Open access peer-reviewed chapter

Loop-like Solitons

By Vyacheslav O. Vakhnenko, E. John Parkes and Dmitri B. Vengrovich

Submitted: February 23rd 2019Reviewed: April 29th 2019Published: September 27th 2019

DOI: 10.5772/intechopen.86583

Downloaded: 106

Abstract

The physical phenomena that take place in nature generally have complicated nonlinear features. 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. One remarkable feature of the VE is that it possesses loop-like soliton solutions. Loop-like solitons are a class of interesting wave phenomena, which have been involved in some nonlinear systems. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE). The VPE can be written in Hirota bilinear form. The Hirota method not only gives the N-soliton solution but enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering transform (IST) method. This method is the most appropriate way of tackling the initial value problem (Cauchy problem). The standard procedure for IST method is expanded for the case of multiple poles, specifically, for the double poles with a single pole. In recent papers some physical phenomena in optics and magnetism are satisfactorily described by means of the VE. The question of physical interpretation of multivalued (loop-like) solutions is still an open question.

Keywords

  • nonlinear evolution equations
  • solutions
  • Vakhnenko equation
  • Hirota method
  • Bäcklund transformation
  • inverse scattering problem
  • N-soliton solution
  • spectral data
  • PACS: 00.30.Lk
  • 02.30.Jr
  • 05.45.Yv

1. The high-frequency perturbations in a relaxing medium

From the nonequilibrium thermodynamic standpoint, models of a relaxing medium are more general than equilibrium models. 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. 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 p, mass velocity uand density ρ. 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 ξ(the inner parameter) appears in the state equation

p=pρξE1

and defines the completeness of the relaxation process. There are two limiting cases with corresponding sound velocities:

  1. Lack of relaxation (inner interaction processes are frozen) for which ξ=1:

p=pρ1pfρ,cf2=dpf/;E2

  1. Relaxation which is complete (there is local thermodynamic equilibrium) for which ξ=0:

p=pρ0peρ,ce2=dpe/.E3

Slow and fast processes are compared by means of the relaxation time τp.

To analyse the wave motion, we use the following hydrodynamic equations in Lagrangian coordinates:

Vt1ρ0ux=0,ut+1ρ0px=0.E4

The following dynamic state equation is applied to account for the relaxation effects:

τpdpdtcf2dt+ppe=0.E5

Here Vρ1is the specific volume and xis the Lagrangian space coordinate. Clearly, for the fast processes ωτp1, we have relation (2), and for the slow ones ωτp1, we have (3).

The closed system of equations consists of two motion equations (4) and 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 [1, 2]. 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 ceand cfand relaxation time τp. These are very common characteristics and they can be found experimentally. Hence, it is not necessary to know the inner exchange mechanism in detail.

Combining the relationships (4) and (5), we obtain for low-frequency perturbations (τpω1) the Korteweg-de Vries-Burgers (KdVB) equation:

pt+cepx+αece3ppxβe2px2+γe3px3=0,βe=ce2τp2cf2cf2ce2,γe=ce3τp28cf4cf2ce2cf25ce2,E6

whilst for high-frequency waves (τpω1), we have obtained the following equation:

2px2cf22pt2+αfcf22p2x2+βfpx+γfp=0,βf=cf2ce2τpce2cf,γf=cf4ce42τp2ce4cf2.E7

Equation (6) with (βe=0) is the well-known the Korteweg-de Vries (KdV) equation. The investigation of the KdV equation in conjunction with the nonlinear Schrodinger (NLS) and sine-Gordon equations gives rise to the theory of solitons [4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

We focus our main attention on (7). It has a dissipative term βfp/xand a dispersive term γfp. Without the nonlinear and dissipative terms, we have a linear Klein-Gordon equation.

Let us write down (7) in dimensionless form. In the moving coordinate system with velocity cf, after factorization the equation has the form in the dimensionless variables x˜=γf2xcft, t˜=γf2cft, u˜=αfcf2p(tilde over variables x˜, t˜and u˜is omitted)

xt+uxu+αux+u=0.E8

The constant α=βf/2γfis always positive. Equation (8) without the dissipative term has the form of the nonlinear equation [14, 15]:

xt+uxu+u=0.E9

Historically, (9) has been called the Vakhnenko equation (VE), and we will follow this name.

