On the Rosenau equation: Lie symmetries, periodic solutions and solitary wave dynamics

In this paper, we first consider the Rosenau equation with the quadratic nonlinearity and identify its Lie symmetry algebra. We obtain reductions of the equation to ODEs, and find periodic analytical solutions in terms of elliptic functions. Then, considering a general power-type nonlinearity, we prove the non-existence of solitary waves for some parameters using Pohozaev type identities. The Fourier pseudo-spectral method is proposed for the Rosenau equation with this single power type nonlinearity. In order to investigate the solitary wave dynamics, we generate the solitary wave profile as an initial condition by using the Petviashvili's method. Then the evolution of the single solitary wave and overtaking collision of solitary waves are investigated by various numerical experiments.

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1 Introduction

The vibrations of a one dimensional anharmonic lattice associated with the birth of the soliton are modeled in terms of the discrete lattices. If the lattice is dense and weakly anharmonic, the well-known Korteweg-de Vries (KdV) equation is derived. However, KdV equation cannot model the wave to wave and wave to wall interactions for the dynamics of dense discrete systems. To overcome this difficulty, the Rosenau equation

(1.1)

is derived to describe the dynamics of dense discrete systems considering higher order effects by Rosenau rosenau1988dynamics .

In this paper, we study the Rosenau equation with a single power-type nonlinearity

(1.2)

where with . The Rosenau equation satisfies the conservation law Park1990 :

(1.3)

Park Park1990 studied the global existence and uniqueness of the initial value problem for the Rosenau equation of the form

(1.4)
(1.5)

where with and . It was proved that the Cauchy problem (1.4)-(1.5) admits a unique global solution for the initial data . Park also proved that the solution with small initial data decays like in Park1992 . In the multidimensional case, the global existence and uniqueness result of the IBVP with Dirichlet boundary conditions for the equations

(1.6)

and

(1.7)

are given in Park1993 .

From the numerical point of view, the finite element Galerkin approximate solutions of (1.1

) and the error estimates are studied in

chungha . Fully discrete schemes, backward Euler, Crank-Nicolson and two step backward methods, are proposed for the initial-boundary value problem (IBVP) of the Rosenau equation

(1.8)

where is a bounded set in , in Chunk2001 . The authors obtain optimal estimates using Galerkin approximations and derive a priori estimates for discrete schemes. When and , a discontinuous Galerkin method and stability analysis are investigated in Choo2008 . The initial value problem (1.4)-(1.5) is solved in Danumjaya2019 via discontinuous Galerkin finite element methods. A conservative unconditionally stable finite difference scheme is used for the equation for (1.1) in Omrani2008 . A second order splitting combined with orthogonal cubic spline collocation method is employed in manickam . The authors of Erbay2021 consider a nonlocal nonlinear PDE, which reduce to Eq. (1.2) for a special case of the kernel function. They establish a numerical scheme based on truncated discrete convolution sums applicable to ”geniunely nonlocal” Erbay2021 case, where usual finite-difference schemes will not work.

In the current work, we analyze the Rosenau equation (1.2) through several different approaches. We start with the quadratic nonlinearity, namely with Eq. (1.1). As the current literature does not contain any result regarding the Lie symmetry algebra of (1.1

), we identify the symmetry algebra in this case and the equation is reduced to ordinary differential equations. After this we aim at finding exact solutions of traveling wave type, which was achieved with obtaining several periodic solutions. Section 2 is devoted to this analysis. The main outcome of Section 2 is the analytical solutions obtained in terms of the elliptic functions. Some of these solutions are smooth, and some have singularities. To the best of our knowledge, these solutions have been obtained the first time in the existing literature with this work.

