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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 2916582, 39 pages
http://dx.doi.org/10.1155/2016/2916582
Review Article

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

1Institute of Geophysics, National Academy of Sciences of Ukraine, Kyiv 01054, Ukraine
2Department 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) [46]: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, 1218].

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.

Figure 1: Traveling wave solutions with .
Figure 2: Traveling wave solutions with .

A remarkable feature of (11) is that it has a solitary wave (30) which has loop-like form; that is, it is a multivalued function (see Figure 1). Whilst loop solitary waves (30) are rather intriguing, it is the solution to the initial value problem that is of more interest in a physical context. An important question is the stability of the loop-like solutions. Although the analysis of stability does not link with the theory of solitons directly, however, the method applied in Section 3.3 is instructive, since it is successful in a nonlinear approximation.

We note that the notion of a “soliton” will be defined later. We will prove (see Section 5.4) that the solitary wave (30) is, in fact, a soliton. Now we point only out that the soliton is a local traveling wave pulse with remarkable stability and particle-like properties.

3.3. Stability and Interpretation of the Loop-Like Solutions

From a physical viewpoint, the stability or otherwise of solutions is essential for their interpretation. Some methods for the investigation of the stability of nonlinear waves were discussed by Infeld and Rowlands in Chapter 8 of [25] and references therein. One such method is the so-called -expansion method. It is restricted to long wavelength perturbations of small amplitude. It has been applied successfully to a variety of generic nonlinear evolution equations (see [26], e.g.) and specific physical systems (see [27], e.g.). A particularly informative description of the method is given in [28] in the context of the Zakharov-Kuznetsov equation. Some criticism was levelled at the work in [28] by Das et al. [29]; however, after a detailed reinvestigation of the problem, Das et al. [29] vindicated the method used in [28].

The -expansion method was applied to the VE (11) in [21] and is outlined as follows. We assume a perturbed solution of (21) in the formwhere is the periodic solution given by (27) and (28), is a complex function with period given by (29), is a real constant, is a constant (possibly complex), and denotes complex conjugate. Substitution of (32) into (20) and linearization with respect to yieldwhere the linear operator and are given byrespectively. As (21) implies that , we may deduce that, for (33) to have periodic solutions, the conditionmust be satisfied, where denotes an integration over the wavelength .

Formally, the solution of (33) iswhereand is a constant determined fromAs appears on the right-hand side of (36) via , we solve (36) iteratively by assuming that is small in comparison with (so that the perturbations in (32) have long wavelength) and introduce the expansionsso that At zero order in , condition (35) is satisfied identically, (38) gives , and then, from (37), is constant. Hence, from (36), we may take . At first order, condition (35) is again satisfied identically. It is straightforward (see [21]) to find from (38) and from (37); use of these expressions in (35) at second order leads to the desired nonlinear dispersion relation for the perturbations in the formThe coefficients , , and depend on , , and as defined in (22). It turns out that the dispersion relation (41) has real roots for for both the families of solutions (corresponding to and , resp.) derived in Section 3.2. Consequently, it is predicted that both families of solutions are stable to long wavelength perturbations. For the loop-like solutions, the existence of singular points at which the derivatives tend to infinity casts some doubts on the validity of the method. However, in [21] it is argued that as the method depends on the average behaviour over a wavelength, the method is indeed valid.

The ambiguous structure of the loop-like solutions is similar to the loop soliton solution to an equation that models a stretched rope [30]. Loop-like solitons on a vortex filament were investigated by Hasimoto [31] and Lamb [32]. From the mathematical point of view, an ambiguous solution does not present difficulties whereas the physical interpretation of ambiguity always presents some difficulties. In this connection the problem of ambiguous solutions is regarded as important. The problem consists in whether the ambiguity has a physical nature or is related to the incompleteness of the mathematical model, in particular to the lack of dissipation.

We will consider the problem related to the singular points when dissipation takes place. At these points the dissipative term tends to infinity. The question arises: are there solutions of (18) in a loop-like form? The fact that the dissipation is likely to destroy the loop-like solutions can be associated with the following well-known fact [8]. For the simplest nonlinear equation without dispersion and without dissipation, namely,any initial smooth solution with boundary conditionsbecomes ambiguous in the final analysis. When dissipation is considered, we have the Burgers equation [33]:The dissipative terms in this equation and in (7) for low frequency are coincident. The inclusion of the dissipative term transforms the solutions so that they cannot be ambiguous as a result of evolution. The wave parameters are always unambiguous. What happens in our case for high frequency when the dissipative term has the form (see (18))? Will the inclusion of dissipation give rise to unambiguous solutions?