We note that (9) follows as a particular limit of the following generalized Korteweg-de Vries equation:

xut+uuxβ3ux3=γuE10

derived by Ostrovsky [16] to model small-amplitude long waves in a rotating fluid (γuis induced by the Coriolis force) of finite depth. Subsequently, (9) 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 (9) has interest from the viewpoint of the investigation of the propagation of high-frequency perturbations.

2. Loop-like stationary solutions

The travelling wave solutions 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 (9) it is convenient to introduce a new dependent variable zand new independent variables ηand τdefined by

z=uv/v,η=xvt/v1/2,τ=tv1/2,E11

where vis a nonzero constant [15]. Then the VE becomes begin equation:

zητ+zzηη+z+c=0,E12

where c=±1corresponding to v0. We now seek stationary solutions of (12) for which zis a function of ηonly so that zτ=0and zsatisfies the ODE:

zzηη+z+c=0.E13

After one integration (13) gives

12zzη2=fz,fz=13z312cz2+16A=13zz1zz2zz3.E14

where Ais a constant, and for periodic solutions z1, z2and z3are real constants such that z1z2z3. On using results 236.00 and 236.01 of [17], we may integrate (14) to obtain

η=6z1z3z1Fφm+6z3z1Eφm,E15
sinφ=z3zz3z2,m=z3z2z3z1.E16

where Fφmand Eφmare incomplete elliptic integrals of the first and second kind, respectively. We have chosen the constant of integration in (15) to be zero so that z=z3at η=0. The relations (15) give the required solution in parametric form, with zand ηas functions of the parameter φ.

For c=1(i.e., v>0), there are periodic solutions for 0<A<1with λ<0, z210and z30,0.5; an example of such a periodic wave is illustrated by curve 2 in Figure 1. Here we introduce a new independent variable ζdefined by

Figure 1.

Travelling wave solutions with v > 0.

=z.E17

A=1gives the solitary wave limit:

u=32vsech2ζ/2,η=ζ+3tanhζ/2E18

as 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 c=1(i.e., v<0), there are periodic waves for 1<A<0with λ>0, z201and z311.5; an example of such a periodic wave is illustrated by curve 2 in Figure 2. When A=0and λ=6, then the periodic wave solution simplifies to

Figure 2.

Travelling wave solutions with v < 0.

uη/v=16η2+12,3η3,uη+6=uη.E19

This is shown by curve 1 in Figure 2. For A1the solution has a sinusoidal form (curve 3 in Figure 2). Note that there are no solitary wave solutions.

A remarkable feature of the equation (9) is that it has a solitary wave (18) which has a loop-like form, i.e., it is a multivalued function (see Figure 1). Whilst loop solitary waves (18) 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, the method applied in [15] is instructive, since it is successful in a nonlinear approximation. Stability of the loop-like solutions has been proved in [15]. From a physical viewpoint, the stability or otherwise of solutions is essential to their interpretation.

3. The Vakhnenko-Parkes equation

The multivalued solutions obtained in Section 2 obviously mean that the study of the VE (9) in the original coordinates xtleads 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 the 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 XTby the transformation

φdT=dxudt,X=t.E20

The function φis to be obtained. It is important that the functions x=θXTand u=UXTturn out to be single-valued. In terms of the coordinates XT, the solution of the VE (9) 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 multivalued solutions [3]. The transformation (20) is similar to the transformation between Eulerian coordinates xtand Lagrangian coordinates XT. We require that T=xif there is no perturbation, i.e., if uxt0. Hence φ=1when uxt0.

The function φis the additional dependent variable in the equation system (22), (24) to which we reduce the original Eq. (9). We note that the transformation inverse to (20) is

dx=φdT+UdX,t=X,UXTuxt.E21

It follows that

xX=U,xT=φ,tX=1,tT=0.

