Benney-Lin and Kawahara equations: a detailed study through Lie symmetries and Painlev\'{e} analysis

TL;DR

This paper investigates the integrability of Benney-Lin and Kawahara equations using Lie symmetry and Painlevé analysis, revealing integrable traveling-wave solutions and algebraic solutions for these PDEs.

Contribution

It provides a comprehensive symmetry and singularity analysis of these equations, identifying their integrable structures and solutions.

Findings

Existence of integrable traveling-wave solutions

Application of Painlevé analysis to PDEs

Derivation of algebraic solutions for the equations

Abstract

We perform a detailed study on the integrability of the Benney-Lin and KdV-Kawahara equations by using the Lie symmetry analysis and the singularity analysis. We find that the equations under our consideration admit integrable travelling-wave solutions. The singularity analysis is applied for the partial differential equations and the generic algebraic solution is presented.

Tables

Table 1. Table 1: Lie Brackets of the Lie point symmetries of the Benney-Lin equation

$[,]$	${𝐗}_1$	${𝐗}_2$	${𝐗}_3$
${𝐗}_1$	$0$	$0$	$X_2$
${𝐗}_2$	$0$	$0$	$0$
${𝐗}_3$	$-X_2$	$0$	$0$

Table 2. Table 2: Adjoint representation for the Lie point symmetries of the Benney-Lin equation

$Ad\left(\exp \left(\epsilon X_i\right)\right)X_j$	$X_1$	$X_2$	$X_3$
$X_1$	$X_1$	$X_2$	$-\epsilon X_2+X_3$
$X_2$	$X_1$	$X_2$	$X_3$
$X_3$	$X_1+\epsilon X_2$	$X_2$	$X_3$

Equations

$$u_t+u{u}_x+u_{xxx}+\beta \left(u_{xx}+u_{xxxx}\right)+\alpha u_{xxxxx}=0$$

$$t^{\prime }$$

$$x^{\prime }$$

$$u$$

$$X=\frac{\partial t^{\prime }}{\partial \epsilon }{\partial }_t+\frac{\partial x^{\prime }}{\partial \epsilon }{\partial }_x+\frac{\partial u^{\prime }}{\partial \epsilon }{\partial }_u,$$

$$\mathcal{H}\left(u,u_t,u_x,u_{tt},u_{xx},u_{tx},u_{xxx},...\right)=0;$$

$$\Phi \left(\mathcal{H}\left(u,u_t,u_x,u_{tt},u_{xx},u_{tx},u_{xxx},...\right)\right)=0.$$

$$X^{\left[n\right]}\left(\mathcal{H}\right)=0,$$

$$X^{\left[n\right]}=X+{\eta }_i^{\left[1\right]}{\partial }_{u_i}+...+{\eta }^{\left[n\right]}{\partial }_{u_{i_i{i}_j...i_n}},$$

$${\eta }_i^{\left[n\right]}=D_i{\eta }^{\left[n-1\right]}-u_{i{i}_2...i_{n-1}}{D}_j\left({\xi }^j\right),i⪰1.$$

$$\frac{dt}{{\xi }^t}=\frac{dx}{{\xi }^x}=\frac{du}{\eta },$$

$$X_1={\partial }_t,X_2={\partial }_x{X}_3=t{\partial }_x+{\partial }_u$$

$$Ad\left(\exp \left(\epsilon X_i\right)\right)X_j=X_j-\epsilon \left[X_i,X_j\right]+\frac{1}{2}{\epsilon }^2\left[X_i,\left[X_i,X_j\right]\right]+...\text{.}$$

$$X_1,X_2,X_3,X_1+c{X}_2\text{and }{X}_1+\gamma X_3\text{.}$$

$$t\text{and }u=\frac{{\gamma }_{2}x}{{\gamma }_{2}t+{\gamma }_1}+v.$$

$$\left({\gamma }_1+{\gamma }_{2}t\right)v_t+{\gamma }_{2}v=0,$$

$$v\left(t\right)$$

$$s=x-ct,u.$$

$$\alpha u^{\prime \prime \prime \prime \prime }+\beta \left(u^{\prime \prime \prime \prime }+u^{\prime \prime }\right)+u^{\prime \prime \prime }+u^{\prime }\left(u-c\right)=0,$$

