I] J un Group Classification of a Higher-Order Boussinesq Equation
Y. Hasano˘glu, C. ¨Ozemir
Department of Mathematics, Faculty of Science and Letters,Istanbul Technical University, 34469 Istanbul, Turkey ∗ June 29, 2020
Abstract
We consider a family of higher-order Boussinesq equations with an arbitrary nonlin-earity. We determine the classes of equations so that a certain type of Lie symmetryalgebra is admitted in this family. In case of a quadratic nonlinearity we provide severalexact solutions, some of which are in terms of elliptic functions.
The aim of this manuscript is to classify higher-order Boussinesq (HBq) equations of the form u tt = η u xxtt − η u xxxxtt + ( f ( u )) xx (1.1)according to the Lie symmetry algebras the equation admits depending on the formulation ofthe nonlinearity f ( u ) and to study possible reductions of this equation to find exact solutions.More explicitly, we will determine the classes of functions f ( u ) for which the equation has finite-dimensional Lie symmetry algebras. Among these classes, we shall concentrate on a specificfamily, which is widely concerned in literature, to find exact traveling wave solutions. Here weassume η , η are nonzero constants and f uu = 0.The derivation of (1.1) appears in [1], when the approximation of the equations of motionof a 1-dimensional lattice to the continuum requires considering higher order effects. It is alsoderived in [2] for the propagation of longitudinal waves in an infinite elastic medium within thecontext of nonlinear non-local elasticity. The authors also investigate the well-posedness of theCauchy problem. As a recent literature, we see Eq. (1.1) in [3] where the authors study thelocal and global existence and blow-up of solutions to the initial and boundary value problemof the equation. In this literature, η and η are positive constants and f ( u ) is considered to bean arbitrary nonlinearity. HBq equations of [2] and [3] are obtained from (1.1) when we replace f ( u ) → u + f ( u ).The Lie symmetry algebra of the Boussinesq equation u tt + uu xx + ( u x ) + u xxxx = 0 (1.2) ∗ e-mail: [email protected], [email protected]
1s the Lie algebra of the vector fields D = x∂ x + 2 t∂ t − u∂ u , P = ∂ x , P = ∂ t , (1.3)which generate translations and dilations, see Refs. [4, 5, 6, 7]. Classical and non-classicalsimilarity reductions of the Boussinesq equation u tt + au xx + b ( u ) xx + cu xxxx = 0 (1.4)are obtained in [5] and these nonclassical reductions are given a group-theoretical frameworkin the context of conditional symmetries in [4].In connection with classification problem in Lie theory, [8] performs the symmetry classifi-cation of the generalized Boussinesq equation u tt = u xxxx + ( f ( u )) xx . (1.5)In [9], the authors perform Lie symmetry analysis of the equation u tt − u xx + u xxxx + ( f ( u )) xx = 0 . (1.6)Ref. [10] handles the double-dispersion equation u tt = u xx + au xxtt − bu xxxx + du xxt + ( f ( u )) xx (1.7)and exhibits the functional forms of f ( u ) so that the equation enjoys Lie symmetry algebras.Ref. [11] studies the symmetry algebra and reductions of the equation u tt − ∆ u − ∆ u tt + ∆ u + k ∆ u t = ∆ f ( u ) (1.8)where x ∈ R and f is a power-type nonlinearity. [12] considers this equation for n = 1, in theform u tt − u xx + au xxxx − bu xxtt = ( f ( u )) xx , (1.9)and for n = 2, to derive conservation laws. Ref. [13] considers symmetry algebras of theequation u tt = cu xx + bu xxxx + au xxxxxx + ( f ( u )) xx (1.10)and derives the conservation laws of this