Hence

φX=UTE22

and

X=t+ux,T=φx.E23

By using (23), we can write Eq. (9) in terms of φXTand UXT, namely,

UXT+φU=0.E24

Equations (22) and (24) are the main system of equations. It can be reduced to a nonlinear equation (27) in one unknown Wdefined by

WX=U.E25

From (22), (25) and the requirement that φ=1when U0, we have

φ=1+WT.E26

Then, by eliminating φand Ubetween (24), (25) and (26), we arrive at a transformed form of the VE (9), namely,

WXXT+1+WTWX=0.E27

Alternatively, by eliminating φbetween (22) and (24), we obtain

UUXXTUXUXT+U2UT=0.E28

Furthermore it follows from (21) that the original independent coordinates xtare given by

x=θXT=x0+T+W,t=X,E29

where x0is an arbitrary constant. Since the functions θXTand UXTare single-valued, the problem of multivalued solutions has been resolved from the mathematical point of view.

Alternatively, in a pure mathematical approach, we may start by introducing new independent variables Xand Tdefined by

x=T+XUXTdX+x1,t=X,E30

where x1is an arbitrary constant. From (30), we obtain (23) but with

φXT=1+XUTdX.E31

Now, on introducing (25), (30) and (31) may be identified with (29) and (26), respectively. The derivation of (27) and (28) proceeds as before.

The transformation into new coordinates, as has already been pointed out, was obtained by us independently of each other; nevertheless, we published the result together [18, 19]. Following the papers [20, 21, 22, 23] hereafter, Eq. (27) (or in alternative form (28)) is referred to as the Vakhnenko-Parkes equation (VPE).

The travelling wave solution (15) and (16) for Equation (9) is also a travelling wave solution when written in terms of the transformed coordinates (X,T). In order to do this, we need to express the independent variable ζ, as introduced in (17), in terms of Xand T.

From the expressions for zin (11) and (17), we obtain

=UvvE32

so that

vη=U.E33

From the definition of ηin (17), and the expressions for xand tgiven by (29), we obtain

vη=v1/2WvXVT,whereVv1.E34

The expressions for vηin (33) and (34) are equivalent if

ζ=v1/2Z,whereZXVTX0E35

and X0is an arbitrary constant, so that

W=UdZandU=WZ.E36

Hence, from (34), it follows that

W=vpz1+cw+z3z1E(wm)+W0,E37

where w=pvZand W0is an arbitrary constant. Then

Uv=c+z3z3z2sn2wm,wherew=pvZ.E38

Eqs. (37) and (38) give the travelling wave solutions to the VPE in the forms (27) and (28), respectively. Eq. (38) is also the travelling wave solution of the VE (9) expressed in terms of the new coordinates (X,T). In the limiting case m=1, (38) gives a solitary wave in the following two forms: For v>0

U/v=32sech212vZE39

and, for v<0,

U/v=1+32sech212vZ.E40

These two solutions are illustrated by curve 1 in Figures 3 and 4, respectively. The other curves illustrate examples of the solution given by (38) when m1. Curves 1 and 2 in Figure 3 relate to curves 1 and 2, respectively, in Figure 1. Curves 1, 2 and 3 in Figure 4 relate to curves 1, 2 and 3, respectively, in Figure 2.

Figure 3.

Travelling wave solutions with v > 0 in coordinates (X, T).

Figure 4.

Travelling wave solutions with v < 0 in coordinates (X, T).

There are two important observations to be made. Firstly, all the travelling wave solutions in terms of the new coordinates are single-valued. Secondly, the periodic solution shown by curve 1 in Figure 2, i.e., the solution consisting of parabolas, is not periodic in terms of the new coordinates. Hence, we reveal some accordance between curve 1 in Figure 3 and curve 1 in Figure 4. These features are important for finding the solutions by the inverse scattering method [24, 25, 26, 27, 28, 29, 30].

4. From Hirota method to the inverse scattering method

The Hirota method gives the N-soliton solution as well as enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering method [24]. Thus, the Hirota method allows us to formulate the inverse scattering method which is the most appropriate way of tackling the initial value problem (Cauchy problem).

In the Hirota method the equation, in our case the VPE (27), under investigation should be transformed into the Hirota bilinear form [9, 24]:

DTDX3+DX2ff=0,E41

with

W=6lnfX,E42

The Hirota bilinear D-operator is defined as (see Section 5.2 in [9])

DTnDXmab=TTnXXmaTXb(TX)T=T,X=X.E43

Now we present a Bäcklund transformation for VPE (27) written in the bilinear form (41). This type of Bäcklund transformation was first introduced by Hirota [31] and has the advantage that the transformation equations are linear with respect to each dependent variable. This Bäcklund transformation can be transformed to the ordinary one [24]:

DX3λff=0,E44
3DXDT+1+μDXff=0,E45

where λ=λXis an arbitrary function of X and μ=μTis an arbitrary function of T.