By direct integration of (10) (written in terms of variables (19)) within the neighborhood of singular points , where and , it can be derived (see [4]) that the dissipative term, with dissipation parameter less than some limit value , does not destroy the loop-like solutions. Now we give a physical interpretation to ambiguous solutions.

Since the solution to the VE has a parametric form (23), (24) or (27), (28), there is a space of variables in which the solution is a single-valued function. Hence, we can solve the problem of the ambiguous solution. A number of states with their thermodynamic parameters can occupy one microvolume. It is assumed that the interaction between the separated states occupying one microvolume can be neglected in comparison with the interaction between the particles of one thermodynamic state. Even if we take into account the interaction between the separated states in accordance with the dynamic state equation (5) then, for high frequencies, a dissipative term arises which is similar to the corresponding term in (8), but with the other relaxation time. In this sense the separated terms are distributed in space, but describing the wave process we consider them as interpenetrable. A similar situation, when several components with different hydrodynamic parameters occupy one microvolume, has been assumed in mixture theory (see, e.g., [34, 35]). Such a fundamental assumption in the theory of mixtures is physically impossible (see [34, page 7]), but it is appropriate in the sense that separated components are multivelocity interpenetrable continua.

Consequently, the following three observations show that, in the framework of the approach considered here, there are multivalued solutions when we model high-frequency wave processes: (1) All parts of loop-like solution are stable to perturbations. (2) Dissipation does not destroy the loop-like solutions. (3) The investigation regarding the interaction of the solitons has shown that it is necessary to take into account the whole ambiguous solution, not just the separate parts.

4. The Vakhnenko-Parkes Equation

The multivalued solutions obtained in Section 3.3 obviously mean that the study of the VE (11) in the original coordinates leads to certain difficulties. These difficulties can be avoided by writing down the VE in new independent coordinates. We have succeeded in finding these coordinates. Historically, working separately, we (Vyacheslav Vakhnenko in Ukraine and John Parkes in UK) independently suggested such independent coordinates in which the solutions become one-valued functions. It is instructive to present the two derivations here. In one derivation a physical approach, namely, a transformation between Euler and Lagrange coordinates, was used whereas in the other derivation a pure mathematical approach was used.

Let us define new independent variables by the transformationThe function is to be obtained. It is important that the functions and turn out to be single-valued. In terms of the coordinates the solution of the VE (11) is given by single-valued parametric relations. The transformation into these coordinates is the key point in solving the problem of the interaction of solitons as well as explaining the multiple-valued solutions [4]. Transformation (45) is similar to the transformation between Eulerian coordinates and Lagrangian coordinates . We require that if there is no perturbation, that is, if . Hence when .

The function is the additional dependent variable in the equation system (47) and (48) to which we reduce the original equation (11). We note that the transformation inverse to (45) isThen, by taking into account the condition that is an exact differential, we obtainThis equation, together with (11) rewritten in terms of and , namely,is the main system of equations. The equation system (47) and (48) can be reduced to a nonlinear equation in one unknown defined byWe study solutions that vanish as or, equivalently, solutions for which tends to a constant as . From (47) and (49) and the requirement that as we have ; then, by eliminating from (48) we arrive at the transformed form of the VE (11)or, in equivalent form,

Furthermore it follows from (46) that the original independent space coordinate is given bywhere is an arbitrary constant. Since the functions and are single-valued, the problem of multivalued solutions has been resolved from the mathematical point of view.

Alternatively, in a pure mathematical approach, we introduce new independent variables , defined bywhere and is a constant. From (53) it follows thatso thatFrom (11) and (54) we obtainBy eliminating between (55) and (56) we obtain (51) or, on introducing , (50).

The transformation, as has already been pointed out, was obtained by us independently of each other; nevertheless, we published the result together [3638]. Following [3942], hereafter (50) (or in alternative form (51)) is referred to as the Vakhnenko-Parkes equation (VPE).