equation which admits a Hamiltonian form whenwritten as a system.Let us finally mention two references which consider the closest family of equations to theone we consider. Ref. [14] considers u tt = u xx + u xxtt − u xxxxtt − cu xxxx + ( f ( u )) xx (1.11)in the case c = 0 and finds exact solutions to this equation in terms of trigonometric, hyperbolicand elliptic functions when f ( u ) has some certain forms. Let us note that the case c = 0 is notconsidered in that article separately in the search of the Lie symmetry algebra, therefore theydo not cover our results.Classification of the family of equations (1.1) in the case η = 0; explicitly, the family u tt = δu ttxx + ( f ( u )) xx , (1.12)2ccording to symmetry algebras the equation admits is studied in [15]. Clearly, our mainequation (1.1) is an extension of this family to the sixth-order. According to the results of [15],the Lie symmetry algebra of an equation from the class (1.12) can be at most three-dimensional.However, we find that for a specific form of f ( u ), (1.1) has a four-dimensional symmetry algebraand this result is also valid when η = 0, namely, for Eq. (1.12). We state this after our maintheorem as a Remark, which actually serves as a complementary result to those of [15].Our analysis is consisting of two parts. First we perform the Lie algebra classification ofEq. (1.1). After we produce some exact solutions for a specific form of f ( u ). In what follows we assume that η = 0, η = 0, f uu = 0. The infinitesimal generator is of theform V = φ ( t, x, u ) ∂ t + φ ( t, x, u ) ∂ x + φ ( t, x, u ) ∂ u . (2.1)We find V = τ ( t ) ∂ t + ξ ( x ) ∂ x + φ ( t, x, u ) ∂ u , (2.2)where φ = Q ( x, t ) + (cid:0) τ t + 32 ξ x + φ (cid:1) u, (2.3a) τ ttt = 0 , (2.3b)2 η ξ x − η ξ xxx = 0 , (2.3c)8 ξ x − η ξ xxx + 3 η ξ xxxxx = 0 , (2.3d)2( ξ x + τ t ) f u + φf uu = 0 , (2.3e) ξ xx f u + φ x f uu = 0 , (2.3f) Q tt − η Q xxtt + η Q xxxxtt − ( Q xx + 32 uξ xxx ) f u = 0 , (2.3g)where φ is an arbitrary constant. If we differentiate (2.3e) with respect to x and subtract itfrom (2.3f), we get ξ xx = 0 and φ x = 0, and hence Q x = 0. Eq. (2.3c) gives ξ x = 0, so ξ ( x ) = ξ ,a constant. After these, the infinitesimal generator is of the form V = τ ( t ) ∂ t + ξ ∂ x + φ ( t, u ) ∂ u (2.4)with φ = Q ( t ) + (cid:0) τ t + φ (cid:1) u, (2.5a) τ ttt = 0 , Q tt = 0 , (2.5b)2 τ t f u + φf uu = 0 . (2.5c)It is seen that when f is arbitrary, we have the two symmetries X = ∂ t , X = ∂ x (2.6)and the symmetry algebra is the Abelian two-dimensional Lie algebra. One can proceed andsolve the system of determining equations above for different cases. We take another approach3nd play a little bit on (2.5c) to get an equation involving only f . We differentiate (2.5c) withrespect to u to get 2 τ t f uu + φ u f uu + φf uuu = 0 . (2.7)Using (2.5c) and (2.7) we can eliminate the term with τ t and obtain φf u f uuu + φ u f u f uu − φf uu = 0 . (2.8)Again we differentiate (2.8) with respect to u and find2 φ u f u f uuu + φf u f uuuu − φf uu f uuu = 0 . (2.9)We eliminate φ between (2.8) and (2.9) and hence obtain f u f uu f uuuu + f uu f uuu − f u f uuu = 0 , (2.10)which is exactly the same equation for f ( u ) that was obtained in [15]. Compatible with theirfindings, Eq. (2.10) is solved by the following different forms of f :(a) f ( u ) = αe βu + γ, (2.11a)(b) f ( u ) = α ln( βu + δ ) + γ, (2.11b)(c) f ( u ) = α ( βu + δ ) n + γ, n = 0 , . (2.11c)Here α, β, γ, δ, n are arbitrary constants where αβ = 0. Actually, the constant γ has no signif-icance when