The inverse scattering transform (IST) method is arguably the most important discovery in the theory of solitons. The method enables one to solve the initial value problem for a nonlinear evolution equation. Moreover, it provides a proof of the complete integrability of the equation.

The essence of the application of the IST is as follows. The initial equation VPE (27) is written as the compatibility condition for two linear equations. These equations are presented in (47) and (48). Then WX0is mapped into the scattering S0for (47). It is important that since the variable WXTcontained in the spectral equation (47) evolves according to (27), the spectrum λalways retains constant values. The time evolution of STis simple and linear. From a knowledge of ST, we reconstruct WXT.

The use of the IST is the most appropriate way of tackling the initial value problem. In order to apply the IST method, one first has to formulate the associated eigenvalue problem. This can be achieved by finding a Bäcklund transformation associated with the VPE.

Now we will show that the IST problem for the VPE in the form (27) has a third-order eigenvalue problem that is similar to the one associated with a higher-order KdV equation [32, 33], a Boussinesq equation [33, 34, 35, 36, 37] and a model equation for shallow water waves [9, 38].

Introducing the function

ψ=f/f,E46

and taking into account (42), we find that (44) and (45) reduce to

ψXXX+WXψXλψ=0,E47
3ψXT+1+WTψ+μψX=0,E48

respectively, where we have used results similar to (X.1)–(X.3) in [9].

From (47) and (48), it can be shown that

3λψT+1+WTψXXWXTψX+WXXT+1+WTWX+μλψ=0E49

and

WXXT+1+WTWXXψ+3ψT+μψλX=0.E50

In view of (27), (49) becomes

3λψT+1+WTψXXWXTψX+λμψ=0,E51

and (50) implies that λX=0so the spectrum λof (47) remains constant. Constant λis what is required in the IST problem. Equation (50) yields the equation WXXT+1+WTWX=hT, where hTis an arbitrary function of T. Now, according to (62) and (72), the inverse scattering method restricts the solutions to those that vanish as X, so hTis to be identically zero. Thus, the pair of equations (47) and (48) or (47) and (50) can be considered as the Lax pair for the VPE (27).

Since (47) and (48) are alternative forms of Eqs. (44) and (45), respectively, it follows that the pair of equations (47) and (48) is associated with the VPE (27) considered here. Thus, the IST problem is directly related to a spectral equation of third order, namely, (47). The inverse problem for certain third-order spectral equations has been considered by Kaup [33] and Caudrey [34, 35]. As expected, (47) and (48) are similar to, but cannot be transformed into, the corresponding equations for the Hirota-Satsuma equation (HSE) (see Eq. (A8a) and (A8b) in [39]). Clarkson and Mansfield [40] note that the scattering problem for the HSE is similar to that for the Boussinesq equation which has been studied comprehensively by Deift et al. [37].

5. The inverse scattering method for a third-order equation

5.1 Example of the use of the IST method to find the one-soliton solution

Consider the one-soliton solution of the VPE by application of the IST method. Let the initial perturbation be

WX0=6k1+tanhη,η=kX+α.E52

For convenience we introduce new notation ξ1and β1instead of parameters kand αby

k=32ξ1,α=12lnβ1/23ξ1E53

then

WX0=63ξ1Xln1+β123ξ1exp3ξ1XE54

is the initial condition for the VPE.

The first step in the IST method is to solve the spectral equation (47) with spectral parameter λfor the given initial condition WX0. In our example it is (54). The solution is studied over the complex ζ-plane, where ζ3=λ. One can verify by direct substitution of (55) in (47) that the solution ψX0ζof the linear ODE (47), normalized so that ψX0ζexpζX1at X, is given by

ψX0ζexpζX=1β1exp3ξ1X1+β1exp3ξ1X23ξ1ω2iω2ξ1ζ+ω3iω3ξ1ζ,E55

where ωj=ei2πj1/3are the cube roots of 1 (j=1,2,3). The constants β1and ξ1, as we will show, are associated with the local spectral data.

