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n (ut + αux u ux + βuxx) + uyy =0. (2) − x which is the natural generalization of the generalized Benjamin-Ono equation [19]. For the two equations (1), (2) we determine the Lie point symmetries while we prove the existence of travel-wave similarity solutions for every value of parameter n 1 and arbitrary depth α. The plan of the paper is as follows. ≥ In Section 2, for the convenience of the reader we briefly discuss the basic properties and definitions of the theory of Lie point symmetries. Sections 3 and 4, include the main new material of our analysis, where we present the algebraic properties for equations (1) and (2). Finally in Section 5 we draw our conclusions.

2 Preliminaries

In this section, we briefly discuss the theory of Lie point symmetries of differential equations which is the main mathematical tool that we apply in the following.

Consider function Φ to describe the map of a one-parameter point transformation such as u′ (t′, x′,y′) = Φ (u (t,x,y); ε) where the infinitesimal transformation is expressed as follows

t t′ = t + εξ (t,x,y,u) (3) x x′ = x + εξ (t,x,y,u) (4) y y′ = y + εξ (t,x,y,u) (5) u = u + εη (t,x,y,u) (6)

where ε is the infinitesimal parameter, that is, ε2 0. → From the latter one-parameter point transformation we can define the infinitesimal generator ∂t ∂x ∂y ∂u X = ′ ∂ + ′ ∂ + ′ ∂ + ′ ∂ , (7) ∂ε t ∂ε x ∂ε y ∂ε u from where the map Φ can be written as follows

Φ (u (t,x,y); ε)= u (t,x,y)+ εX (u (t,x,y)) , (8)

2 that is, Φ (u (t,x,y); ε) u (t,x,y) X (u (t,x,y)) = lim − . (9) ε 0 ε → The latter expression defines the Lie derivative of the function u (t,x,y) with respect to the vector field X, also noted as LX u. When

LX u =0 (10) then we shall say that u (t,x,y) is invariant under the action of the one-parameter point transformation with generator the vector field X. In terms of differential equations, i.e.

(u,ut,ux,uy, ..) = 0; (11) H then the symmetry condition reads [n] LX ( )=0or X ( )=0, (12) H H where X[n] describes the nth prolongation/extension of the symmetry vector in the jet-space of variables

u,ut,ux, ... defined as { } X[n] X η[1]∂ + ... + η[n]∂ , = + i ui uiiij ...in

∂u i where ui = ∂zi , z = (t,x,y) and

[n] [n 1] j η = Diη − uii2...i −1 Dj ξ , i 1. (13) i − n   The main application of the Lie point symmetries is based on the determination of the Lie invariants which are used to define similarity transformations and simplify the given differential equation. The exact solutions which follow by the application of the Lie point symmetries are called similarity solutions. If X is an admitted Lie point symmetry, the solution of the associated Lagrange’s system, dt dx dy du = = = , (14) ξt ξx ξy η

provides the zeroth-order invariants, U A[0] (t,x,u) which are applied to reduce the number of independent variables in partial differential equations, or the order in the case of ordinary differential equations. For more details on the symmetry analysis of differential equations we refer the reader to the standard references [20–22].

3 Point symmetries of the Camassa-Choi equation

From the symmetry condition (12) for the CC equation (1) and for the one-parameter point transformation 1 2 3 with generator X = ξ (t,x,y,u) ∂t + ξ (t,x,y,u) ∂x + ξ (t,x,y,u) ∂y + η (t,x,y,u) ∂u, we find the following

3 system of differential equations

1 2 3 1 3 ξ,u =0 , ξ,u =0 , ξ,u =0 , ξ,x =0 , ξ,x =0 , η,uu =0 , (15)

