Advances in Mathematical Physics

Volume 2016 (2016), Article ID 2916582, 39 pages

http://dx.doi.org/10.1155/2016/2916582

## Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation

^{1}Institute of Geophysics, National Academy of Sciences of Ukraine, Kyiv 01054, Ukraine^{2}Department of Mathematics & Statistics, University of Strathclyde, Glasgow G1 1XH, UK

Received 13 September 2015; Accepted 12 November 2015

Academic Editor: Andrei D. Mironov

Copyright © 2016 V. O. Vakhnenko and E. J. Parkes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. The VE, which arises in modelling the propagation of high-frequency waves in a relaxing medium, has periodic and solitary traveling wave solutions some of which are loop-like in nature. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE), by a change of independent variables. The VPE has an -soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises -loop-like solitons. Aspects of the inverse scattering transform (IST) method, as applied originally to the KdV equation, are used to find one- and two-soliton solutions to the VPE even though the VPE’s spectral equation is third-order and not second-order. A Bäcklund transformation for the VPE is used to construct conservation laws. The standard IST method for third-order spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to -soliton solutions and -mode periodic solutions, respectively. Interactions between these types of solutions are investigated.

#### 1. Introduction

The physical phenomena and processes that take place in nature generally have complicated nonlinear features. This leads to nonlinear mathematical models for the real processes. There is much interest in the practical issues involved, as well as the development of methods to investigate the associated nonlinear mathematical problems including nonlinear wave propagation. An early example of the latter was the development of the inverse scattering method for the Korteweg-de Vries (KdV) equation [1] and the subsequent interest in soliton theory. Now soliton theory is applied in many branches of science.

The modern physicist should be aware of aspects of nonlinear wave theory developed over the past few years. This paper focuses on the connection between a variety of different approaches and methods. The application of the theory of nonlinear evolution equations to study a new equation is always an important step. Based on our experience of the study of the Vakhnenko equation (VE), we acquaint the reader with a series of methods and approaches which may be applied to certain nonlinear equations. Thus we outline a way in which an uninitiated reader could investigate a new nonlinear equation.

#### 2. A Model for High-Frequency Waves in a Relaxing Medium

Starting from a general idea of relaxing phenomena in real media via a hydrodynamic approach, we will derive a nonlinear evolution equation for describing high-frequency waves. To develop physical models for wave propagation through media with complicated inner kinetics, notions based on the relaxational nature of a phenomenon are regarded to be promising. From the nonequilibrium thermodynamics standpoint, models of a relaxing medium are more general than equilibrium models. Thermodynamic equilibrium is disturbed owing to the propagation of fast perturbations. There are processes of the interaction that tend to return the equilibrium. The parameters characterizing this interaction are referred to as the inner variables unlike the macroparameters such as the pressure , mass velocity , and density . In essence, the change of macroparameters caused by the changes of inner parameters is a relaxation process.

We restrict our attention to barotropic media. An equilibrium state equation of a barotropic medium is a one-parameter equation. As a result of relaxation, an additional variable (the inner parameter) appears in the state equationand defines the completeness of the relaxation process. There are two limiting cases with corresponding sound velocities:(i)Lack of relaxation (inner interaction processes are frozen) for which :(ii)Relaxation which is complete (there is local thermodynamic equilibrium) for which :Slow and fast processes are compared by means of the relaxation time .

To analyze the wave motion, we use the following hydrodynamic equations in Lagrangian coordinates:The following dynamic state equation is applied to account for the relaxation effects:Here is the specific volume, and is the Lagrangian space coordinate. Clearly, for the fast processes , we have relation (2), and for the slow ones we have (3).

The closed system of equations consists of two motion equations (4) and the dynamic state equation (5). The motion equations (4) are written in Lagrangian coordinates since the state equation (5) is related to the element of mass of the medium.

The substantiation of (5) within the framework of the thermodynamics of irreversible processes has been given in [2, 3]. We note that the mechanisms of the exchange processes are not defined concretely when deriving the dynamic state equation (5). In this equation the thermodynamic and kinetic parameters appear only as sound velocities , and relaxation time . These are very common characteristics and they can be found experimentally. Hence it is not necessary to know the inner exchange mechanism in detail.

Let us consider a small nonlinear perturbation . Combining the relationships (4) and (5) we obtain the following nonlinear evolution equation in one unknown (the dash in is omitted) [4–6]:A similar equation has been obtained by Clarke [2], but without nonlinear terms. In [4] it is shown by the multiscale method [7] that for low-frequency perturbations () (6) is reduced to the Korteweg-de Vries-Burgers (KdVB) equation:while for high-frequency waves () we have obtained the following equation:Equation (7) is the well-known KdVB equation. It is encountered in many areas of physics to describe nonlinear wave processes [8]. In [9] it was shown how hydrodynamic equations reduce to either the KdV or Burgers equation according to the choices for the state equation and the generalized force when analyzing gasdynamical waves, waves in shallow water [9], hydrodynamic waves in cold plasma [10], and ion-acoustic waves in cold plasma [11].

As is known, the investigation of the KdV equation () in conjunction with the nonlinear Schrödinger (NLS) and sine-Gordon equations gives rise to the theory of solitons [1, 8, 9, 12–18].