For example, we will rewrite the solutions (23) and (24) for (11) in the transformed coordinates (,), that is, find the traveling wave solutions for (11) in new coordinates. Differentiating the relationship (23) with respect to , we takeorThen after integration, we obtainTogether with this relationship (59) determines the desired dependence in parametrical form. Thus, we have the solution for (11) in new coordinates .

The solutions for in coordinates are illustrated in Figure 3. Curves 1 and 2 in this figure relate to curves 1 and 2 in Figure 1. The solutions in coordinates for are plotted in Figure 4. Curves 1, 2, and 3 in this figure relate to curves 1, 2, and 3 in Figure 2.

Figure 3: Traveling wave solutions with in coordinates .
Figure 4: Traveling wave solutions with in coordinates .

On one hand, we have attained the goal; namely, we have found the solutions in new coordinates in which the solutions become one-valued functions. On the other hand, it is important that periodical solution shown by curve 1 in Figure 3, that is, the solution consisting of parabolas, becomes not periodical in new coordinates. Hence, we reveal some accordance between curve 1 in Figure 1 and curve 1 in Figure 2. This feature is important for finding the solutions by inverse scattering method [38, 4347].

5. Hirota Method

Now let us define the notion “soliton” more precisely. Apart from the fact that a soliton is a stable solitary wave with particle-like properties, a soliton must possess additional properties. One property is that two such solitary waves may pass through each other without any loss of identity. Consider two solitons with different speeds, the faster one chasing the slower one. The faster soliton will eventually overtake the slower one. After the nonlinear interaction, two solitons again will emerge, with the faster one in front, and each will regain its former identity precisely. The only interaction memory will be a phase shift; each soliton will be centered at a location different from where it would have been had it traveled unimpeded. However, this property is still not sufficient in order that the solitary wave be a soliton. There are equations which possess solutions which are a nonlinear superposition of two solitary waves but which do not have all the properties enjoyed by soliton equations. A soliton equation, when it admits solitary wave solutions, must possess a solution which satisfies the “-soliton condition” (see Section 5.3). The solitary wave with these properties defines a soliton. The term “soliton” was originally coined by Zabusky and Kruskal in 1965 [48].

One of the key properties of a soliton equation is that it has an infinite number of conservation laws. These soliton equations satisfy the Hirota condition (“-soliton condition”) and are exactly integrable.

The Hirota method not only gives the -soliton solution, but also enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering method. Thus the Hirota method, which can be applied only for finding solitary wave solutions or traveling wave solutions, allows us to formulate the inverse scattering method which is the most appropriate way of tackling the initial value problem (Cauchy problem). Consequently, in this case, the integrability of an equation can be regarded as proved.

5.1. The -Operator and -Soliton Solution

Various effective approaches have been developed to construct exact wave solutions of completely integrable equations. One of the fundamental direct methods is undoubtedly the Hirota bilinear method [14, 15, 49, 50], which possesses significant features that make it practical for the determination of multiple soliton solutions.

In the Hirota method the equation under investigation should first be transformed into the Hirota bilinear form [14]where is a polynomial in and . Each equation has its own polynomial. The Hirota bilinear -operator is defined as (see Section  5.2 in [14])If the polynomial satisfies conditions (see and in [14])then the Hirota method can be applied successfully.

According to [14], the -soliton solution reads as follows:whereThe connection between and is found by the dispersion relationsIn (63), means the summation over all possible combinations of or 1, or or 1, and means the summation over all possible combinations of elements under the condition .

Moreover, for there to be an -soliton solution (NSS) to (60) with arbitrary, must satisfy the “-soliton condition” (NSC) [14]; namely,whereand, for ,In (68) are given in terms of by the dispersion relations (), means the summation over all possible combinations of , , and is a function of that is independent of the summation indices .

From (67) it follows that (66) is satisfied for . If , then (66) is satisfied for . However, whether or not (66) is satisfied for depends on the particular form of , that is, on the original equation being studied.