ξ3 +2ξ1 =0 , ξ3 +3ξ2 =0 , 2ξ2 ξ1 =0 , 2ξ2 ξ3 =0, (16) ,x y ,y ,x ,x − ,t ,y − ,t η + (α u) η + η + η =0 , (17) ,xxx − ,xx ,yy ,tx η ξ1 =0 , 2η ξ3 =0 , (18) ,xu − ,yy ,yu − ,yy 3η 3ξ2 + (α u) ξ2 ξ2 η =0 , (19) ,xu − ,xx − ,x − ,t − 3η + (α u) η ξ2 η =0 , (20) ,xuu − ,uu − ,x − ,u 3η ξ2 + (α u) 2η ξ2 + η ξ2 2η ξ2 =0. (21) ,xxu − ,xxx − ,xu − ,xx ,tu − ,yy − ,x − ,tx  1 3 The generic solution of the latter system is X = (c + c (2t)) ∂t+ c x + c φ (t) c ψ (t) y ∂x+ c y + c ψ (t) ∂y+ 1 2 2 3 − 2 4 t 2 2 4 1 c (α u) c φ (t)+ c ψ (t) y ∂u , where c ,c ,c ,c are constants of integration and φ (t), ψ (t) are ar-  2 − − 3 t 4 2 tt 1 2 3 4 bitrary functions.  Therefore, the Lie point symmetries of the CC equation (1) are 3 X = ∂t , X =2t∂t + x∂x + y∂y (u α) ∂u , (22) 1 2 2 − − 1 1 X (φ)= φ (t) ∂x φ (t) ∂u , X (ψ)= ψ (t) ∂y ψ (t) y∂x + ψ (t) y∂u . (23) 3 − t 4 − 2 t 2 tt Surprisingly, the CC equation admits infinity Lie point symmetries. The commutators of the Lie point symmetries are

[X1,X2] = 2X1 , [X1,X3 (φ)]=(X3 (φt)) , [X1,X4 (ψ)] = (X4 (ψt)) , (24) 3 [X ,X (φ)] = X (φ 2tφ ) , [X ,X (ψ)] = X ψ 2tψ , [X (φ) ,X (ψ)]=0 , (25) 2 3 3 − t 2 4 4 4 − t 3 4 1 [X (φ) ,X′ (χ)] = 0 , [X (ψ) ,X (ξ)] = (X (ξψ ψξ )) . (26) 3 3 4 4 2 3 t − t from where we observe that they form an infinity-dimensional . The existence of the infinity number of symmetries it is not a real surprise. From X we determine the similarity transformation u = (ln φ (t)) x + 3 − ,t 1 φ,tt 2 U (t,y) where U (t,y) = 2 φ y + U1 (t) y + U0 (t) solves the reduced equation, functions U1 (t) ,U0 (t) are arbitrary functions. In the special case where φ (t) and ψ (t) are constants, without loss of generality we assume that φ (t) = ψ (t) = 1, the Lie point symmetries are simplified as 3 X′ = ∂t , X′ =2t∂t + x∂x + y∂y (u α) ∂u , X′ = ∂x , X′ = ∂y (27) 1 2 2 − − 3 4

4 Table 1: Commutators for Lie point symmetries of CC which form a finite-dimensional Lie algebra

[ , ] X1′ X2′ X3′ X4′

X1′ 0 2X1′ 0 0 3 X2′ 2X1′ 0 X3′ 2 X3′

X′ 0 X′ 0 0 3 − 3 3 X′ 0 X′ 0 0 4 − 2 3

with commutators

[X1′ ,X2′ ] = 2X1′ , [X1′ ,X3′ ]=0 , [X1′ ,X4′ ]=0 , (28) 3 [X′ ,X′ ] = X′ , [X′ ,X′ ]= X′ , [X′ ,X′ ]=0. (29) 2 3 3 2 4 2 3 3 4 However, there are not any finite-dimensional closed Lie algebras for arbitrary functions of φ (t) and ψ (t). The commutators of the latter finite-dimensional Lie algebra are presented in Table 1.