$${\left(a{v}^{\prime \prime \prime \prime }+\beta \left(v^{\prime \prime \prime }+v^{\prime }\right)+v^{\prime \prime }+\frac{v^2}{2}\right)}^{{}^{\prime }}=0,$$

$$a{v}^{\prime \prime \prime \prime }+\beta \left(v^{\prime \prime \prime }+v^{\prime }\right)+v^{\prime \prime }+\frac{v^2}{2}=\delta ,$$

$$a\frac{d}{dz}\left(y{\left(\frac{dy}{dz}\right)}^2+2y^3\left(\frac{d^{2}y}{d{z}^2}\right)\right)+2\beta y\left(\left(\frac{d^{2}y}{d{z}^2}\right)+y^2{\left(\frac{dy}{dz}\right)}^2+1\right)+2\frac{d}{dz}\left(y^2\right)+z^2-2\delta =0,$$

$$y{\left(\frac{dy}{dz}\right)}^2+2y^3\left(\frac{d^{2}y}{d{z}^2}\right)+2y^2+\frac{z^3}{3}-2\delta z+\kappa =0.$$

$$e^{-\frac{1}{2y^2}}\frac{dy}{dz}+\frac{2}{3}{y}^3+\frac{z^4}{12}-\delta z^2+\kappa z+\lambda =0.$$

$$s=\left(x-ct\right)-\frac{\gamma }{2}{t}^2,u=\gamma t+w.$$

$$\alpha w^{\prime \prime \prime \prime \prime }+\beta \left(w^{\prime \prime \prime \prime }+w^{\prime \prime }\right)+w^{\prime \prime \prime }+w^{\prime }\left(w-c\right)+\gamma =0$$

$$\alpha v^{\prime \prime \prime \prime \prime }+\beta \left(v^{\prime \prime \prime \prime }+v^{\prime \prime }\right)+v^{\prime \prime \prime }+v^{\prime }v+\gamma =0$$

$$a{v}^{\prime \prime \prime \prime }+\beta \left(v^{\prime \prime \prime }+v^{\prime }\right)+v^{\prime \prime }+\frac{v^2}{2}+\gamma s=\delta$$

$$ap\left(p-1\right)\left(p-2\right)\left(p-3\right)s^{p-4}+p\beta \left(\left(p-1\right)\left(p-2\right)s^{p-3}+s^{p-1}\right)+p\left(p-1\right)s^{p-2}+\frac{v_0}{2}{s}^{2p}-\frac{\delta }{v_0}=0.$$

$$v\left(s\right)=-1680\alpha s^{-4}+m{s}^{-4+r}$$

$$\left(r+1\right)\left(r-12\right)\left(r^2-11r+70\right)=0,$$

Full text

Keywords: Lie symmetries; Singularity analysis; Benney-Lin equation; KdV-Kawahara equation

1 Introduction

We investigate the integrability of the Benney-Lin partial differential equation (PDE) [1, 2]

[TABLE]

with $\alpha\neq 0,$ by applying the theory of symmetries of differential equations and Painlevé analysis.

The Benney-Lin equation follows from the theory of fluid dynamics and more specifically describes the propagation of long waves on thin film. For $\beta=0$, the equation becomes the Kawahara equation [3] while when $\alpha=0$ equation (1) takes the form of the derivative of the KdV equation. The Benney-Lin equation has been widely studied in the literature [4, 5, 6, 7, 8, 9, 10]. However, in this work we prove the integrability of the Benney-Lin equation and of the Kawahara equation for the similarity solutions by applying the theory of Lie symmetries and the singularity analysis.

The theory of symmetries for differential equations it is a powerful mathematical tool for the study of nonlinear differential equations, and consequently of the study of nonlinear phenomena, for instance see [11, 12, 13, 14, 15, 16, 17] and references therein. The novelty of Lie symmetries is that they provide invariant surfaces which can be used to simplify the nonlinear differential equation or to determine conservation laws which are necessary for the integration of differential equations.

