Symmetry Reductions for a Generalized Fifth Order KdV Equation

Nonlinear equations have a great interest in Mathematics and Physics because almost all of the physics problems are nonlinear due to its nature. Furthermore, a big amount of physics models are described by time evolution equations that in fact, also are nonlinear partial differential equations. That is why, nonlinear evolution equations are widely used as models to describe many important complex physical phenomena in various fields of science, such as the Einstein field equations that describe gravitational fields [16], Navier-Stokes equations of fluid dynamics [8,24] or the Koterweg de Vries equations (KdV) which model waves on shallow water surfaces [17,25].

In the present paper we are going to focus our attention in Koterweg de Vries equations. The KdV equation

ut+λuux+μuxxx=0$$ \begin{equation} u_t+\lambda u u_x + \mu u_{xxx}=0 \end{equation} $$(1)

was arised to model shallow water waves with weak nonlinearities and it is probably the most studied nonlinear evolution equation due to its wide applicability.

The KdV equation (1) was generalized to a standard fifth order equation of the form

ut+au2ux+buxuxx+cuuxxx+duxxxxx=0$$ \begin{equation} u_t + a u^2u_x + bu_x u_{xx} + cu u_{xxx} + d u_{xxxxx}=0 \end{equation} $$(2)

where a, b, c and d are non zero parameters and u(x, t) is a sufficiently smooth function.

As the original KdV, this equation (2) models shallow water waves, however its applications are including shallow-water waves near critical value of surface tension and waves in nonlinear LC circuit with mutual inductance between neighbouring inductors [7].

In the present paper we take the first step to generalized the standard fifth order KdV (2) and extend the obtained results to a more general case in which appears an unknown function of u. Particularly, we have studied the following form of KdV of fifth order

ut+uxuxx+f(u)ux+buuxxx+uxxxxx=0.$$ \begin{equation} u_t+u_xu_{xx}+f(u) u_x+buu_{xxx}+u_{xxxxx}=0. \end{equation} $$(3)

where b is a constant and f a function.

First of all, Lie point symmetries have been obtained applying the Lie classical method [11, 13, 20]. Several well-know researchers obtain symmetries because they can be use to obtain systematically exact solutions of the equations. Remark that these solutions are usually with soliton or compacton structures, which are attracting lots of interest nowadays [18, 19, 21]. Lie symmetries also play an important role simplifying models and helping to understand bifurcations of nonlinear systems.

Furthermore, once the Lie symmetries and their corresponding group invariant solutions have been obtained we have transformed the equation (3) into an ordinary differential equation and we have applied the double reduction method given by Sjöberg [23]. That way, we have solved the ordinary differential equation (ODE), whose solutions provide solutions of the original partial differential equation (3). Examples about the method are in [12,6]

The double reduction method emphasizes and uses the relation between symmetries and conservation laws. In fact, its first step is to obtained the conservation laws for the ODE, which have been obtained applying the direct method of the multipliers [3, 4] proposed by Anco and Bluman. This method have extensively applied to partial differential equation because allow to obtain all conservation laws for the corresponding equation. Some examples can be found in [9, 10, 14].

The structure of the work is as follows. In Section 2, we have applied the Lie group method of infinitesimals transformations to the generalized fifth order KdV (3) and we have reported its reductions obtained from the optimal system of subalgebras. Moreover, in Section 3 the double reduction method have been applied to the ODE already obtained in the previous section and, of course, their conservation laws have been obtained. Finally, exact solutions for the ODE and, consequently, for the studied generalized fifth order KdV have been showed.

To apply the classical method to (3) we consider the one-parameter Lie group of infinitesimal transformations in (x, t, u) given by

where ε is the group parameter. Then one requires that this transformation leaves invariant the set of solutions of (3). This yields to an overdetermined, linear system of equations for the infinitesimals ξ(x, t, u), τ(x, t, u) and η(x, t, u). The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form

Χ=ξ(x,t,u)∂∂x+τ(x,t,u)∂∂t+η(x,t,u)∂∂u.$$ \begin{equation} X=\xi(x,t,u) \frac{\partial}{\partial x}+ \tau(x,t,u)\frac{\partial}{\partial t}+\eta(x,t,u)\frac{\partial}{\partial u}. \end{equation} $$(5)

Having determined the infinitesimals, the symmetry variables are found by solving the invariant surface condition

Φ≡ξ∂u∂x+τ∂u∂t−η=0.$$ \begin{equation}\label{sur}\Phi \equiv \xi\frac{\partial u}{\partial x}+ \tau \frac{\partial u}{\partial t}-\eta=0. \end{equation} $$(6)

We consider the classical Lie group symmetry analysis of equation (3). Invariance of equation (3) under a Lie group of point transformations with infinitesimal generator (5) leads to a set of 63 determining equations for the infinitesimals ξ(x, t, u), τ(x, t, u) and η(x, t, u). The solutions of this system depends on b and f(u).