ω1t ω2t Let us demonstrate that by assuming φ (t)= φ1 + φ2e and ψ (t)= ψ1 + ψ2e . Then from (22), (23) it follows that the CC equation admits six Lie point symmetries which are the vector fields

′ 2 ω1t ω1t ω2 ω2 X′ , X′ , X′ , X′ , X′ = e (∂x ω ∂u) , X = e ∂y y∂x + y∂u (30) 1 2 3 4 5 − 1 6  − 2 2  with commutators (28), (29) and

ω1t [X′ ,X′ ] = ω X′ , [X′ ,X′ ]= ω X′ , [X′ ,X′ ]= e ((1 ω t) ∂x + ω (1+2ω t) ∂u) , (31) 1 5 1 5 1 6 2 6 2 5 − 1 1 1 ω2t (3 4ω2t) (1+4ω2t) (5+4ω2t) 2 [X′ ,X′ ] = e − ∂y + ω y∂x ω y∂u , (32) 2 6  2 4 2 − 4 2  ′ ′ ω2 ω2t [X3′ ,X5′ ] = 0 , X3,X6 =0 , [X4′ ,X6′ ]= e (∂t ω2∂u) . (33) h i 2 − from where it is clear that the symmetry vectors (30) do not form a closed Lie algebra. We want to constraint functions φ (t) , and ψ (t) such that the admitted Lie symmetries to form a closed Lie algebra of five-dimension with a different basis. In particular we focus on the case where the coefficients of the commutators (24)-(26) are constants. Thus we end up with the system of equations

φ = c φ , φ = c φ 2tφ or φ = c′ φ , φ = c′ (φ 2tφ ) , (34) { 1 t 2 − t} { 1 t t 2 − t } and 3 3 ψ = c ψ , ψ = c ψ 2tψ or ψ′ = c ψ , ψ′ = c ψ 2tψ , (35)  3 t 4 4 − t  3 t t 4 4 − t with constraint equations

ξ = ξψ ψξ , where ξ = φ, or ξ = φ or ξ = (φ 2tφ ) . (36) t − t t − t

5 Therefore, from (34), (35) and (36) it follows that the unique possible admitted five-dimensional Lie algebra

is that of (27) for φ (t) = const. and ψ (t) = ψ0 + ψ1t. Of course there are additional finite dimensional Lie algebras, for instance any set of generators constructed by X3 form a Lie algebra; however this specific five- dimensional Lie algebra has the novelty that it can provide a plethora of different similarity transformations,

while for instance the similarity transformations which follow by X3 are all of the same family.

Proposition 1 The CC equation (1) is invariant under infinity Lie point symmetries which form the Lie

algebra A2,1 s A s A in the Morozov-Mubarakzyanov classification scheme [25–28]. However, there { ⊗ ∞ ⊗ ∞} 1 exists a five-dimensional subalgebra consisted by the vector fields X ,X ,X′ ,X′ ,X = t∂y y∂x and form 1 2 3 4 5 − 2 ab  the Lie algebra A5,19 in the Patera-Winternitz classification scheme [29,30]. This five-dimensional Lie algebra provides the maximum number of alternative families of similarity transformations.

As we shall see in the following, this five-dimensional Lie algebra plays a significant role in the study of the Lie point symmetries for the GCC equation (2). We proceed with the application of the Lie point symmetries for the derivation of similarity solutions.

3.1 Similarity Solutions for the Camassa-Choi equation

Let us not apply the Lie point symmetries found in the previous section in order to find similarity solutions for the CC equation (1). The CC equation is a third equation of three independent variables. By applying the Lie point symmetries in partial differential equations we reduce the number of the independent variables. Hence, in order to reduce the CC equation to an ordinary differential equation we should apply two symmetry vectors. However, not all the symmetry vectors survive through the reduction process. In particular, if a given

differential equation admits the two symmetry vectors Γ1, Γ2 with commutator [Γ1, Γ2] = cΓ2, where c may

be zero, then reduction of the differential equation with respect to the symmetry vector Γ2 provides that the reduced equation inherits the symmetry vector Γ1, while reduction with Γ1 provides a differential equation where Γ is not a point symmetry when c = 0 [24]. It is clear, that if we want to perform a second reduction 2 6 for the differential equation we start by considering the symmetry vector Γ2.

Therefore, by using the results of Table 1 we find that the reduction with the symmetry vectors X′ ,X′ ,X′ ,X′ + X′ { 1 3 4 3 4} gives reduced equations which inherits some of the symmetries of the original equation. However, the applica- tion of the symmetry vectors X′ ,X′ ,X′ gives time-independent or static solutions which are not solutions { 1 3 4} of special interests solutions. Hence, we focus on the reduction which follows by the symmetry vector X3′ + X4′ .