Singularity analysis is another major tool for the study of the integrability of differential equations. In contrary to the symmetry analysis, singularity analysis is based on the existence of movable singularities in the differential equation. Singularity analysis is associated with the French school led by Painlevé in the last years of the nineteenth century and the early years of the twentieth century [18, 19, 20] which was actually inspired by the successful application to the determination of the third integrable case of Euler’s equations for a spinning top by Kowalevskaya [21]. Modern applications of the singularity analysis in nonlinear analysis are presented in [22, 23, 24, 25] and references therein.

In this work investigate the integrability of the Benney-Lin and Kawahara equations for the PDEs by using the Lie point symmetries and the singularity analysis. Specifically we prove that similarity transformations provide integrable ordinary differential equations, while the two PDEs of our consideration passes the singularity analysis. The plan of the paper is as follows.

The main properties and definitions of the mathematical tools that we apply in this work are presented in Section 2. In Section 3, we present the detailed Lie symmetry analysis for the Benney-Lin equation. We find that the Benney-Lin equation admits three Lie point symmetries which form the $A_{3,1}$ Lie algebra. The seven possible reductions of the Benney-Lin equation with the use of the Lie invariants are studied. The resulting differential equations are studied again in terms of symmetries, or with the use of singularity analysis. The latter study is performed in Section 4. More specifically we find that the ordinary differential equations (ODEs) which follow from the similarity transformations of the Benney-Lin equations pass the singularity test, only after we apply firstly a recently proposed technique in the theory of singularity analysis. In Section 5, we extend our analysis in the case of a generalized Benney-Lin equation. In Section 6 we apply the Painlevé analysis directly for the PDEs without considering any reduction with the Lie symmetries. We find that the Benney-Lin equation and the generalized Benney-Lin equations pass the Painlevé test. The generic solution is given in a Right Painlevé Series. Our discussion and conclusions are presented in Section 7.

2 Preliminaries

In this section, we present the basic mathematical material and definitions which we use in this work. More specifically we discuss the theory of symmetries for differential equations and the singularity analysis.

2.1 Lie symmetries of differential equations

We briefly discuss the theory of Lie symmetries of differential equations and the application of the differential invariants for the construction of similarity solutions.

Let $\Phi$ describe the map of an one-parameter point transformation such as u^{\prime}\left(t^{\prime},x^{\prime}\right)=\Phi\left(u\left(t,x\right);\varepsilon\right)\with infinitesimal transformation

[TABLE]

and generator

[TABLE]

where $\varepsilon$ is an infinitesimal parameter.

Consider function $u\left(x^{i}\right)$ to be a solution of the differential equation

[TABLE]

then under the map $\Phi$, function $u^{\prime}\left(x^{i\prime}\right)=\Phi\left(u\left(x^{i}\right)\right)$ remains a solution for the differential equation iff the differential equation $\mathcal{H\,}=0$ remains invariant under the action of the map $\Phi$, that is

[TABLE]

The vector field $X$ is called a Lie point symmetry for the differential equation $\mathcal{H\,}=0$.

The mathematical formulation of the latter is expressed by the condition

[TABLE]

where $X^{\left[n\right]}$ describes the $n$th prolongation/extension of the symmetry vector in the jet-space of variables $\left\{u,u_{t},u_{x},...\right\}$ defined as

[TABLE]

where $u_{i}=\frac{\partial u}{\partial y^{i}},~{}y^{i}=\left(t,x\right)$ and

[TABLE]

One of the novelties of the existence of a Lie point symmetry is that invariant functions can be determined. If $X$ is an admitted Lie point symmetry, the solution of the associated Lagrange’s system,

[TABLE]

provides the zeroth-order invariants, $U^{A\left[0\right]}\left(t,x,u\right)$ which are applied to reduce the number of independent variables in PDEs or reduce the order for ODE.

When the admitted symmetries of a given differential equation is sufficient to reduce the differential equation into a first-order ODE, or into well-known integrable differential equation we say that the given differential equation is integrable in terms of Lie symmetries. However, symmetry is not the unique method to study the integrability of differential equations. Another major method for the study of the integrability of differential equations is presented in the following lines.