1. For b arbitrary constant and f arbitrary function, the only symmetries admitted by (3) are the group of space and time translations, which are defined by the infinitesimal generators

X1=∂∂x,X2=∂∂t.$$ \begin{equation}\begin{array}{lll} {\bf X}_1=\displaystyle\frac{\partial}{\partial x},&\hspace{0.5cm} &{\bf X}_2=\displaystyle\frac{\partial}{\partial t}.\end{array} \end{equation} $$

In this case, we obtain travelling wave reductions

z=x−λt,u=h(z),$$ \begin{equation}\begin{array}{lll} z=x-\lambda t,&\hspace{0.5cm} &u=h(z),\end{array} \end{equation} $$

where h(z) satisfies

h′′′′′+bhh‴+h′h″+f(h)h′−<!-- - -->λ<!-- ? -->h′=0.$$ \begin{equation} h'''''+b h h'''+h' h''+f(h)h'-\lambda h'=0. \end{equation} $$(7)

2. For b arbitrary constant and f(u) = au² + c, we obtain one extra symmetry

X31=(x+4ct)∂∂x+5t∂∂t−2u∂∂u.$$ \begin{equation}{\bf X}_3^1=(x+4 c t) \frac{\partial}{\partial x}+ 5 t \frac{\partial}{\partial t}-2 u \frac{\partial}{\partial u}. \end{equation} $$

3. For b = 0, f(u) = au² + d u + c and a≠0, we obtain one extra symmetry, and this symmetry is defined by the following infinitesimal generator:

X32=(x+4ct−d2at)∂∂x+5t∂∂t−(2u+da)∂∂u.$$ \begin{equation}{\bf X}_3^2=(x+4ct-\frac{d^2}{a}t)\frac{\partial}{\partial x}+5t\frac{\partial}{\partial t}-(2u+\frac{d}{a})\frac{\partial}{\partial u}. \end{equation} $$

4. For b = 0, f(u) = a u + c, we obtain one extra symmetry

X33=at∂∂x+∂∂u.$$ \begin{equation}{\bf X}_3^3=a t \frac{\partial}{\partial x}+ \frac{\partial}{\partial u}. \end{equation} $$

5. For b = 0, f(u) = c, we obtain X₁, X₂, X31 ${\bf X}_3^1$ and X33 |a=0 $\bf \text {X}^3_3|_{a=0}.$ .

In order to determine solutions of PDE (3) that are not equivalent by the action of the group, we must calculate the one-dimensional optimal system [22]. Next we construct a table showing the separate adjoint actions of each element in Xij ${X^j_i}$ , i = 1, …, 3 and j = 1, 2. This construction is done by summing the Lie series. The commutator table, and its adjoint table are as follows.

Table 1

Commutator table for the Lie algebra {X1,X2,X31} $\{X_1, X_2, X_3^1 \}$ .

where

Y1=e5εX2+4cεX1(1+∑p=1∞(εp(p+1)!∑q=0p5q)).$$ \begin{equation} Y_1= \displaystyle e^{5\epsilon} X_2+ 4 c \epsilon X_1 \left(1+ \sum_{p=1}^{\infty} \left( \frac{\epsilon^p}{(p+1)!} \sum_{q=0}^p 5^q \right) \right). \end{equation} $$(8)

For b arbitrary constant and f(u) = au² + c, the one-dimensional optimal system of subalgebras is given by the set {X31,λX1+X2} $\left\{X_3^1, \lambda X_1+X_2 \right\}$ . Taking into account X31 $X_3^1$ we obtain,

z=(x−ct)t−1/5,u=h(z)t−2/5,$$ \begin{equation}\begin{array}{lll} z=(x-c t)t^{-1/5},&\hspace{0.5cm} &u=h(z)t^{-2/5},\end{array} \end{equation} $$

where h(z) satisfies

5h′′′′′+5bhh‴+5hzh″+5ah2h′−<!-- - -->zh′−<!-- - -->2h=0.$$ \begin{equation} 5 h'''''+5 b h h'''+5 h_z h''+5 a h^2 h'-z h'-2 h=0 . \end{equation} $$(9)

