Solitary Wave Solutions of the Vakhnenko–parkes Equation

In this paper, two solitary wave solutions are obtained for the Vakhnenko–Parkes equation with power law nonlinearity by the ansatz method. Both topological as well as non-topological solitary wave solutions are obtained. The parameter regimes, for the existence of solitary waves, are identified during the derivation of the solution.


Introduction
The theory of nonlinear evolution equations (NLEEs) is an important area of research in the area of applied mathematics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].A challenging task is to look for solutions of these NLEEs.There are various types of solutions that are available for these equations.Some of them are soliton solutions, solitary wave solutions, cnoidal and snoidal waves, periodic solutions, shock wave solutions as well as various other types.In this paper there will be one such NLEE that will be studied.This is the Vakhnenko-Parkes (VP) equation with power law nonlinearity.The ansatz method will be used to retrieve the topological as well as non-topological solitary wave solution.The domain restrictions will be revealed during the process of obtaining the solutions.c Vilnius University, 2012

Mathematical analysis
The standard form of the VP equation is given by [11] where u(x, t) is a real function of the spatial variable x and the temporal variable t.
In this paper, we are interested in the following family of the VP equation with power law nonlinearity: where a and b are nonzero real constants, while n ∈ Z + .The parameter n indicates the power law nonlinearity parameter.Thus, for a = −1 and b = n = 1, Eq. ( 2) collapses to (1).The purpose of this paper is to calculate the exact topological and non-topological solitary wave solutions for this model, exhibiting power law nonlinearity.The importance of the results presented here is two-fold.First, exact solitary wave solutions to a family of the considered equation and the conditions for their existence are obtained for a general case of power nonlinearity law in a simple way.Importantly, the finding of explicit solutions of a given NLEE with any value of the exponent in the nonlinearity term is very interesting since it offers some knowledge on the general dynamical behavior of the wave propagation so that special cases are truly meaningful both from the physical and mathematical point of view.Second, these results confirm the existence of non-topological solitary waves for any exponent n > 1/2, while topological solitary waves exist only in the case when n = 3/2.Furthermore, closed form solitary wave solutions exist only for a = −1, which corresponds exactly to the value of this coefficient in the standard form of the VP equation (1).To achieve our goal we will use the solitary wave ansatz method which has recently been applied successfully to several NLEEs with constant and variable coefficients.

Non-topological solitary wave
In order to solve (2), the starting hypothesis is [7][8][9][10][11][12][13] and p > 0 for solitary waves to exist.Here, in (3) and ( 4), A is the amplitude of the solitary wave while v is the velocity of the solitary wave and B is the inverse width.The exponent p is unknown at this point and its value will fall out in the process of deriving the solution of this equation.From the ansatz (3), one can find that Inserting the expressions ( 6)-( 8) into ( 2) yields From ( 9), equating the exponents 2p + 2 and p(2n + 1) gives 2p + 2 = p(2n + 1), Taking p > 0 as a necessary condition for the existence of the solitary wave solution (3) implies that we must have n > 1/2 in (10).Now, from ( 9), setting the coefficients of the linearly independent functions tanh τ / cosh 2p+j τ to zero, where j = 0, 2, gives

Solving the above equations yields
The substitution of ( 10) and ( 11) into (12) gives the inverse width of the solitary wave as 1/2 (13) which shows that the solitary waves will exist for b > 0 as long as n > 1/2 which is guaranteed from ( 5) and (10).The width of the solitary wave given by ( 12) or ( 13) is the same by virtue of (10).Thus, the solitary wave solution of the VP equation ( 2) with power law nonlinearity is given by u where the inverse width of the solitary wave B is given by ( 12) or (13).Finally, we would like to note that the solution ( 14) exists under the conditions n > 1/2, b > 0 and a = −1.This solitary wave given by ( 14) is a generalized version of the solitary wave solution that was obtained earlier.In particular when n = 1, (14) collapses to Eqs. (3.4) of [11], (2) of [7], (3) of [8] and (2) of [10].

Topological solitary wave
In this section, we will calculate the topological solitary wave solution of the family of VP equation ( 2) with power law nonlinearity, using the solitary wave ansatz.To start off, the hypothesis is taken to be u(x, t) = A tanh p τ, where and p > 0 for solitary waves to exist.Here, in ( 1) and ( 16), A and B are free parameters and v is the velocity of the wave.Also, the unknown exponent p will be determined during the course of the derivation of the solitary wave solution to (2).Therefore from (15),we have Substituting the expressions (17)-( 19) into (2), we obtain From (20), equating the exponents 2p + 3 and p(2n + 1) + 1 gives 2p + 3 = p(2n + 1) + 1, Nonlinear Anal.Model.Control, 2012, Vol.17, No. 1, 60-66

Conclusions
In this paper, the topological and non-topologial solitary wave solution to the VP equation, with power law nonlinearity, was obtained.In addition, the constraint conditions were also obtained in order for the solitary wave solutions to exist.In the future, this VP equation will be investigated further.The time-dependent coefficients will be considered.Also, several other solutions, using a variety of other methods, will be obtained.These results will be reported in future.