The second step in the IST method is to obtain the evolution of β1and ξ1. The time dependence of the solution ψXTis described by Eq. (48). Analysing Eq. (48), we may assume that

ξ1T=ξ10=const.,β1T=β10exp13ξ1T.E56

Below, the assumption of these relationships will be justified. Indeed, we know that the spectrum λin (47) remains constant if WXTevolves according to Eq. (27). Therefore, as will be proved, the spectrum data evolve as in (70). In notations (77) and (78), from (70) we obtain the relations (56).

The final step in IST method is to select the solution WXTfrom (55) with ξ1T, β1Tas in (56). According to Eq. (2.7) in [33], we expand ψXTζas an asymptotic series in ζ1to obtain

ψX0ζexpζX=113ζWXW+Oζ2,E57

i.e., WXW=limζ3ζ1ψexpζX. Taking into account the functional dependence (56), we find the required one-soliton solution of the VPE in form

WXT=63ξ1Xln1+β123exp3ξ1X13ξ1T+const.E58

Thus, for the example of the one-soliton solution, we have demonstrated the IST method.

5.2 The direct spectral problem

Let us consider the principal aspects of the inverse scattering transform problem for a third-order equation. The inverse problem for certain third-order spectral equations has been considered by Kaup [33] and Caudrey [34, 35]. The time evolution of ψis determined from (48) or (51).

Following the method described by Caudrey [34], the spectral equation (47) can be rewritten

Xψ=Aζ+B(Xζ)ψE59

with

ψ=ψψXψXX,A=010001λ00,B=0000000WX0.E60

The matrix Ahas eigenvalues λjζand left and right eigenvectors v˜jζand vjζ, respectively. These quantities are defined through a spectral parameter λas

λjζ=ωjζ,λj3ζ=λ,vjζ=1λjζλj2ζ,v˜jζ=λj2ζλjζ1,E61

where, as previously, ωj=e2πij1/3are the cube roots of 1 (j=1,2,3). Obviously the λjζare distinct, and they and v˜jζand vjζare analytic throughout the complex ζ-plane.

The solution of the linear equation (47) (or equivalently (59)) has been obtained by Caudrey [34] in terms of Jost functions ϕjXζwhich have the asymptotic behaviour:

ΦjXζexpλjζXϕjXζvjζasX.E62

Caudrey [34] showed how Eq. (59) can be solved by expressing it as a Fredholm integral equation.

The complex ζ-plane is to be divided into regions such that, in the interior of each region, the order of the numbers Reλiζis fixed. As we pass from one region to another, this order changes, and hence, on a boundary between two regions, Reλiζ=Reλjζfor at least one pair ij. The Jost function ϕjis regular throughout the complex ζ-plane apart from poles and finite singularities on the boundaries between the regions. At any point in the interior of any region of the complex ζ-plane, the solution of Eq. (59) is obtained by the relation (2.12) from [34]. It is the direct spectral problem.

5.3 The spectral data

The information about the singularities of the Jost functions ϕjXζreside in the spectral data. First let us consider the poles. It is assumed that a pole ζikin ϕiXζis simple, does not coincide with a pole of ϕjXζand jiand does not lie on a boundary between two regions. Then, as proven in [34], the residue is

ResϕiXζik=j=1jinγijkϕjXζikE63

and it can be found because we know the solution (47) in any regular regions from solving the direct problem (see Section 5.2). Note that, for ϕjXζik, the point ζiklies in the interior of a regular region. The quantities ζikand γijkconstitute the discrete part of the spectral data.

Now we consider the singularities on the boundaries between regions. However, in order to simplify matters, we first make some observations. The solution of the spectral problem can be facilitated by using various symmetry properties. In view of (47), we need only consider the first elements of

ϕiXζ=ϕiXζϕiXζXϕiXζXX,E64

whilst the symmetry

ϕ1Xζ/ω1=ϕ2Xζ/ω2=ϕ3Xζ/ω3E65

means we need only to consider ϕ1Xζ. In our case, for ϕ1Xζ, the complex ζ-plane is divided into four regions by two lines (see Figure 5) given by

Figure 5.