Table 3

Commutator table for the Lie algebra {X1,X2,X32} $\{X_1, X_2, X_3^2 \}$ .

where Y2=e5εX2+(4c−d2a)εX1(1+∑p=1∞(εp(p+1)!∑q=0p5q)) $Y_2= \displaystyle e^{5\epsilon} X_2+ \left( 4 c-\tfrac{d^2}{a}\right) \epsilon X_1 \left(1+ \sum_{p=1}^{\infty} \left( \frac{\epsilon^p}{(p+1)!} \sum_{q=0}^p 5^q \right) \right)$ .

For b = 0, f(u) = au² + d u + c and a≠0, the one-dimensional optimal system of subalgebras is given by the set {X32,λX1+X2} $\left\{X_3^2, \lambda X_1+X_2 \right\}$ . Taking into account X32 $X_3^2$ we obtain,

z=(x+(d24a−c)t)t−1/5,u=h(z)t−2/5−d2a,$$ \begin{equation}\begin{array}{lll} z=\left(x+\left( \tfrac{d^2}{4a}-c\right) t \right)t^{-1/5},&\hspace{0.5cm} &u=h(z)t^{-2/5}-\tfrac{d}{2a},\end{array} \end{equation} $$

where h(z) satisfies

5h′′′′′+5h′h″+5ah2h′−<!-- - -->zh′−<!-- - -->2h=0.$$ \begin{equation} 5 h'''''+5 h' h''+5 a h^2 h'-z h'-2 h=0 . \end{equation} $$(10)

Table 5

Commutator table for the Lie algebra {X1,X2,X33} $\{X_1, X_2, X_3^3 \}$ .

For b = 0, f(u) = a u + c and a≠0, the one-dimensional optimal system of subalgebras is given by the set {λX1+X2,X2+μX33} $\left\{\lambda X_1+X_2, X_2 +\mu X_3^3 \right\}$ . We obtain the reduction for X2+μX33 $X_2 +\mu X_3^3$

z=x−aμt22,u=h(z)+μt,$$ \begin{equation}\begin{array}{lll} z=x-\frac{a \mu t^2}{2},&\hspace{0.5cm} &u=h(z)+\mu t ,\end{array} \end{equation} $$

where h(z) satisfies

h′′′′′+h′h″+ahh′+ch′+μ=0.$$ \begin{equation} h'''''+ h' h''+a h h'+c h'+\mu=0 . \end{equation} $$(11)

Table 7

Commutator table for the Lie algebra {X1,X2,X31,X33 |a=0} $\{X_1, X_2, X_3^1, X_3^3 \left|_{a=0} \right. \}$ .

where Y₁ is given by (3). The one-dimensional optimal system of subalgebras is given by the set {X31,λX1+X2+μX33 |a=0 } $\left\{X_3^1, \lambda X_1+X_2+\mu X_3^3 \left|_{a=0} \right. \right\}$ . For λX1+X2+μX33 |a=0 $\lambda X_1+X_2+\mu X_3^3|_{a=0}$

z=x−λt,u=h(z)+μt,$$ \begin{equation}\begin{array}{lll} z=x-\lambda t,&\hspace{0.5cm} &u=h(z)+\mu t,\end{array} \end{equation} $$

where h(z) satisfies

h′′′′′+h′h″+c−λh′+μ=0.$$ \begin{equation} h'''''+ h' h''+ \left( c -\lambda \right) h'+\mu=0. \end{equation} $$(12)

Another powerful application of conservation laws taking into account the relationship between Lie symmetries and conservation laws it is the so called double reduction method given by Sjöberg [23]. Sjöberg introduced this method in order to get solutions of a qth partial differential equation from the solutions of an ordinary differential equation of order q−1.

In [5] by using the general method of conservation law multipliers Anco developed symmetry properties of conservation laws of partial differential equations. The author proved that conservation laws that are symmetry invariant or symmetry homogeneous have at least one important application: any symmetry-invariant conservation law will reduce to a first integral for the ODE obtained by symmetry reduction of the given PDE when symmetry-invariant solutions u(t, x) are sought. This provides a direct reduction of order of the ODE.