we consider the HBq equation (1.1). By a transformation u = δ ¯ u + δ , ¯ x = µx ,¯ t = λt and relabeling the constants, Eq. (1.1) with the above forms of f ( u ) can be convertedto an equation with (A) f ( u ) = αe u , α = ∓ , (2.12a)(B) f ( u ) = α ln( u ) , α = ∓ , (2.12b)(C) f ( u ) = αu n , α = ∓ , R ∋ n = 0 , . (2.12c)We shall concentrate on these simplified forms of the nonlinearity f ( u ). Let us note that, inthe remaining part of the paper, for all of the cases (A), (B) and (C), we did not restrict theconstant α to ∓ α . Case A: f ( u ) = α e u , α = ∓ Equation (1.1) is of the form u tt = η u xxtt − η u xxxxtt + α ( e u ) xx . (2.13)The Lie algebra L A of this equation is three dimensional, L A = { X , X , X } , generated by thevector fields X = ∂ t , X = t∂ t − ∂ u , X = ∂ x . (2.14)The nonzero commutation relation is [ X , X ] = X , (2.15)4herefore the Lie algebra has the structure L A = A ⊕ A = { X , X } ⊕ X . The optimal systemof one-dimensional subalgebras of A ⊕ A is given in [16]. Therefore, the optimal system ofone-dimensional subalgebras of L A is { X } , { X + ǫX } , {− X cos θ + X sin θ } (2.16)with ǫ = ∓ ≤ θ < π . The reductions through the last subalgebra should be analyzedcarefully. (i) When θ = π , we have the generator X = ∂ x . Solutions invariant under thegroup of transformations generated by this subalgebra are time-dependent ones, u = u ( t ). Notonly for (2.13), but for any form of f in (1.1), these solutions are found from u tt = 0 hence u = at + b , trivially. This subalgebra will not be considered in any of the subcases. (ii) When θ ∈ [0 , π ) − { π } , we have {− X cos θ + X sin θ } ≃ { X − (tan θ ) X } , for which we simplywrite { X + cX } , c ∈ R . We observe that when c = 0, the reduction obtained is 2 less in theorder than the order of the reduced equation that is obtained when c = 0; therefore, for thissubalgebra, we consider the cases c = 0 and c = 0 separately. (i) The Subalgebra X = ∂ t . The solutions will have the form u = u ( x ) and from (2.13) we get u = ln( ax + b ), where a, b are arbitrary constants. (ii) The Subalgebra X + ǫX = ∂ t + ǫ∂ x , ǫ = ∓ . The invariant solution will have the form u ( x, t ) = F ( ξ ) = F ( x − ǫt ). This generator produces traveling wave solutions, and will appearin other forms of the nonlinearity f ( u ). Instead of working on (2.13), let us do the reductionfor (1.1), which will be useful for other cases of f ( u ). (Furthermore, see that since ∂ t and ∂ x aresymmetries of (1.1) for any form of f ( u ), so is the generator ∂ t + ǫ∂ x .) Substituting u = F ( ξ ), ξ = x − ǫt in (1.1), it reduces to F ′′ = η F (4) − η F (6) + [ f ( F )] ′′ (2.17)which is integrated to η F (4) − η F ′′ + F − f ( F ) = K ξ + K . (2.18)Here K , K are arbitrary constants and the derivatives are with respect to the variable ξ .Therefore, for f ( u ) = αe u , the reduced equation is η F (4) − η F ′′ + F − αe F = K ξ + K . (2.19) (iii) The Subalgebra X = t∂ t − ∂ u . The invariant solution is of the form u = − t + F ( x ),of which substitution into (2.13) gives F ( x ) = ln( 1 α x + ax + b ) and hence u ( x, t ) = ln h t (cid:0) α x + ax + b (cid:1)i . (2.20) (iv) The Subalgebra X + cX = t∂ t + c∂ x − ∂ u . The group-invariant solution will have theform u = − t + F ( ξ ), ξ = x − c ln t. From (2.13) we get, after a further integration, η c F (5) + cη F (4) − η c F (3) − η cF ′′ + c F ′ − α ( e F ) ′ + cF + 2 ξ = K (2.21)with K being the integration constant. 5able 1: Subalgebras and reduced equations for Cases B and CSubalgebra u tt = η u xxtt − η u xxxxtt + ( f ( u )) xx Similarity variable