From the Lie point symmetry X34 = X3′ + X4′ we calculate the invariants

t , w = y x, u = U (t, w) . (37) − By using the latter invariant functions equation (1) is reduced to the following partial differential equation

2 Uwww + (Uw) (1 U + h ) Uww + Uwt =0. (38) − − 0

6 In order to proceed with the reduction we should derive the Lie point symmetries of (38). Hence, by applying the Lie symmetry condition we find that equation (38) is invariant under the Lie point symmetries

Z = ∂t,Z =2t∂t + w∂w + (h +1 U) ∂U , (39) 1 2 0 − 2 Z = t ∂t + tw∂w + [(h +1 U) t + w] ∂U , (40) 3 0 − Z4 = φ (t) ∂w + φt∂U . (41)

Vector fields Z1,Z2 and Z4 are reduced symmetries, while Z3 is a new symmetry for the reduced equation (38).

It is important to mention that Z4 describes an infinity number of symmetries, hence the reduced equation (38) admits infinity number of Lie point symmetries as the “mother” equation (1). On the other hand, Lie point symmetries Z ,Z ,Z form a closed Lie algebra, known as SL (2, R). { 1 2 3} φt The application of Z4 in (38) provides the linear second-order ODE φtt = 0, where U (t, w)= U0 (t)+ φ w, where U0 (t) is an arbitrary function. Therefore, the similarity solution is derived to be

φ1 U (t, w)= U0 (t)+ w. (42) φ1t + φ0

Reduction with respect the symmetry vector Z1 of equation (38) provides the third-order ODE

2 Ywww + (Yw) Y Yww =0 ,U (t, w)= Y (w)+1+ h , w = x (43) − 0

which admit two point symmetries the reduced symmetries Z2, and Z3. Equation (43) can be integrated as follows

Yww + Y Yw + Y0 =0, (44)

where the latter equation can be solved in terms of quadratics. Indeed for the integration constant Y0 = 0, the general solution is w w Y (w)= Y tanh − 0 , (45) 0  2c  while in general equation (44) becomes 1 Y + Y 2 + Y w + Y =0. (46) w 2 0 1

The application of the Lie symmetry vector Z2 provides the reduced third-order ODE

2Y¯σσσ + σ 2 1+ Y¯ Y¯σσ 2Y¯ =0 , (47) − −  where now U (t, w)=1+ h + Y (σ) , σ = w . The latter equation can be easily integrated as follows 0 √t √t

2Y¯σσ Y¯ 2Y¯ σ Y¯ + Y¯ = 0 (48) − − − 0  or 2 2Y¯σ + Y¯ σY¯ + Y¯ σ + Y¯ =0. (49) − 0 1

7 In a similar way, the reduction with respect to the Lie symmetry vector Z3 gives the solution w Y (λ) w U (t, w)= + h +1+ ′ , λ = , (50) t 0 t t where Y ′ (λ) is given by the following first-order ODE

1 2 Y ′ + (Y ′) + Y λ + Y =0. (51) λ 2 0 1 It comes as no surprise that the reduction with the three elements of the SL (2, R) provides similar reduced equations. That is because the three symmetry vectors are related with similarity transformations as well as also the reduced equations are related, for more details we refer the reader to [23].

Finally, reduction with the vector field Z1 + Z4, for φ (t) = 1, provides travel-wave solution and the reduced equation is that of (43) where w = t x. − Similarly, the reduction of CC equation (1) with respect the symmetry vector X14 = X1′ + X4′ , provides a travel-wave solution, as before. Therefore, we conclude that travel-wave solutions exist for the CC equation. We proceed our analysis by studying the invariant point transformations for the GCC equation (2).