In Ref. [3,4,22] the authors show that all non-trivial conservation laws arise from multipliers. Specifically, when we move off of the set of solutions of equation (12), every non-trivial local conservation law is equivalent to one that can be expressed in the characteristic form

DzX˜=(h‴″+bhh‴+h′h″+f(h)h′−λh′)Q,$$ \begin{equation} D_z \tilde X= \left(h'''''+b h h'''+h' h''+f(h)h'-\lambda h'\right)Q, \end{equation} $$(13)

where Q(z, h, h′, h″, …) is the multiplier. In general, a function Q(z, h, h′, …) is a multiplier if it is non-singular on the set of solutions h(z) of equation (12), and if its product with equation (3) is a divergence expression with respect to z. There is a one-to-one correspondence between non-trivial multipliers and non-trivial conservation laws in characteristic form.

The determining equation to obtain all multipliers is

δδh((h‴″+bhh‴+h′h″+f(h)h′−λh′)Q)=0.$$ \begin{equation} \frac{\delta}{\delta h}\Big( (h'''''+b h h'''+h' h''+f(h)h'-\lambda h')Q \Big) =0. \end{equation} $$(14)

This equation must hold off of the set of solutions of equation (12). Once the multipliers are found, the corresponding non-trivial conservation laws are obtained either by using a homotopy formula Ref. [2,3,4].

We will now find all multipliers Q, and we will obtain corresponding conservation laws. The determining equation (14) splits with respect to the variables h‴, h″″, h″‴, h″″″. This yields a linear determining system for Q which can be solved by the same algorithmic method used to solve the determining equation for infinitesimal symmetries.

For the multiplier Q(z, h, h′, h″) we construct the following conservation laws

For b arbitrary constant and f arbitrary function, for the multiplier

Q=1,$$ \begin{equation*}\nonumber Q=1, \end{equation*} $$

we obtain the following conservation law:

Dz[∫(​f(h)−λ)dh+1/2(−b+1)h′2+bhh″+h‴′]=0.$$ \begin{equation} \begin{array}{ll} & D_z[\int( \!f \left( h \right) -\lambda)\,{\rm d}h+1/2\, \left( -b +1 \right) {h'}^{2}+bhh''+h'''']=0. \end{array} \end{equation} $$(15)

For b = 1/2, from (15) and (16) we obtain the system

∫[f(h)−λ]dh+1/4h′2+1/2hh″+h″″+k1=0,$$ \begin{eqnarray} \int [\!f \left( h \right) -\lambda]\,{\rm d}h+1/4 {h'}^{2}+1/2\,h\,h''+h''''+k_1&=&0,\end{eqnarray} $$(18)

hh‴′−h′h‴+1/2h2h″+1/2h″2−∫​h(λ−f(h))dh−k2=0.$$ \begin{eqnarray}hh''''-h'h'''+1/2\,{h}^{2}h''+1/2 \,{h''}^{2}-\int \! h \left( \lambda-f \left( h \right) \right) \,{\rm d}h-k_2&=&0. \end{eqnarray} $$(19)

From equation (18) we deduce h″″ and substituting into equation (19) we derive the equation

−<!-- - -->h∫<!-- ? -->[fh]dh+λ<!-- ? -->h2−<!-- - -->1/4hh′2−<!-- - -->hk1−<!-- - -->h′h‴+1/2h″2+∫<!-- ? -->(hfh−<!-- - -->hλ<!-- ? -->)dh+k2=0.$$ \begin{equation} -h\int [\!f \left( h \right)] \,{\rm d}h+\lambda\,{h}^{2}-1/4\,h{h'}^{2}-h{k_1}-h'h''' +1/2\,{h''}^{2}+\int \!(hf \left( h \right) -h \lambda)\,{\rm d}h+{k_2}=0. \end{equation} $$(20)

For f(h) = λ, k₁ = k₂ = 0 we deduce the follows solution of equation (20)

h(z)=e∫​g(ζ)dζ+k3,$$ \begin{equation}h(z)={{\rm e}^{\int \!g \left( \zeta \right) \,{\rm d}\zeta +{k_3}}}, \end{equation} $$

where ddζg(ζ)=1/8(15ζ2+2)(g(ζ))3ζ+7/2(g(ζ))2+1/2g(ζ)ζ ${\frac {\rm d}{{\rm d}\zeta}}g \left( \zeta \right) =1 /8\,{\frac {\left( 15\,{\zeta}^{2}+2 \right) \left( g \left( \zeta \right) \right) ^{3}}{\zeta}}+7/2\, \left( g \left( \zeta \right) \right) ^{2}+1/2\,{\frac {g \left( \zeta \right)}{\zeta}}$ .