For more details on the symmetry analysis of differential equations we refer the reader to the standard references [26, 27, 28], while a review on the application of Lie symmetries to the theory of turbulence can be found in [29].

2.2 Singularity analysis

An alternate tool to study the integrability of differential equations is the Painlevé analysis, also known as singularity analysis. The modern treatment of singularity analysis is summarized in the ARS algorithm [30, 31, 32]. ARS stands for Ablowitz, Ramani and Segur.

There are three main steps to the algorithm which summarized as: (a) determine the leading-order term, and show if a moveable singularity exists, (b) determine the resonances which indicates the position of the constants of integration and (c) write a Painlevé Series with exponent and step as determined in the previous step and study if it is a solution of the original differential equation, the latter it is called the consistency test.

There are various criteria which should be satisfied for the values of the exponent of the leading-order term or for the values of the resonances these are summarized in the review article of Ramani et al. [33]. In the following line we discuss in details the three steps of the ARS algorithm.

Let $H\left(x,y\left(x\right),y^{\prime}\left(x\right),...\right)=0$ describe an ODE, where obviously $x$ denotes the independent variable and $y\left(x\right)$ the dependent variable. In order to determine the leading-order term we substitute $y=a(x-x_{0})^{p}$ into the differential equation, where $x_{0}$ is the location of the  movable singularity. The dominant powers are selected in order that the terms to share a common scaling symmetry. In order for the singularity to be a pole, the exponent $p$  should take negative integer values. However, $p$ can also take fractional exponents, even positive ones as the repeated differentiation of a positive fractional exponent eventually gives a negative exponent and so a singularity.

The resonances are determined by substituting $y=a(x-x_{0})^{p}+m(x-x_{0})^{p+r}$ into the differential equation for given $a$ and $p$. The dominant terms linear in $m$ provide a polynomial in $r,$ the zeros of which are the resonances. One of the zeros should be $-1$, which is important for the existence of the singularity. The acceptable values of the resonances are rational numbers. When the rest of the resonances are positive, the possible solution may be written by a Right Painlevé Series (Laurent expansion), when the resonances are negative, the possible solution is expressed by a Left Painlevé Series, while, when there are positive and negative resonances, the possible solution is expressed by a Mixed Painlevé Series (full Laurent expansion).

Finally, in the consistency test, the constants of integration are determined while the coefficients of the Laurent expansions are derived, which provide the exact form of the algebraic solution for the given differential equation.

3 Point symmetries of the Benney-Lin and Kawahara equations

We apply Lie’s theory for the Benney-Lin equation (1) for $\beta\neq 0$ and $\beta=0$ and we derive the following Lie point symmetries

[TABLE]

with Lie Brackets presented in Table 1. We infer that these three vector fields form the $A_{3,1}$ algebra in the Morozov-Mubarakzyanov classification scheme [34, 35, 36, 37]. The admitted Lie point symmetries are fewer than the number of symmetries required to reduce the fifth-order PDE into a first-order ODE. However, for every possible reduction we search again for the admitted Lie point symmetries. In the following we remark when there are different results for the special case with $\beta=0$.

We continue by calculating the adjoint representation for the Lie algebra in order to determine the one-dimensional optimal system. For every element of the admitted Lie algebra, the adjoint representation is defined as [38]

[TABLE]

In Table 2 the adjoint representation for the Lie symmetries of the Benney-Lin equation is presented.  Hence from Table 2 the one-dimensional optimal systems are determined to be

[TABLE]

3.1 Reduction by use of Lie invariants

From the admitted Lie point symmetries, the possible reductions that we can perform are  with the use of the symmetry vectors: (i) $X_{1},~{}$(ii) $X_{2}$, (iii) $X_{3}$, (iv) $X_{1}+cX_{2},~{}$(v)$~{}X_{1}+\gamma X_{3}$.

Reduction (i) provides the static solution, reduction (ii) provides the stationary solution, reduction (iii) is a scaling solution, while reduction (iv) provides the travelling-wave solution which is of special interest. The constant $c$ denotes the wave speed. The remaining reductions provide combinations of the previously discussed solutions.