Case B f ( u ) = α ln u, α = ∓ u tt = η u xxtt − η u xxxxtt + α (ln u ) xx L B = { X , X , X } X = ∂ t , X = t∂ t + 2 u∂ u , X = ∂ x Reduction by X u ( x ) = ae bx X + ǫX , ǫ = ∓ η F (4) − η F ′′ + F − α ln F = K ξ + K u = F ( ξ ) = F ( x − ǫt ) X + cX , c ∈ R η c F (6) − η cF (5) + (2 η − η c ) F (4) + 3 η cF (3) u = t F ( ξ )+( c − η ) F ′′ − α (ln F ) ′′ − cF ′ + 2 F = 0 ξ = x − c ln tX η F (4) − η F ′′ − α (ln F ) ′′ + F = 0 u = t F ( x ) Case C.1 f ( u ) = αu n , α = ∓ , n = 0 , , n ∈ R The equation u tt = η u xxtt − η u xxxxtt + α ( u n ) xx L C. = { X , X , X } X = ∂ t , X = t∂ t + 21 − n u ∂ u , X = ∂ x Reduction by X u ( x ) = ( ax + b ) /n X + ǫX , ǫ = ∓ η F (4) − η F ′′ + F − αF n = K ξ + K u = F ( ξ ) = F ( x − ǫt ) X + cX , c ∈ R η c ( n − F (6) + η c ( n − n + 3) F (5) + (cid:2) η ( n + 1) − η c ( n − (cid:3) F (4) − η c ( n − n + 3) F (3) + 2( n + 1) F u = t / (1 − n ) F ( ξ )+ h c ( n − − η ( n + 1) i F ′′ ξ = x − c ln t − α ( n − ( F n ) ′′ + c ( n − n + 3) F ′ = 0 X n + 1) (cid:0) η F (4) − η F ′′ + F (cid:1) − α ( n − ( F n ) ′′ = 0 u = t / (1 − n ) F ( x ) Case C.2 f ( u ) = αu − , α = ∓ u tt = η u xxtt − η u xxxxtt + α ( u − ) xx L C. X = ∂ t , X = t∂ t + u∂ u , = { X , X , X , X } X = t ∂ t + tu∂ u , X = ∂ x Reduction by X u ( x ) = ( ax + b ) − / X u ( t ) = at + bX + cX η c F (6) − ( η + 4 η c ) F (4) u = t / F ( ξ )( c ≥
0) +( η + 4 c ) F ′′ − α ( F − ) ′′ − F = 0 ξ = x − c ln tX η F (4) − η F ′′ + 4 α ( F − ) ′′ + F = 0 u = t / F ( x ) − X + X + dX η d F (6) − ( η + η d ) F (4) + ( η + d ) F ′′ u = p | t − | F ( ξ )( d ∈ R ) − α ( F − ) ′′ − F = 0 ξ = x + d tanh − tX + ǫX , ǫ = ∓ η F (4) − η F ′′ + F − αF − = K ξ + K u = F ( ξ ) = F ( x − ǫt )6 ase B: f ( u ) = α ln u, α = ∓ . We summarize the results in Table 1. For this case of f ( u ), the Lie algebra L B of theequation is again three-dimensional, and the basis of the algebra is presented in Table 1. Letus note that the nonzero commutation relation for this algebra is exactly the same as (2.15);therefore, the same Lie algebra is realized as the Case A by the vector fields that generate thegroup of transformations of the related equation. Case C: f ( u ) = α u n , α = ∓ , R ∋ n = 0 , . This case has two different branches.
C.1: n = −
3. In that case, the Lie symmetry algebra L C. is 3-dimensional, with the generatorsgiven in Table 1. The structure of the Lie algebra is the same with L A and L B . The nonzerocommutation relation is as in (2.15). C.2: n = − , f ( u ) = αu − . The symmetry generators of L C. are also admitted in this case.Besides, there arises a new symmetry generator and the equation admits a 4-dimensional Liealgebra L C. = { X , X , X , X } , and the basis of the Lie algebra is presented in Table 1. Thenonzero commutation relations are[ X , X ] = X , [ X , X ] = 2 X , [ X , X ] = X . (2.22)In [16] we see this algebra as A . ⊕ A and the optimal system of one-dimensional subalgebrasis { X } , { X } , { X + cX } , {− X + X + dX } , { X + ǫX } , (2.23)where c ≥ d ∈ R . We present the ODEs that are satisfied by the group-invariant solutionsof (1.1) under the transformations generated by these one-dimensional subalgebras in Table 1.We present the main result of this article in the following Theorem. Theorem 2.1