For b = 1/2 and f=340h2+c1 $f=\frac{3}{40}h^2+c_1$ , from (15) and (17) we obtain the system

1/40h3+hc1−<!-- - -->hλ<!-- ? -->+1/4h′2+1/2h″h+h⁗+k1=0,$$ \begin{eqnarray} 1/40\, h^{3}+h {c_1}-h \lambda+1/4\, \left( h' \right) ^{2}+1/2\, h'' h +h'''' +{k_1} =0, \end{eqnarray} $$(21)

9h54000+200c1−<!-- - -->200λ<!-- ? -->+300h″h34000+3h2h⁗20+−<!-- - -->1200h′h‴+1600h″2h4000+1200h′2+4000h⁗h″4000+2000c1−<!-- - -->2000λ<!-- ? -->h′24000−<!-- - -->1/2h‴2+k3=0.$$ \begin{eqnarray} {\frac {9\,{h}^{5}}{4000}}+{\frac {\left( 200\,{c_1}-200\, \lambda+300\,h'' \right) {h}^{3}}{4000}}+{\frac {3\,{h}^{2}h''''}{20}}+{\frac {\left( -1200\,h'h'''+ 1600\,{h''}^{2} \right) h}{4000}}&&\nonumber\\+{\frac {\left( 1200\,{h'}^{2}+4000\,h'''' \right) h''}{4000}} +{\frac {\left( 2000\,{c_1}-2000\,\lambda \right) {h'}^{2}}{4000}}-1/2\,{h'''}^{2}+k_3&=&0. \end{eqnarray} $$(22)

From equation (21) we deduce h″″ and substituting into equation (22) we derive the equation

−3h52000−1/10c1h3+1/10λh3−1/40h3h″−3h2(h′)280−3h2k120−3/10hh′h‴−1/10h(h″)2−c1hh″+λhh″+1/20h″(h′)2−k1h″+1/2(h′)2c1−1/2λ(h′)2−1/2(h‴)2+k3=0.$$ \begin{equation}\label{eqq}\begin{array}{l}-{\frac {3\, h ^{5}}{2000}}-1/10\,{ c_1}\, h^{3}+1/10\,\lambda\, h^{3}-1/40\, h^{3}h'' -{\frac {3\, h^{2} \left( h'\right) ^{2}}{80}}-{\frac {3\, h^{2}{ k_1}}{20}}-3/10\,h h'h''' -1/10 \,h \left( h'' \right) ^{2}-{ c_1}\,h h''\\ +\lambda\,h h'' +1/20 \, h'' \left( h' \right)^{2}-{k_1}\,h'' +1/2\, \left(h' \right)^{2}{c_1}-1/2\,\lambda\, \left( h' \right) ^{2}-1/2\, \left( h''' \right) ^{2}+{k_3} =0.\end{array}\end{equation} $$(23)

If c₁ = k₁ = k₃ = 0, we obtain the following solution for the equation (23)

h(z)=−<!-- - -->40WeierstrassP(z+z1,(1/2)λ<!-- ? -->,z2)$$ \begin{equation*}\nonumber h(z) = -40\,\mbox{WeierstrassP}(z+z_1, (1/2)\lambda, z_2) \end{equation*} $$

where WeierstrassP is the Weierstrass elliptic function, which is defined by WeierstrassP(z, g², g³) = 1/z² + ∑(1/(z−ω)²−1/ω², ω) where sums and products range over ω = 2 m₁ω₁ + 2 m₂ω₂ such that m₁, m₂ ∈ (Z × Z)−(0, 0). Quantities g² and g³ are known as the invariants and are related to ω₁ and ω₂ by g2=60∑ω(1ω4) $g2=60\displaystyle\sum_{\omega}\left(\frac{1}{\omega^4}\right)$ , g3=140∑ω(1ω6) $g3=140\displaystyle\sum_{\omega}\left(\frac{1}{\omega^6}\right)$ , [1].

In this paper, we have studied a generalized fifth-order KdV equation from Lie symmetries view point. We have established a symmetry classification of equation (3) in terms of the arbitrary constants b and the arbitrary function f(u). By using the adjoint representation of the symmetry group on its Lie algebra, we have constructed an optimal system of one-dimensional subalgebras. We have obtained the similarity-reduced equations for each element of optimal system as well as some group invariant solutions. Moreover, double reduction method is used to obtain some exact solutions.