Moreover, reductions (ii), (iii) and (vi) provide a linear first-order ODE as reduced equation, while a nonlinear fifth-order ODE is provided in the reductions (i), (iv), (v) and (vii).

3.1.1 Reductions (ii) and (iii)

The Lie invariants which correspond to the Lie point symmetry $\gamma_{1}X_{2}+\gamma_{2}X_{3}$ are

[TABLE]

When we consider $t$ to be the new independent variable and $v$ the new dependent variable, equation (1) becomes

[TABLE]

with exact solutions

[TABLE]

3.1.2 Reduction (i) and (iv)

From the symmetry vector $X_{1}+cX_{2}$ we calculate the Lie invariants

[TABLE]

Now, when we consider $s$ to be the new independent variable and $u=u\left(s\right)$ the dependent variable, the Benney-Lin equation, (1), is reduced to the fifth-order ODE

[TABLE]

which provides the travelling-wave solution or the stationary solution for $c=0$, and prime denotes total derivative with respect to $s$, i.e. $u^{\prime}=\frac{du}{ds}$.

Easily we observe that any travelling-wave solution $u\left(s\right)$ of equation (17) describes a stationary solution $v\left(s\right)$, where $v\left(s\right)=u\left(s\right)-c$. Moreover, equation (17) in the new variable $v\left(s\right)$ can be written as follows

[TABLE]

or equivalently

[TABLE]

which is a fourth-order ODE.

Equation (19) admits only the autonomous Lie point symmetry $\bar{X}=\partial_{s}\,,~{}$ which is an inherited symmetry. The latter symmetry vector is applied and equation (19) is written in the form of the following third-order nonlinear differential equation

[TABLE]

where $z=v\left(s\right)$ and  $y=v^{\prime}$. This third-order ODE does not admit Lie point symmetries. Hence other method should be considered to study the integrability of this ODE. In our study we consider the singularity analysis. However, before we proceed with such an analysis we observe that for the Kawahara equation expression (20) is a total derivative, that is, $y\left(z\right)$ is a solution of the second-order ODE

[TABLE]

This equation admits no Lie point symmetries. However, it can be integrated as follows

[TABLE]

Before we study equation, (19), with the use of singularity analysis we study the reductions (v) and (vii).

3.1.3 Reduction (v)

The zeroth-order invariants of the Lie symmetry vector $X_{1}+cX_{2}+\gamma X_{3}$ are

[TABLE]

Consequently, in the new variables, the Benney-Lin equation is simplified as

[TABLE]

which reduces to (17) for $\gamma=0$. In a similar way, the reduced equation (24) can be written as

[TABLE]

in which $v\left(s\right)=w\left(s\right)-c$.  By following the steps for the travelling-wave reduction, this fifth-order equation can be reduced to a third-order equation, or a first-order equation when $\beta=0$.

Equation (25) provides the fourth-order equation

[TABLE]

which does not admit any symmetry for $\gamma\neq 0$ and cannot be reduced to a third-order as can be equation (19).

In the following Section, we apply the singularity analysis to study the integrability for the travelling-wave solutions (19) and (26).

4 Singularity analysis

We proceed our analysis by applying the ARS algorithm for the fourth-order ODE (19) which provides the travelling-wave solution.

We substitute $v\left(s\right)=v_{0}s^{p}$ into (19) and obtain

[TABLE]

Consequently from this expression it follows necessarily that the leading-order terms are $p-4=2p$, which means that $p=-4$. Hence, in order that the leading-order terms cancel coefficient $v_{0}$ has the value $v_{0}=-1680\alpha$.

The second step is to determine the resonances. We replace

[TABLE]

in (19) and we linearize around the $m\rightarrow 0$, the requirement that the coefficient of the leading-order terms be zero provides the algebraic equation

[TABLE]

with solutions

[TABLE]

Because two of the resonances are imaginary we conclude that equation (19) does not pass the Painlevé test. Hence no conclusion for the integrability of the equation can be made.