The Lie symmetry algebra L of the higher order Boussinesq equation (1.1) canbe 2-dimensional, 3-dimensional, or 4-dimensional.(i) The Abelian two-dimensional Lie algebra A is admitted as the invariance the algebra ofEq. (1.1) for any f ( u ) , and is realized by the Lie algebra with basis { X , X } = { ∂ t , ∂ x } .(ii) The three-dimensional Lie algebra A ⊕ A (where A is the one-dimensional Lie algebraand A is the two-dimensional non-Abelian algebra) is admitted as the symmetry algebraof Eq. (1.1) if f ( u ) respects one of the forms given in (2.11) , or, equivalently, (2.12) . Therelated generators of the Lie algebras for these cases are given in, respectively, for case Ain (2.14) , and in case B and case C.1 of Table 1 for the latter two. In all of these cases,the Lie algebra has the decomposition { X , X } ⊕ X .(iii) If f ( u ) = α ( βu + δ ) − + γ , or,equivalently, if f ( u ) = αu − , α = ∓ , then L is 4-dimensional, which is denoted as case C.2 in Table 1. The symmetry algebra has thestructure L C. = { X , X , X } ⊕ X ≃ sl(2 , R ) ⊕ R (2.24) which contains the simple algebra sl(2 , R ) as a subalgebra. iv) According to these results, maximal dimension of the Lie algebra of a higher-order Boussi-nesq equation belonging to the class (1.1) can be 4. Remark 2.1
Let us have a more close look to the case C.2, i.e., when f ( u ) = αu − . The Liealgebra with the basis L C. = { X = ∂ t , X = t∂ t + 12 u∂ u , X = t ∂ t + tu∂ u , X = ∂ x } (2.25) is the symmetry algebra of the equation u tt = η u xxtt − η u xxxxtt + α ( u − ) xx (2.26) regardless of the values of η , η and α . Therefore, when η = 0 , the symmetry algebra of theequation u tt = η u xxtt + α ( u − ) xx (2.27) is also 4-dimensional. Eq. (2.27) falls into the class (1.12) , the generalized modified Boussinesqequation, analyzed in [15]. In their classification of the symmetry algebras of Eq. (1.12) , theyarrive at the same forms of f ( u ) given in (2.11) , and, according to their results, for theseforms of f ( u ) the symmetry algebras are at most three-dimensional. As far as we can see, thiswork does not consider the case n = − , f ( u ) = αu − separately and seems to miss the fourthsymmetry generator X = t ∂ t + tu∂ u appearing.Therefore, we should state that, the above Theorem is also valid when η = 0 , hence themaximal dimension of the symmetry algebra of the generalized modified Boussinesq equation (1.12) , studied in [15], is equal to 4. The simple algebra L C. with the decomposition (2.24) isalso admitted as an invariance algebra in the case f ( u ) = αu − .This remark should be considered as a complementary result of our main Theorem to thefindings in [15]. In this Section we present some exact solutions to the equation (1.1). We consider the nonlin-earity f ( u ) = αu + u , which gives rise to the equation u tt = u xx + η u xxtt − η u xxxxtt + α ( u ) xx (3.1)where α = 0 is any constant. The reason for the inclusion of the term u xx is obvious, as onecan see from the literature review. For our analysis above, we had considered the u xx term tocovered by the nonlinearity f ( u ), just on a purpose of bookkeeping. The quadratic nonlinearity u can be interpreted like that one considers the stress-strain function of the physical model tobe having a quadratic nonlinearity; see [2].We aim at finding traveling wave solutions to (3.1), therefore we assume u = F ( ξ ) with ξ = kx − ct (which amounts to finding the group-invariant solutions under the action of thetransformation produced by the generator c∂ x + k∂ t ). Putting this ansatz in (3.1) and integratingthrice, we obtain η k c h F ′′′ F ′ −