The equations constructed by adding some terms to Rosenau equation such as Rosenau-KdV equation, Rosenau-Kawahara equation Zuo09 have exact solitary wave solutions. There have been many studies focusing on the solitary wave dynamics of these equations. To the best of our knowledge, there is no exact solitary wave solution for the Rosenau equation with single power type nonlinearity. The solitary wave solution was first generated in erkip by using Petviashvili’s method, numerically. The non-existence result of solitary wave solutions for some parameters is given in Section . We use Pohozaev type identities to show the non-existence of solitary wave. The existence and stability of the solitary wave solutions of the equation (1.2) for is discussed in zeng . To the best of our knowledge, existence and stability of solitary waves for is an open problem. The present study addresses the existence of solitary wave solutions for numerically by using Petviashvili method. For the time evolution of the constructed solution we need an efficient numerical method. We therefore propose a numerical method combining a Fourier pseudo-spectral method for the space discretization and a fourth-order Runge-Kutta scheme for time discretization. Section 4 is devoted the fully-discrete Fourier pseudo-spectral scheme and show how to formulate it for the Rosenau equation. We also discuss the evolution of the single solitary wave solutions and interaction of two solitary waves for the Rosenau equation. As far as we know, the solitary wave dynamics of the Rosenau equation with single power type nonlinearity has never been investigated before in literature. In Section 5, the numerical scheme is tested for accuracy and convergence rate. The solitary wave dynamics of the Rosenau equation by considering various problems, like propagation of single solitary wave, collision of two solitary waves are discussed.

2 The Symmetry Algebra and Reductions of the Rosenau Equation

In this Section we will determine the Lie symmetry algebra of (1.2) when , i.e., in case of a quadratic nonlinearity. This invariance algebra is going to provide us several reductions of the equation to ordinary differential equations. One of these reductions, which determines the traveling solutions of (1.2), will be solved in terms of elliptic functions.

When , if we replace , (1.2) takes the form

(2.1)

Applying the well-known procedure for finding the Lie symmetries of a differential equation, we obtain the following result.

Theorem 2.1

Lie symmetry algebra of the Rosenau equation (2.1

) is 3-dimensional and is generated by the vector fields

(2.2)

with the only nonzero commutation relation . The algebra has the direct sum structure . The optimal system of one-dimensional subalgebras of is given in patera1977subalgebras as

(2.3)

where and .

In the following we briefly study the reductions of (2.1) to ODEs making use of these one-dimensional subalgebras.

(i) Reduction through . The solutions invariant under the transformations generated by have the form , hence we obtain the trivial constant solutions .

(ii) Reduction through . First suppose . Then , therefore one looks for solutions , which again gives the trivial solutions. Let . Then , for which we simply write , . In order to determine the solutions which are invariant under the group of transformations generated by , we solve the invariant surface condition

(2.4)

and find the two invariants and . Considering as the new independent variable and as the new dependent variable, group-invariant solutions in this case will be searched according to and hence

(2.5)

of which replacement in (2.1) gives the reduction

(2.6)

Observe that when , the order of this reduction is one less:

(2.7)

(iii) Reduction through . The reduction with the generator determines the traveling wave solutions: When we plug in (2.1) we get

(2.8)

where and is a constant.

In the following subsection, by proposing a suitable expansion for the solution, we are going to find some exact solutions for the reduction (2.8) that yields traveling solutions. For the reductions (2.6) and (2.7) similar treatments can be performed, which we kept beyond the scope of this article.

2.1 Elliptic type solutions

We shall try to find some exact solutions of traveling wave type. We can work on (2.8), but we prefer to write down the reduction for the equivalent traveling wave ansatz: , . Plugging this in (2.1) and integrating twice we obtain

(2.9)

For (2.9) we propose

(2.10a)
(2.10b)
(2.10c)

The expansion in (2.10c) is suggested so as to obtain elliptic function solutions and if possible the limiting cases trigonometric and hyperbolic ones. The form of the expansion (2.10a) is chosen so that a balancing between the terms is possible when inserted in the equation. In our case, it appears that this balance is only possible when the degree of the expansion in (2.10a) is 4. Upon this substitution, in the resulting expression we express all derivatives of in terms of using (2.10c). Afterwards, we look for the possibility that coefficients of , vanish.