4 Point symmetries of the generalized Camassa-Choi equation

The Lie point symmetries of the GCC equation (2) are 3 1 Y = ∂t , Y =2t∂t + x∂x + y∂y u∂u (52) 1 2 2 − n

Y = ∂x , Y = ∂y , Y =2t∂y y∂x , (53) 3 4 5 − when α = 0 and 3 1 Y¯ = ∂t , Y¯ = ∂x , Y¯ =2t∂t + (x + αt) ∂x + y∂y u∂u (54) 1 2 2 2 − n

Y¯ = ∂x , Y¯ = ∂y , Y¯ =2t∂y y∂x , (55) 3 4 5 − for α = 0. 6 The corresponding commutators for the admitted Lie symmetries are presented in Table 2. We observe that the two admitted Lie algebras are different. For α = 0 the Lie symmetries form the Lie algebra Ab and for 6 5,23 ab α = 0, the Lie symmetries form the Lie algebra A5,19 in the Patera-Winternitz classification scheme [29, 30]. When the parameter α is zero, the Lie point symmetries Y , Y , Y , Y are these which form a finite- { 1 2 3 4} dimensional Lie algebra for the CC equation (1), that is, vector fields (27). However, When α = 0 things are 6 different. The fifth symmetry Y5 is a case of X4 (ψ) with ψ (t)= t. Indeed, the admitted Lie point symmetries by the GCC are those which form the maximum finite-dimensional Lie algebra for the CC equation. We continue our analysis by applying the Lie point symmetries to determine similarity solutions for the GCC equation.

8 Table 2: Commutators of the admitted Lie point symmetries by the GCC

[ , ] Y¯1 Y¯2 Y¯3 Y¯4 Y¯5

Y¯1 0 2Y1 + αY3 0 0 2Y¯4 Y¯ 2Y¯ + αY 0 Y¯ 3 Y¯ 1 Y¯ 2 − 1 3 − 3 − 2 4 2 5  Y¯3 0 Y¯3 0 0 0 Y¯ 0 3 Y¯ 0 0 Y¯ 4 2 4 − 3 Y¯ 2Y¯ 1 Y¯ 0 Y¯ 0 5 − 4 − 2 5 3

4.1 Similarity Solutions for the generalized Camassa-Choi equation

As in the case of the CC we consider the similarity transformation provided by the vector field Y34 = Y3 + Y4, because it is the similarity transformation which provides a reduced equation which inherits symmetry vectors. Hence, we find that the GCC equation (2) is reduced to

n 1 2 n Uwww + Uwt + nU − (Uw) + (U +1 α) Uww =0, (56) − where u (t,x,y)= U (t, w) and w = x y. We observe that equation (56) reduces into (38) when n = 1. − For n =1, we calculate the Lie point symmetries of (56) which are we found to be 6 1 Z¯ = ∂t, Z¯ = ∂w and Z¯ =2t∂t + (t (1 + α) w) ∂w U∂u. 1 2 3 − − n

The application of the Lie symmetry vector Z¯12 = ∂t + ∂w in (56) provides the travel-wave solution

n 1 2 n Yσσσ + nY − (Yσ) + (Y α 2) Yσσ =0 ,U (t, w)= Y (σ) , σ = w t. (57) − − − The latter equation can be easily integrated by quadratures as follows 1 Y + Y n+1 +(2+ A) Y + Y σ + Y =0. (58) σ n +1 1 0 On the other hand, the reduction of (56) with respect to the similarity transformation provided by the vector

field Z¯3 provides 1 w + t (1 + α) U (t, w)= H (ζ) t− 2n , ζ = (59) √t where H (ζ) satisfies the third-order ordinary differential equation

n 2 n 2nHHζζζ + nH (2H ζ) Hζζ (n + 1) H 2n H Hz Hz =0. (60) − − −  Equation (60) can be integrated as follows

1 n ζ Hζζ H + H H Hζ + H =0. (61) − 2n  − 2  1

9 n=2 n=2 n=5

Figure 1: Qualitative evolution of H(ζ) as it is given by the differential equation (61) for initial conditions

H (0) = 1 and Hζ (0) = 0.5. The plots are for H = 0 and n = 2 (red line), n = 3 (blue line) and n = 5 − 1 (yellow line).

The latter equation does not admit any point symmetry and we cannot perform further reduction. However in 1 we present some numerical solutions. What is also important to mention is that in equation (61) parameter α plays no role. Hence the same reduction holds and for the case α = 0.