12 ( F ′′ ) i − η k c F ′ ) − αk F + c − k F = K . (3.2)8e chose the coefficients of the first two integrations as zero and kept only the last one, K .Since this equation does not contain the independent variable ξ , it can be integrated once bysetting F ′ = W ( F ) and treating W as the dependent variable and F as independent. However,the resulting equation is so complicated that we could not proceed with it further.At this point, let us briefly outline the results of [17] in their Section 4, in which theyconsider a 2 + 1-dimensional Boussinesq type equation U tt − U xx − U yy − α ( U ) xx − αU xxxx − αǫ U xxxxxx = 0 . (3.3)In order to find traveling wave solutions of this equation, they propose the following ansatz: U = a + a ϕ ( ζ ) + a ϕ ( ζ ) + a ϕ ( ζ ) + a ϕ ( ζ ) , (3.4a) ζ ( x, y, t ) = x + y − κ t, (3.4b) (cid:16) dϕdζ (cid:17) = c + c ϕ ( ζ ) + c ϕ ( ζ ) + c ϕ ( ζ ) + c ϕ ( ζ ) . (3.4c)Notice that, if successful, this ansatz will produce trigonometric, hyperbolic or elliptic typesolutions due to the Eq. (3.4c) that ϕ ( ζ ) satisfies. It is easy to see that, under the travelingwave ansatz, Eq. (3.1) with u = F ( kx − ct ) and Eq. (3.3) with U = U ( x + y − κ t ) reduce toordinary differential equations which are the same up to coefficients. Therefore we adapt themethodology in [17] find the exact solutions to (3.1). Let us stress that we obtained some moresolutions which were not mentioned there. Therefore, for (3.2) we propose F ( ξ ) = a + a ϕ ( ξ ) + a ϕ ( ξ ) + a ϕ ( ξ ) + a ϕ ( ξ ) , ξ ( x, t ) = kx − ct, (3.5a) (cid:16) dϕdξ (cid:17) = c + c ϕ ( ξ ) + c ϕ ( ξ ) + c ϕ ( ξ ) + c ϕ ( ξ ) = P ( ϕ ( ξ )) . (3.5b)Upon this substitution, in the resulting expression we express all derivatives of ϕ ( ξ ) in termsof ϕ using (3.5b). Afterwards, we look for the possibility that coefficients of ϕ j , j = 0 , , , ... vanish. Below are the several cases we examined. We assume c = c = c = 0 and a = a = a = 0. We find two main branches for theremaining constants a , a , c , c , k and c .The first set of parameters is a = 0 , (3.6a) a = 840 c c (169 η − η )169 α , (3.6b) c = 13 η c (169 η − η ) , (3.6c) k = c (cid:16) − η η (cid:17) (3.6d)9nd the second set of possible parameters is a = − η α (169 η + 36 η ) , (3.7a) a = 840 c c (169 η + 36 η )169 α , (3.7b) c = 13 η c (169 η + 36 η ) , (3.7c) k = c (cid:16) η η (cid:17) . (3.7d)In both cases, c and c are arbitrary. Equation (3.5b) reduces to dϕ | ϕ | p c + c ϕ = dξ , andit is integrated in three different ways depending on the signs of c and c . Observe that c = η / (52 k η ). Although the physical derivation of (1.1) gives η , η > c < Case I.a
In case c > c >
0, we obtain ϕ ( ξ ) = (cid:18) c c (cid:19) / cosech (cid:16) ǫ √ c ( ξ − ξ ) (cid:17) (3.8)and the solution to (3.1) is u ( x, t ) = a + a c c cosech (cid:16) ǫ √ c ( kx − ct − ξ ) (cid:17) (3.9)with ǫ = ∓
1. The set of four constants a , a , c , c can be chosen as in (3.6) or (3.7). Case I.b If c > c <
0, we obtain ϕ ( ξ ) = (cid:18) c − c (cid:19) / sech (cid:16) ǫ √ c ( ξ − ξ ) (cid:17) (3.10)and hence u ( x, t ) = a + a c c sech (cid:16) ǫ √ c ( kx − ct − ξ ) (cid:17) . (3.11)with ǫ = ∓ a = 0, this result is in the same form with the exact solutionpresented in [3]. Case I.c