We assume and . We find the following values for the remaining constants , , , , and ;

(2.11a)
(2.11b)
(2.11c)
(2.11d)
(2.11e)
(2.11f)

where and and are arbitrary. Let us note that numerical values of and will not be important in the solutions we shall determine below.

Now we need to integrate (2.10c), which takes the form

(2.12)

and find and hence . Evaluation of the integral of (2.12) depends on the factorization of the polynomial . Assume that , , and are roots of the equation . Let us call . Then we need to solve

(2.13)

keeping in mind from (2.11d) that . The discriminant of this equation is . There are two cases.

Case I.  This implies that . Eq. (2.10c) takes the form

(2.14)

where . Therefore we find

(2.15)

The solution becomes

(2.16)

Case II. Then and . There are two possibilities for this case: If the signs of and are opposite then the equation has four distinct real roots , , which we name as

(2.17)

If the signs of and are the same then the equation has four distinct complex roots ,

Let and . In order that (2.12) makes sense, the right hand side must be nonnegative. Thus we should consider the intervals (a) , (b) and (c) when integrating (2.12).

Case II.a  Let us first write

(2.18)

where . In the first hand, when , using the results available in the handbook byrd2013handbook , we obtain

(2.19)

for the integration of the left hand side of (2.18). Using the substitution

(2.20)

where

(2.21)

for which we calculate that , gives rise to the elliptic function solution to (2.12)

(2.22)

where . Hence the solution to (2.1) can be written as follows

(2.23)

Since , we removed in passing from (2.22) to (2.23). For the following cases II.b, II.c, II.d and II.e there are similar calculations, therefore we skip the details of the calculations made by using the transformations available in byrd2013handbook and state only the results.

Case II.b  When and , integrating for we obtain the elliptic function solution to (2.12) as

(2.24)

where

(2.25)

Hence the solution to (2.1) can be written as

(2.26)

Case II.c  When and , integrating over we obtain the solution to (2.1) as

(2.27)

with

(2.28)

Let and . Since the right hand side of (2.12) must be nonnegative, we should consider the intervals (d) and (e) when integrating (2.12).

Case II.d  In case and , employing a suitable substitution on a solution of (2.1) is found to be as follows

(2.29)

where

(2.30)

Case II.e  For and , working with a substitution when we obtain

(2.31)

where

(2.32)

Case II.f  Now, we consider the case when the polynomial has four distinct complex zeros. In order that (2.12) makes sense, the right hand side must be nonnegative. Therefore, we should consider the case and . (2.12) takes the form

(2.33)

Therefore we have

(2.34)

with the substitution

(2.35)

This gives rise to the elliptic function solution to (2.12),

(2.36)

Hence the solution to (2.1) can be written as follows

(2.37)
Figure 1: Plots of the periodic solution (a), (b) . For both of the solutions, we choose the free parameters as , . Note that there is a singular characteristic in the solution whereas (2.26) is smooth.

We depict the solutions given in (2.23) and (2.26) for cases II.a and II.b, respectively. For both of the cases, we choose the arbitrary constants as

(2.38)

This determines the roots

(2.39)

For both of the solutions, we have , , , , and .

For the solution (2.23) of II.a, for any choice of the arbitrary constants , , and ; therefore, (2.23) has discontinuities. This is illustrated in Figure 1(a). On the contrary, for case II.b, for any value of the arbitrary constants, therefore (2.26) is smooth, which is well illustrated in Figure 1(b). Both solutions are periodic.

The other solutions (2.27), (2.29), (2.31) and (2.37) of the remaining cases give similar pictures to these two.

3 Solitary wave solutions

In this section, we first establish the non-existence of solitary waves of the Rosenau equation (1.2) for some parameters. To find the localized solitary wave solutions of the Rosenau eq. (1.2), we use the ansatz with which leads to the ordinary differential equation

(3.1)

Here denotes the derivative with respect to . Integrating the equation (3.1) and, we have

(3.2)

The following theorem shows the non-existence of solitary waves for some parameters.