Recently in [39] a new technique proposed on the ARS algorithm in order to extend the application range of the ARS algorithm. The idea is based that the movable singularity in finite time where the ARS algorithm required can be coordinate dependent.

Indeed from (27) it is clear that the exponent $p-4$ can be a leading term for $p=1,$ $p=2$ or $p=3$ . However, because $p>0,~{}$we do not have any movable singularity. That it is not true for the equation which follows for the the coordinate transformation $V\left(s\right)=\left(v\left(s\right)\right)^{-1}$ in (19).

By replacing $V\left(s\right)=V_{0}s^{p}$ in the new equation we find the following exponents and corresponding coefficients

[TABLE]

[TABLE]

We apply the second step of the ARS algorithm and the resonances are determined to be

[TABLE]

There are four possible solutions. Solution with $p_{1}$ is expressed by a Right Painlevé Series, solutions with exponent $p_{3}$ and $p_{4}$ by Left Painlevé Series, while the solution with exponent $p_{2}$ is expressed by a Mixed Painlevé Series. Before we conclude about the validity of the solutions, the consistency test should be performed.

For the exponent $p_{1}$ the solution is written as

[TABLE]

where from the consistency test we find that $V_{0},~{}V_{1}$ and $V_{2}$ are arbitrary while the rest coefficients are expressed as functions

[TABLE]

with first coefficient $V_{3}$ to be

[TABLE]

For the two Left Painlevé Series, and the Mixed Painlevé Series, the consistency test fails. Nevertheless, we conclude that the fourth-order ODE (19) is integrable according to the singularity analysis. We proceed by studying if the fifth-order ODE (25) passes the Painlevé test.

For equation (25)  in order the ARS algorithm to succeed we apply first the coordinate transformation $v\left(s\right)\rightarrow V^{-1}\left(s\right)$, from where we find the possible exponents to be

[TABLE]

From these exponents only the solution correspond to the $p_{1}$ passes all the steps of the ARS algorithm with resonances

[TABLE]

and solution given by a Right Painlevé Series with step one and the first four coefficients as integration constants. We remark that the analysis is valid for arbitrary parameter $\beta$.

We conclude that there exists similarity solutions for the Benney-Lin equation (1). We continue by extending our analysis for a generalization of the Benney-Lin equation.

5 The generalized Benney-Lin equation

The generalized Benney-Lin and generalized Kawahara equation proposed in [40] is

[TABLE]

with$~{}\alpha\neq 0~{},~{}n\neq 1,2$.

We apply the symmetry analysis and we find that equation (34) is invariant under the two-dimensional Lie algebra $2A_{1}~{}$comprised of the symmetry vector fields $X_{1}$ and $X_{2}$. The application of the Lie invariants (16) derived by the generic Lie symmetry $X_{1}+cX_{2}$ provide the fifth-order ODE

[TABLE]

which can be integrated as

[TABLE]

In a similar way as with (1) the generalized Benney-Lin equation is reduced to the third-order ode

[TABLE]

or to the first-order ODE for $\beta=0$,

[TABLE]

in which $z=v\left(s\right)$ and  $y=v^{\prime}$ and $\delta,~{}\kappa,~{}\lambda$ are constants of integration.  We are interested in the application of the singularity analysis for equation (36) in order to study the integrability of the generalized Benney-Lin equation for $\beta\neq 0$. Hence, we continue with the application of the ARS algorithm.

We substitute $u\left(s\right)=u_{0}s^{p}$ and we find that the only dominant term has exponent $p=-\frac{4}{n}$ for $n>0,$ while for the coefficient $u_{0}$ we find

[TABLE]

The second step of the ARS provides the resonances

[TABLE]

Because for $n>0,$ it holds $\left(15n^{2}+80n+64\right)>0$, resonances $r_{3}$ and $r_{4}$ are complex which means that (36) fails to pass the Painlevé test. Note that for $n=1$ resonances (30) are recovered.

We follow the same procedure with the previous section, i.e. we perform the change of variable $u\rightarrow U^{-1}$ and we find that the Right Painlevé Series

[TABLE]

is a solution of equation (36), where $U_{0},~{}U_{1}$ and $U_{2}$ are constants of integration and the remaining coefficient constants are expressed as $U_{I}=U_{I}\left(U_{0},U_{1},U_{2},n,\alpha,\beta,c\right)$.