5.2. Bilinear Form of the Vakhnenko-Parkes Equation

In order to find soliton solutions to VPE (50)by using Hirota’s method [14] we need to express (50) in Hirota form [36]. Transformation (53) of the independent variables in the original equation (11) is a key step in finding an exact explicit -soliton solution to (50) by use of the Hirota method and hence an exact implicit -soliton solution to (11). By takingwe find thatand the bilinear form of the VPE is as follows:

In passing we note that the Hirota-Satsuma equation (HSE) for shallow water waves [51]may be written asor in bilinear formClearly (74) and (75) are similar to, but cannot be transformed into, (50) and (72), respectively. Hence solutions to the HSE cannot be transformed into solutions of the VPE. The solution to the HSE by Hirota method is given in [51].

The Hirota method can be applied successfully if we can prove “-soliton condition” (NSC) (66)–(68) for (72). Let us present this proof [37].

5.3. The “-Soliton Condition” for the VPE

Since, for (72), we have , then (66) is satisfied for .

With given by (72), the dispersion relations give and (66) may be written:The presence of the first product term in (76) ensures that is a homogeneous polynomial in .

In passing we remark that previous work suggests, but does not prove, that (72) does have an NSS for all . The expression for given by (72) is a special case of one proposed by Ito (see Equation in [52]). Ito claimed that this satisfies 3SC. Hietarinta [53] performed a search for bilinear equations of the form (72) that have that satisfies 3SC. One such was found to be the one given by (72). Hietarinta [54] later claimed that this also passed 4SC. The bilinear equation (72) is a special case of one given in Grammaticos et al. (see Equation in [55]); they showed that this equation has the Painlevé property. According to Hietarinta [54] a bilinear equation that has 4SS and the Painlevé property is almost certainly integrable. All this evidence suggests that it is highly likely that (72) does have an NSS for all . Here we remove any doubt by using induction to prove that condition (66) is satisfied with given by (76).

We need the following properties of (as given by (76)) for :(i).(ii).(iii)        . (We established property (iii) by adapting the argument used to obtain equation in [56] in the context of a shallow water wave equation.) Furthermore, because of summation in (76), is an odd, symmetric function of . As already noted, the condition (66) is satisfied for and . We now assume that the condition is satisfied for all , where ; then the properties of imply that it may be factorized as follows:where is a homogeneous polynomial. It follows that the degree of is at least . On the other hand, from (76) the degree of is at most . As for , it follows that . It now follows by induction that the NSC is satisfied.

5.4. The -Soliton Solution of the VPE

With given by (72) for the VPE the dispersion relations (65) () give and thenAlso, without loss of generality, we may take and thenso that .

Consequently, the relationship (63) with (79) and (80) gives for the VPE. Finally, substitution of (63) into (70) gives the -soliton solution of the VPE [37].

However, following Moloney and Hodnett [57], it is more convenient to express in the formfor a given with , whereThen we may write the -soliton solution for VPE (50) in the formFrom (83) and the relationship , the -soliton solution for VPE (51) isWith and given by (83) and (84), respectively, and by using (45), (46), and (52), we may write the -loop soliton solution to VE (11) in the parametric form:

5.5. The One-Loop Soliton Solution

The solution to (72) corresponding to one soliton is given byand , , and are constants. The dispersion relation (65) is from which we find that and thenSubstitution of (86) into (70) givesso thatThe one-loop soliton solution to the VE is given by (85) with (88) and (89). From (85) with we haveClearly, from (89) and (90), and are related by the parameter so that is a soliton that travels with speed in the positive -direction. The fact that this soliton is a loop may be shown as follows. From (54) we have , and on using (87) and (89) we also have and . HenceThus, as goes from to in (90), so that goes from to , changes sign once and remains finite whereas given by (91) changes sign three times and goes infinite twice. The one-loop soliton solution may be written in terms of the parameter aswith and arbitrary. Solution (92) is essentially the one-loop soliton solution given by (30) (see [20, 21] too).

Usually it is assumed that the value is real in order that the solution is a real function. However, the real solution is obtained also at ( is real) (see Sections 9.1 and 11.2 and Appendix B). In this case the soliton solution (singular soliton solution) is discontinuous [58]

5.6. The Two-Loop Soliton Solution

The solution to (72) corresponding to two solitons is given byand , , and are constants. The dispersion relation is from which we find that and thenWithout loss of generality we may take and thenSubstitution of (94) into (70) gives the two-soliton solution of the VPE. Following Hodnett and Moloney [57, 59], we may write in the formIt follows that may be written:The two-loop soliton solution to the VE is given by (85) with (98) and (100) [36].