Finally, when c < c > ϕ ( ξ ) = (cid:18) − c c (cid:19) / sec (cid:16) ǫ √− c ( ξ − ξ ) (cid:17) . (3.12)(3.12) also appears in [17]. The solution to (3.1) becomes u ( x, t ) = a + a c c sec (cid:16) ǫ √− c ( kx − ct − ξ ) (cid:17) . (3.13)In the solutions (3.9), (3.11) and (3.13) the valid sets of parameters are those given in (3.6)and (3.7). 10 .2 Elliptic type solutions We assume c = 0 and a = a = a = 0. We find the following values for the remainingconstants a , a , c , c and c ; a = 169 η ( c − k + 42 c k c c η ) − c η k αη , (3.14a) a = 105 c c k η α , (3.14b) c = 4 c η k c η , (3.14c) c = ε Rc , (3.14d) c = η k η , (3.14e) R = p c − k ) η − c η √ c k η , (3.14f)where ε = ± c is arbitrary.Now that we have determined the constants appearing in (3.5) successfully, we need to integrate(3.5b), which takes the form ˙ ϕ = c + c ϕ + c ϕ + c ϕ = P ( ϕ ) , (3.15)and find ϕ ( ξ ) and hence u ( x, t ). Evaluation of the integral of (3.15) depends on the factorizationof the polynomial P ( ϕ ). Assume that ϕ , ϕ and ϕ are zeros of the equation P ( ϕ ) = 0, forwhich the discriminant is∆ = 18 c c c c + c c − c c − c c − c c . (3.16)Making use of (3.14) we obtain∆ = − εR (80 η + 7943 εk Rη η + 2856100 k R η )714025 c k η . (3.17)∆ = 0 if R = 0. When we analyze this branch, the coefficients in (3.14) give results the sameas in Case I.Let ε = −
1. The sign of ∆ is determined by the sign of the term inside the paranthesisin (3.17). When we consider this term as a second-degree polynomial in R and calculate itsdiscriminant, we see it is negative, therefore the polynomial is always positive. Hence ∆ > ϕ , ϕ , ϕ . Then we can factorize(3.15) as ˙ ϕ = P ( ϕ ) = c ( ϕ − ϕ )( ϕ − ϕ )( ϕ − ϕ ) . (3.18) Case II.a
Let c >
0. In order that (3.18) makes sense, the right hand side must benonnegative. Therefore we should consider the intervals ϕ > ϕ > ϕ > ϕ and ϕ > ϕ > ϕ >ϕ when integrating (3.18). Let us first write dϕ p c ( ϕ − ϕ )( ϕ − ϕ )( ϕ − ϕ ) = ǫdξ (3.19)11here ǫ = ∓
1. In the first hand, when ϕ > ϕ > ϕ > ϕ , using the results available in thehandbook [18], we obtain Z ϕϕ dτ p c ( τ − ϕ )( τ − ϕ )( τ − ϕ ) = 1 √ c g sn − (cid:16)r ϕ − ϕ ϕ − ϕ , m (cid:17) (3.20)for the integration of the left hand side of (3.19), where g = 2 √ ϕ − ϕ , m = ϕ − ϕ ϕ − ϕ . Thisgives rise to the elliptic function solution ϕ to (3.15), ϕ ( ξ ) = ϕ nc (cid:16) ǫ √ c g ( ξ − ξ ) , m (cid:17) − ϕ tn (cid:16) ǫ √ c g ( ξ − ξ ) , m (cid:17) , (3.21)and hence the solution to (3.1) can be written as follows u ( x, t ) = a + a (cid:20) ϕ nc (cid:16) ǫ √ c g ( kx − ct − ξ ) , m (cid:17) − ϕ tn (cid:16) ǫ √ c g ( kx − ct − ξ ) , m (cid:17)(cid:21) . (3.22) Case II.b
When the coefficient c >
0, for ϕ > ϕ > ϕ > ϕ we obtain Z ϕϕ dτ p c ( ϕ − τ )( ϕ − τ )( τ − ϕ ) = 1 √ c g sn − (cid:16)r ϕ − ϕ ϕ − ϕ , m (cid:17) (3.23)where g = 2 √ ϕ − ϕ , m = ϕ − ϕ ϕ − ϕ . After we find ϕ ( ξ ) = ϕ sn (cid:16) ǫ √ c g ( ξ − ξ ) , m (cid:17) + ϕ cn (cid:16) ǫ √ c g ( ξ − ξ ) , m (cid:17) (3.24)and hence the solution to (3.1) can be written as follows: u ( x, t ) = a + a (cid:20) ϕ sn (cid:16) ǫ √ c g ( kx − ct − ξ ) , m (cid:17) + ϕ cn (cid:16) ǫ √ c g ( kx − ct − ξ ) , m (cid:17)(cid:21) . (3.25) Case II.c
If the coefficient c <
0, working on the interval ϕ > ϕ > ϕ > ϕ we find thefollowing: Z ϕ ϕ dτ p − c ( ϕ − τ )( ϕ − τ )( ϕ − τ ) = 1 √− c g sn − (cid:16)r ϕ − ϕϕ − ϕ , m (cid:17) (3.26)where g = 2 √ ϕ − ϕ , m = ϕ − ϕ ϕ − ϕ . This gives us ϕ ( ξ ) = ϕ nc (cid:16) ǫ √− c g ( ξ − ξ ) , m (cid:17) − ϕ tn (cid:16) ǫ √− c g ( ξ − ξ ) , m (cid:17) (3.27)therefore the solution to (3.1) can be written as follows: u ( x, t ) = a + a (cid:20) ϕ nc (cid:16) ǫ √− c g ( kx − ct − ξ ) , m (cid:17) − ϕ tn (cid:16) ǫ √− c g ( kx − ct − ξ ) , m (cid:17)(cid:21) . (3.28)12 ase II.d For c <