5 Conclusions

In this work, we applied the theory of symmetries of differential equations in order to determine exact similarity solutions for the Camassa-Choi equation (1) and its generalization (2). CC equation describes weakly nonlinear internal waves in a two-fluid system and it can be seen as the two-dimensional generalization of the Benjamin- Ono. For the CC equation we found that it is invariant under an infinity-dimensional Lie algebra, with maximum finite Lie subalgebra of dimension five. That five-dimensional subalgebra is the one which form the complete group of invariant one-parameter point transformations for the GCC equation. We apply the Lie point symmetries and we prove the existence of similarity solutions in the two-dimensional plane x, y . Specifically, we found that the similarity solutions can be expressed in terms of quadratures. { } Surprisingly, the CC equation under the application of similarity transformations can be reduced into a three-dimensional ordinary differential equation which is invariant under the SL (3, R), where all the possible

10 reductions provide similarity solutions related under point transformations. In a future work we plant to study the physical properties of those new similarity solutions.

References

[1] W. Choi and R. Camassa. J. Fluid Mech. 313, 83 (1996)

[2] B. Harrop-Griffiths and J.L. Marzula, Nonlinearity 31, 1868 (2018)

[3] A. Paliathanasis, K. Krishnakumar, K.M. Tamizhmani and P.G.L. Leach, Mathematics 4, 28 (2016)

[4] X. Xin, Appl. Math. Lett. 55, 63 (2016)

[5] X. Xin, Acta Phys. Sin. 65, 240202 (2016)

[6] N. Kallinikos and E. Meletlidou, J. Phys. A: Math. Theor. 46, 305202 (2013)

[7] S. Jamal and A. Paliathanasis, J. Geom. Phys. 117, 50 (2017)

[8] G.M. Webb, J. Phys A: Math. Gen. 23, 3885 (1990)

[9] P.G.L. Leach, J. Math. Anal. Appl. 348, 487 (2008)

[10] M.S. Velan and M. Lakshmanan, Int. J. Non-linear Mech. 31, 339 (1996)

[11] M. Pandey, Int. J. Nonl. Sc. Num. Sim. 16, 337 (2015)

[12] A. Paliathanasis, Zeitschrift f¨ur Naturforschung A, in press [DOI:10.1515/zna-2019-0063]

[13] V.N. Chetverikov, Acta Appl. Math. 56, 121 (1999)

[14] S. Szatmari and A. Bihlo, Comm. Nonl. Sci. Num. Sim. 19, 530 (2014)

[15] A.A. Chesnokov, J. Appl. Mech. Techn. Phys. 49, 737 (2008)

[16] A.A. Chesnokov, Eur. J. Appl. Math. 20, 461 (2009)

[17] X. Xin, L. Zhang, Y. Xia and H. Liu, Appl. Math. Lett. 94, 112 (2019)

[18] A. Paliathanasis, Physica Scripta, in press [DOI: 10.1088/1402-4896/ab32ad]

[19] C.E. Keing, G. Ponce and L. Vega, Transactions of the American Mathematical Society 342, 155 (1994)

[20] G.W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, (1989)

[21] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, (1993)

11 [22] N.H. Ibragimov, CRC Handbook of Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CRS Press LLC, Florida (2000)

[23] S. Jamal, P.G.L. Leach and A. Paliathanasis, Quaestiones Mathematicae, 42, 125 (2018)

[24] K.S. Govinder, J. Math. Anal. Appl. 258, 720 (2001)

[25] V.V. Morozov, Classification of six-dimensional nilpotent Lie algebras, Izvestia Vysshikh Uchebn Zavendeni˘ı Matematika, 5, 161 (1958)

[26] G.M Mubarakzyanov, Izvestia Vysshikh Uchebn Zavendeni˘ıMatematika, 32, 114 (1963)

[27] G.M Mubarakzyanov Izvestia Vysshikh Uchebn Zavendeni˘ıMatematika, 34, 99 (1963)

[28] G.M Mubarakzyanov Izvestia Vysshikh Uchebn Zavendeni˘ıMatematika, 35, 104 (1963)

[29] J. Patera, R.T. Sharp, P. Winternitz and H. Zassenhaus, J. Math. Phys. 17, 986 (1976)

[30] J. Patera and P. Winternitz, J. Math. Phys. 18, 1449 (1977)

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