5.7. Discussion of the Two-Loop Soliton Solution

We now consider the two-loop soliton solution found in Section 5.6 in more detail. First it is instructive to consider what happens in space.

As , we haveFrom (99) and (100) with (101) it follows that, with fixed,Similarly, from (99) and (100) with (102), with fixed,Hence it is apparent that, in the limits , and may be identified as individual solitons moving with speeds and , respectively, in the positive -direction. In contrast to the familiar interaction of two KdV “sech squared” solitons [60], here it is the smaller soliton that overtakes the larger one.

The shifts of the two solitons and in the positive -direction due to the interaction arerespectively. As , the smaller soliton is shifted forwards and the larger soliton is shifted backwards. Since the “mass” of each soliton is given by , where we have used (100), and the shifts satisfy , “momentum” is conserved.

Let and recall that here we are assuming that . (From (103) and (104), is the ratio of the amplitudes of the individual smaller and larger solitons.) Note that for , where is the center of the interaction. For , we have and the two-soliton solution in - space always has two peaks; during interaction the two humps exchange amplitudes. For , we have and the two humps of the individual solitons coalesce into a single hump for part of the interaction; the smaller hump appears to pass through the larger one.

Now let us consider what happens in - space. From (85) with we haveNote that in (103) taking the limits with fixed is equivalent to taking the limits with fixed; also note that from (45). Accordingly from (103) and (106) with we see that, in the limits with fixed, and are related by the parameter . Similarly, from (104) and (106) with , in the limits with fixed, and are related by the parameter . It follows that, in the limits , and may be identified as individual loop solitons moving with speeds and , respectively, in the positive -direction, where . As , the larger loop soliton overtakes the smaller loop soliton.

The shifts, , of the two-loop solitons and in the positive -direction due to the interaction may be computed from (106) as follows. From (103), as , , where ; then and, by use of (101), . Similarly, as , , where ; then and . Use of these results in (106) with givesBy use of (102), (104), and (106) with , a similar calculation yieldsFrom (108) it is found that, for , so that the larger loop soliton is always shifted forwards by the interaction. However, for we find that, (a)for , so the smaller loop soliton is shifted backwards;(b)for , where is the root of , so the smaller loop soliton is not shifted by the interaction;(c)for , so the smaller loop soliton is shifted forwards.

At first sight it might seem that the behaviour in (b) and (c) contradicts conservation of “momentum.” The fact that this is not so is justified as follows. By integrating (11) with respect to we find that ; also, by multiplying (11) by and integrating with respect to we obtain . Thus, in space, the “mass” of each soliton is zero, and “momentum” is conserved whatever and may be. In particular and may have the same sign as in (c), or one of them may be zero as in (b).

Cases (a), (b), and (c) are illustrated in Figures 5, 6, and 7, respectively; in these figures is plotted against for various values of . For convenience in the figures, the interactions of solitons are shown in coordinates moving with speed .

Figure 5: The interaction process for two-loop solitons with and so that and .
Figure 6: The interaction process for two-loop solitons with and so that and .
Figure 7: The interaction process for two-loop solitons with and so that and .
5.8. Discussion of the -Loop Soliton Solution

We now interpret the -loop soliton solution found at beginning of Section 5.4 in terms of individual loop solitons [37].

First it is instructive to consider what happens in space. From (83) and (84) and the fact that we deduce the following behaviour: with fixed and ,with fixed and ,Hence it is apparent that, in the limits , each may be identified as an individual soliton moving with speed in the positive -direction. Smaller solitons overtake larger ones.

The shifts, , of the solitons in the positive -direction due to the interactions between the solitons are given bySince the “mass” of each soliton is given by , where we have used (84), and the shifts satisfy“momentum” is conserved.

Now let us consider what happens in space. From (85) with we haveNote that in (109) and (110) taking the limits with fixed is equivalent to taking the limits with fixed; also note that from (53). Accordingly from (109), (110), and (113), with a given , we see that, in the limits with fixed, and are related by the parameter . It follows that, in the limits , may be identified as an individual loop soliton moving with speed in the positive -direction, where . As , larger loop solitons overtake smaller ones.