It is important to mention the first step of the ARS algorithm provides more exponents $p$ for the differential equation (36) under the transformation $u\rightarrow U^{-1}$. In a similar way with the case $n=1$. However, the Left Painlevé Series and the Mixed Painlevé Series fail the consistency test.

Thereafter, the generalized Benney-Lin equation passes the Painlevé test and it is integrable for any allowed value of the parameter $n$.

6 Singularity analysis for the Benney-Lin equation

So far the Painlevé analysis we have performed for the Benney-Lin equation was based on the reduction of the equation by use of the Lie invariant and perform the singularity analysis for the resulting ODE. Nevertheless it is possible to perform the singularity analysis directly for the PDE (1). The application of Painlevé analysis for PDEs is summarized in [41, 42, 43].

In the case of PDEs the main steps of the ARS remain the same. We demonstrate the main steps in the following well-studied example.

We consider the nonlinear evolution equation [43]

[TABLE]

and we substitute $u=u_{0}\phi\left(t,x\right)^{p}.$ We follow the first step of the ARS algorithm as explained in Section 2. We find the exponent $p=-1$, with $u_{0}=-\frac{u_{x}^{2}}{u_{t}}.$

The resonances are determined by considering the substitution

[TABLE]

in (42) where again the requirement the leading-order terms which are linear in $m$ to be zero gives the algebraic equation $\left(r+1\right)\left(r-1\right)=0$, i.e. $r=-1$ and $r=1$.

The main difference of the algorithm between ODEs and PDEs is in the consistency test. To perform the consistency test we replace in (42) the following expression

[TABLE]

where the first two coefficients in powers of $\phi\left(t,x\right)$ provide the following Painlevé-Backlund equations (PB) [41]

[TABLE]

[TABLE]

The solution of these equations provides the functional form of the coefficient functions. In general an Ansatz should be made for the function $\phi\left(t,x\right)$, one of the most well-known Ansatze is that of Conte [41] where

[TABLE]

For this Ansatz we find that $u_{0}\left(t,x\right)=-kc^{2}\frac{\alpha+\beta e^{k\left(x-ct\right)}}{\left(\gamma+\delta e^{k\left(x-ct\right)}\right)^{2}},$ while $u_{1}\left(t,x\right)$ is arbitrary.

For the Benney-Lin equation we perform the change of variable $u=U^{-1}\left(t,x\right)$ and for the new variable we select $U\left(t,x\right)=U_{0}\phi^{p}\left(t,x\right)$. For the exponent $p$ we find that only the Right Laurent expansion for $p=-1$  satisfies the consistency test. In particular the solution is given by

[TABLE]

where $U_{0},~{}U_{1},~{}U_{2}$ and $U_{4}$ are found to be arbitrary functions and $U_{I}\left(t,x\right)=U_{I}\left(U_{0}\left(t,x\right),U_{1}\left(t,x\right),U_{2}\left(t,x\right),U_{4}\left(t,x\right)\right)$. The same result is found for the generalized Benney-Lin equation (34).

7 Conclusions

In this work we applied two major mathematical tools for the study of the integrability of the Benney-Lin and the KdV-Kawahara equations. In particular, we applied the theory of symmetries of differential equations and the singularity analysis.

The symmetry analysis provided that the Benney-Lin equation is invariant under a three-dimensional Lie algebra, the $A_{3,1}$. We considered the seven possible reductions by using the corresponding zeroth-order invariants. The resulting ODEs have been solved either by applying Lie theory or by considering the ARS algorithm for ODEs. In order for the similarity solutions to be determined with the use of the singularity analysis a change a variable had to be performed. We concluded that the Benney-Lin equation is integrable according to the similarity solutions given by the Lie symmetries. The analysis extended and for the case of the generalized Benney-Lin equation where we ended up with the same conclusion on the integrability.

Finally, we studied the PDE by using the singularity analysis where we find that the Benney-Lin equations passes the Painlevé test and is integrable.