We focus our main attention on (8). It has a dissipative term and a dispersive term . Without the nonlinear and dissipative terms, we have a linear Klein-Gordon equation. At the time we were carrying out our research, it turned out that (8) had not been investigated much. It is likely that this is connected with the fact, noted by Whitham [19], that high-frequency perturbations attenuate very quickly. However in Whitham’s monograph, the evolution equation without nonlinear and dispersive terms was considered.

Note the fact that the dispersion relations for the linearized versions of (7) and (8) are restricted to finite power series in and in , respectively:

Let us write down (8) in dimensionless form. In the moving coordinates system with velocity , after factorization the equation has the form in the dimensionless variables , , (tilde over variables , , is omitted):The constant is always positive. Equation (10) without the dissipative term has the form of the nonlinear equation [20, 21]:Historically, (11) has been called the Vakhnenko equation (VE) and we will follow this name.

It is interesting to note that (11) follows as a particular limit of the following generalized Korteweg-de Vries equation:derived by Ostrovsky [22] to model small-amplitude long waves in a rotating fluid ( is induced by the Coriolis force) of finite depth. Subsequently, (11) was known by different names in the literature, such as the Ostrovsky-Hunter equation, the short-wave equation, the reduced Ostrovsky equation, and the Ostrovsky-Vakhnenko equation depending on the physical context in which it is studied.

The consideration here of (11) has interest not only from the viewpoint of the investigation of the propagation of high-frequency perturbations, but also more specifically from the viewpoint of the study of methods and approaches that may be applied in the theory of nonlinear evolution equations.

#### 3. Loop-Like Stationary Solutions and Their Stability

By investigating (11), we will trace a way in which an uninitiated reader could investigate a new nonlinear equation. As a first step for a new equation, it is necessary to consider the linear analogue and its dispersion relation (these steps for (7) and (8) are described already in Section 2). The next step is, where possible, to link the equation with a known nonlinear equation.

##### 3.1. The Connection of the VE with the Whitham Equation

Now we show how an evolution equation with hydrodynamic nonlinearity can be rewritten in the form of the Whitham equation. The general form of the Whitham equation is as follows [19]:On one hand, (13) has the nonlinearity of hydrodynamic type; on the other hand, it is known (see Section 13.14 in [19]) that the kernel can be selected to give the dispersion required. Indeed, the dispersion relation and the kernel are connected by means of the Fourier transformation:Consequently, for the dispersion relation corresponding to the linearized version of (11), the kernel is as follows:Thus, the VE (11) is related to the particular Whitham equation [19]:Since we can reduce the VE to the Whitham equation, we can assert that the VE shares interesting properties with the Whitham equation; in particular, it describes solitary wave-type formations, has periodic solutions, and explains the existence of the limiting amplitude [19]. An important property is the presence of conservation laws for waves decreasing rapidly at infinity; namely,where by definition .

For (10) the kernel is , where is the Heaviside function. Hence, (10) can be written down asThere is no derivative in the dissipative term of (18).

##### 3.2. The Traveling Wave Solutions

An important step in the investigation of nonlinear evolution equations is to find traveling wave solutions. These are solutions which are stationary with respect to a moving frame of reference. In this case, the evolution equation (a partial differential equation) becomes an ordinary differential equation (ODE) which is considerably easier to solve.

For the VE (11) it is convenient to introduce a new dependent variable and new independent variables and defined bywhere is a nonzero constant [21]. Then the VE becomeswhere corresponding to . We now seek stationary solutions of (20) for which is a function of only so that and satisfies the ODEAfter one integration (21) gives is a constant and for periodic solutions , , and are real constants such that . On using results 236.00 and 236.01 of [23], we may integrate (22) to obtain and are incomplete elliptic integrals of the first and second kind, respectively. We have chosen the constant of integration in (23) to be zero so that at . The relations (23) give the required solution in parametric form, with and as functions of the parameter .

An alternative route to the solution is to follow the procedure described in [24]. We introduce a new independent variable defined byso that (22) becomesBy means of result 236.00 of [23], (26) may be integrated to give , where . Thus, on noting that , where is a Jacobian elliptic function, we haveWith result 310.02 of [23], (25) and (27) givewhere . Relations (27) and (28) are equivalent to (24) and (23), respectively, and give the solution in parametric form with and in terms of the parameter .

We define the wavelength of the solution as the amount by which increases when increases by ; from (23) we obtainwhere and are complete elliptic integrals of the first and second kind, respectively.

For (i.e., ), there are periodic solutions for with , , and ; an example of such a periodic wave is illustrated by curve 2 in Figure 1. gives the solitary wave limitas illustrated by curve 1 in Figure 1. The periodic waves and the solitary wave have a loop-like structure as illustrated in Figure 1. For (i.e., ), there are periodic waves for with , , and ; an example of such a periodic wave is illustrated by curve 2 in Figure 2. When and the periodic wave solution simplifies toThis is shown by curve 1 in Figure 2. For the solution has a sinusoidal form (curve 3 in Figure 2). Note that there are no solitary wave solutions.