In order to calculate the shifts, , of the loop solitons in the positive -direction due to the interactions between the -loop solitons, we need the following results: from (109), as , , wherefrom (110), as , , whereUse of these results in (113) gives

The three-loop soliton solution is discussed in detail in [37]. There, the interaction of three-loop solitons is illustrated in figures. The interaction process is more complicated than that for the two-loop soliton solution [36] given in Sections 5.6 and 5.7. From the three examples illustrated in [37] it is clear that several different types of interaction are possible and it is not always possible to predict what will happen on the basis of the results in [36] alone.

6. The Vakhnenko-Parkes Equation from the Viewpoint of the Inverse Scattering Method for the KdV Equation

Unlike the earlier Sections 5.55.8 where the interaction of the solitons was studied by the Hirota method [14, 15, 49], we use now elements of the inverse scattering transform (IST) method as developed for the KdV equation [38]. The formulation of the IST method is discussed for the Vakhnenko-Parkes equation (50). It is shown that the equation system for the inverse scattering problem associated with the VPE cannot contain the isospectral Schrödinger equation. The results of this section were completed before we made appreciable progress in formulation of the IST problem for the VPE.

As we will prove later in Section 8.1, the spectral problem associated with the VPE is of third order [43, 6163]. At first reading, the present section can be omitted. Nevertheless, methods stated here may be useful in the investigation of a new equation for which the spectral problem is unknown.

6.1. One-Soliton Solutions as Reflectionless Potentials

As was noted previously, the VE (11)and the KdV equationhave the same hydrodynamic nonlinearity and do not contain dissipative terms; only the dispersive terms are different. The similarity between these equations indicates that, in studying the VE and the VPE (50), the application of the IST method should be possible. The IST method is the most appropriate way of tackling initial value problems. The results of applying the IST method would be useful in solving the Cauchy problem for both the VE and the VPE. The study of the VPE is of scientific interest from the viewpoint of the general problem of integrability of nonlinear equations.

The method of the IST is a powerful method as a means for solving the nonlinear differential equations. Let us recall that KdV equation (118) is associated with the system of the equationsEquation system (119) and (120) is a case of the IST method presented in the classic paper [1]. Since system (119) and (120) contains Schrödinger equation (119), we will use the elements of the IST method as applied to the KdV equation in order to analyze the VPE. The known one-soliton solution of KdV equation (118) has the form (without the time-dependence)Here, as an example, we will consider the case .

The results in this paper are based on the assumption that the system of equations associated with VPE (50), which are analogous to (119) and (120), is unknown.

Now let us focus on the fact that (48) is the Schrödinger equationwith the eigenvalue (energy) and potential . Equation (48) determines the dependence on the coordinate , and time appears here as a parameter. However, the time-dependence is determined by (47).

The known one-soliton solution of (50), which we obtained in Section 5.5, has the formIf it is not otherwise noted, for convenience here we will consider and , and then (123) reduces toThe principal fact is that both from (121) and from (124) relate to reflectionless potentials. The general form of the reflectionless potentials is (see Section  2.4 in [8])We have for the potential (121) and for the potential (124). It is known [8, 13] that, for integrable nonlinear equations, reflectionless potentials generate soliton solutions (in the general case, -soliton solutions).

6.2. Two-Level Reflectionless Potential

Let us consider the one-soliton solution of system (119) and (120) in the framework of the IST method for the KdV equation. For this purpose let us analyze the Schrödinger equation with the potential ( is a parameter)For the scattering problem, the solution of  (126) should satisfy the boundary conditionswhere and are the coefficients of reflection and transmission, respectively.

In Section  2.4 in [8], the original method for finding the wave-functions and eigenvalues for the reflectionless potential was described. The general solution of (126) for the potential connects with the general solution for by the relationshipand thenIn our case () (126) has two bound statesThe wave-functions are normalized; that is, , and this conforms to the requirement used in the IST method.

Here the main difference between the VPE and the known integrable nonlinear equations appears. It is connected with the existence of only one bound state for the known equations associated with the isospectral Schrödinger equation, while for the VPE two bound states occur. Indeed, for the known integrable equations, the potential corresponding to the one-soliton solution has the following dependence on the space coordinate (see Equation in [8])It is easy to see that this is related to the case in (125); that is, there is only the one bound state