0, on the interval ϕ > ϕ > ϕ > ϕ we see that we can proceed toobtain Z ϕϕ dτ p − c ( ϕ − τ )( τ − ϕ )( τ − ϕ ) = 1 √− c g sn − (cid:16)s ( ϕ − ϕ )( ϕ − ϕ )( ϕ − ϕ )( ϕ − ϕ ) , m (cid:17) (3.29)where g = 2 √ ϕ − ϕ , m = ϕ − ϕ ϕ − ϕ . This immediately results in ϕ ( ξ ) = ϕ nd (cid:16) ǫ √− c g ( ξ − ξ ) , m (cid:17) − ϕ m sd (cid:16) ǫ √− c g ( ξ − ξ ) , m (cid:17) (3.30)producing the solution to (3.1) as u ( x, t ) = a + a (cid:20) ϕ nd (cid:16) ǫ √− c g ( kx − ct − ξ ) , m (cid:17) − ϕ m sd (cid:16) ǫ √− c g ( kx − ct − ξ ) , m (cid:17)(cid:21) . (3.31)In case ε = 1 we have ∆ <
0. Therefore the polynomial (3.15) has one real zero ϕ and twocomplex conjugate zeros ϕ , ϕ . Case II.e
If the coefficient c > Z ϕϕ dτ p c ( τ − ϕ )[( τ − b ) + a ] = 1 √ c g cn − (cid:16) A + ϕ − ϕA − ϕ + ϕ , m (cid:17) (3.32)where b = ϕ + ϕ a = − ( ϕ − ϕ ) A = ( b − ϕ ) + a , g = 1 √ A , m = A + b − ϕ A .After that we find ϕ ( ξ ) = ϕ + A − cn (cid:0) ǫ √ c g ( ξ − ξ ) , m (cid:1) (cid:0) ǫ √ c g ( ξ − ξ ) , m (cid:1) (3.33)and hence the solution to (3.1) can be written as u ( x, t ) = a + a " ϕ + A − cn (cid:0) ǫ √ c g ( kx − ct − ξ ) , m (cid:1) (cid:0) ǫ √ c g ( kx − ct − ξ ) , m (cid:1) . (3.34) Case II.f
If the coefficient c < Z ϕ ϕ dτ p − c ( ϕ − τ )[( τ − b ) + a ] = 1 √− c g cn − (cid:16) A − ϕ + ϕA + ϕ − ϕ , m (cid:17) (3.35)where b = ϕ + ϕ a = − ( ϕ − ϕ ) A = ( b − ϕ ) + a , g = 1 √ A , m = A − b + ϕ A .After this we get ϕ ( ξ ) = ϕ − A − cn (cid:0) ǫ √− c g ( ξ − ξ ) , m (cid:1) (cid:0) ǫ √− c g ( ξ − ξ ) , m (cid:1) (3.36)and hence the solution to (3.1) turns out to be u ( x, t ) = a + a ϕ − A − cn (cid:16) ǫ √− c g ( kx − ct − ξ ) , m (cid:17) (cid:16) ǫ √− c g ( kx − ct − ξ ) , m (cid:17) . (3.37)13 Conclusion
In this work, we considered a higher-order Boussinesq equation with an arbitrary nonlinearity f ( u ). We determined the canonical forms of f ( u ) so that the equation admits certain finite-dimensional Lie algebras. We proved that, within this family, the maximal dimension of theLie algebra of the equation is equal to 4, and this is realized when f ( u ) assumes some definiteform. This result is also true in the case of a generalized modified Boussinesq equation, when η = 0.After that, we considered the case where f ( u ) is a second degree polynomial in u . Weproduced some exact solutions which were expressed in terms of trigonometric, hyperbolic andelliptic functions. To our knowledge, among the nine solutions we were able to find, except theone in (3.11), all the other eight given in Case I and Case II appear in literature the first timefor the higher-order Boussinesq equation with quadratic nonlinearity.In this manuscript, we restricted ourselves to a subclass of (2.11c), by searching for the exactsolution in the case f ( u ) = u + αu . Actually, the analysis of the reduced equation for n = 2for Case C.1 in Table 1 which were obtained by the infinitesimal generator X + ǫX wouldfollow similar lines to the analysis in Section 3. Mainly due to the complicated nature of thereduced equations, we do not perform a further analysis for the reduced equations in this work.Regarding the families (2.11) or equivalently (2.12), which mean some HBq equations withcertain symmetries, one can ask another question: Do these canonical forms of nonlinearities f ( u ) have any physical meaning? As far as we know, the answer is affirmative when f haspower-type nonlinearities like f ( u ) = u + αu , etc., and that has been the main reason forwriting Section 3 of this manuscript. The analysis of the other reduced equations remains stillopen. References [1] P. Rosenau. Dynamics of dense discrete systems: High order effects.
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