Theorem 3.1

The equation (3.2) does not admit any nontrivial solution if one of the following conditions holds.

i.

and is even.

ii.

, for all .

Proof: Let be any nontrivial solution of the eq. (3.2) in the class . Multiplying the eq. (3.2) by , integrating on and performing integration by parts twice for the first term, we get

(3.3)

The term on the left side of this equation will be negative, a contradiction, when condition is satisfied.

On the other hand, multiplying the eq. (3.2) by and integrating over yields the Pohozaev type identity

(3.4)

Eliminating terms in the above equations gives

(3.5)

The condition implies that the left hand side is non-negative and the right hand side is negative.

In zeng , the author concerned with the class of following equations

(3.6)

where is a real-valued function and

is a Fourier transform operator defined by

(3.7)

with is an even and real valued function. In Section of zeng , existence and stability of solitary waves are proved assuming where is an integer and satisfy the following conditions:

A1. there exist positive constants and  such that for ;

A2. there exist positive constants , and such that  for ;

A3. for all values of ;

A4. is four time differentiable for all non-zero values of , and for each there exist positive constants and such that

and

Choosing , the equation (3.6) becomes the well-known Rosenau equation. The above theorem applies to the Rosenau equation in which , when . To the best of our knowledge, existence and stability of solitary waves for is an open problem. In Section 5, we will answer this question numerically.

4 The numerical method

We solve the Rosenau equation by combining a Fourier pseudo-spectral method for the space component and a fourth-order Runge Kutta scheme (RK4) for time. If the spatial period is normalized to using the transformation , the equation (1.2) becomes

(4.1)

The interval is divided into equal subintervals with grid spacing , where the integer is even. The spatial grid points are given by , . The approximate solutions to is denoted by . The discrete Fourier transform of the sequence , i.e.

(4.2)

gives the corresponding Fourier coefficients. Likewise, can be recovered from the Fourier coefficients by the inversion formula for the discrete Fourier transform (4.2), as follows:

(4.3)

Here denotes the discrete Fourier transform and its inverse. These transforms are efficiently computed using a fast Fourier transform (FFT) algorithm. In this study, we use FFT routines in Matlab (i.e. fft and ifft).

Applying the discrete Fourier transform to the equation (4.1), we obtain the first order ordinary differential equation

(4.4)

In order to handle the nonlinear term we use a pseudo-spectral approximation. We use the fourth-order Runge-Kutta method to solve the resulting ODE (4.4) in time. Finally, we find the approximate solution by using the inverse Fourier transform (4.3).

5 The numerical experiments

In this section, we present some numerical experiments of the Fourier pseudo-spectral method for the Rosenau equation. To the best of our knowledge, there is no exact solitary wave solution for the single power type nonlinearity . In order to investigate the solitary wave dynamics, we first construct the solitary wave profile as an initial condition by using the Petviashvili’s method. Then, by taking this initial condition the evolution of the single solitary wave and overtaking collision of solitary waves are investigated by using Fourier pseudo-spectral method. Since the exact solitary wave solution is unknown, the ”exact” solitary wave solution is obtained numerically with a very fine spatial step size and a very small time step by using Fourier pseudo-spectral method. In order to quantify the numerical results, the -error norm is defined as

(5.1)

5.1 Accuracy test

In order to test our scheme and to investigate dynamics of the solitary waves, we need an initial condition. The initial condition is generated by using the Petviashvili’s iteration method erkip ; petviashvili ; pelinovsky ; yang . The solitary wave solution of the Rosenau equation satisfies the equation (3.2). Applying the Fourier transform to the equation (3.2) yields

The Petviashvili method for the Rosenau eq. is given by

(5.2)

with stabilizing factor

for some parameter . Here is used instead of for simplicity. The Petviashvili’s iteration method for Rosenau eq. was first introduced in erkip . We refer to erkip for detailed information. The overall iterative process is controlled by the error,