The regular regions for Jost functions ϕ1(X, ζ) in the complex ζ -plane. The dashed lines determine the boundaries between regular regions. These lines are lines where the singularity functions Q1j(ζ′) are given. The dotted lines are the lines where the poles appear.

iζ=ω2ξ,whereReλ1ζ=Reλ2ζ,iiζ=ω3ξ,whereReλ1ζ=Reλ3ζ,E66

where ξis real (see Figure 5). The singularity of ϕ1Xζcan appear only on these boundaries between the regular regions on the ζ-plane, and it is characterized by functions Q1jζat each fixed j1. We denote the limit of a quantity, as the boundary is approached, by the superfix ± in according to the sign of Reλ1ζλjζ(see Figure 5).

In [34] (see Eq. (3.14) there) the jump of ϕ1Xζon the boundaries is calculated as

ϕ1+Xζϕ1Xζ=j=23Q1jζϕjXζ,E67

where, from (66), the sum is over the lines ζ=ω2ξand ζ=ω3ξgiven by

iζ=ω2ξ,withQ121ζ0,Q131ζ0,iiζ=ω3ξ,withQ122ζ0,Q132ζ0.

The singularity functions Q1jζare determined by WX0through the matrix BXζ(60) (see Eq. (3.13) in [34])

Q1jζ=1v˜jζvjζv˜jζexpλ1ζλjζzBzζϕ1Xζdz.E68

The quantities Q1jζalong all the boundaries constitute the continuum part of the spectral data.

Thus, the spectral data are

S=ζ1kγ1jkQ1jζj=23k=12m.E69

One of the important features which is to be noted for the IST method is as follows. After the spectral data have been found from BX0ζ, i.e., at initial time, we need to seek the time evolution of the spectral data from Eq. (48). Analysing (48) at Xtogether with (62)

ϕiXTζ=exp3λiζ1TϕiX0ζ,

the T-dependence is revealed as

ζjkT=ζjk0,γ1jkT=γ1jk0exp3λjζ1k1+3λ1ζ1k1T,Q1jTζ=Q1j0ζexp3λjζ1+3λ1ζ1T.E70

The final step in the application of the IST method is to reconstruct BXTζfrom the evaluated spectral data. In the next section, we show how to do this.

5.4 The inverse spectral problem

The final procedure in IST method is that of the reconstruction of the matrix BXTζand WXTfrom the spectral data S.

The spectral data define Φ1Xζuniquely in the form (see Eq. (6.20) in [34]))

Φ1XTζ=1k=1Kj=23γ1jkTexpλjζ1kλ1ζ1kXλ1ζ1kλ1ζΦ1XTωjζ1k+12πij=23Q1jTζexpλjζλ1ζXζζΦ1XTωjζdζ.E71

Eq. (71) contains the spectral data, namely, Kpoles with the quantities γ1jkfor the bound state spectrum as well as the functions Q1jζgiven along all the boundaries of regular regions for the continuous spectrum. The integral in (71) is along all the boundaries (see the dashed lines in Figure 5). The direction of integration is taken so that the side chosen to be Reλ1ζλjζ<0is shown by the arrows in Figure 5 (for the lines (66), ξsweeps from to +).

It is necessary to note that we should carry out the integration along the lines ω2ξ+iεand ω3ξ+iεwith ε>0. In this case condition (62) is satisfied. Passing to the limit ε0, we can obtain the solution which does not satisfy condition (62). However, for any finite ε>0, the restricted region on Xcan be determined where the solution associated with a finite ε>0(for which the condition (62) is valid) and the solution associated with ε=0are sufficiently close to each other. In this sense, taking the integration at ε=0, we remain within the inverse scattering theory [34], and so condition (62) can be omitted. The solution obtained at ε=0can be extended to sufficiently large finite X. Thus, we will interpret the solution obtained at ε=0as the solution of the VPE (27) which is valid for arbitrary but finite X.

By choosing appropriate values for ζ, the left-hand side in (71) can be Φ1XTωjζ1k, or by allowing ζto approach the boundaries from the appropriate sides, the left-hand side can be Φ1XTωjζ. We acquire a set of linear matrix/Fredholm equations in the unknowns Φ1XTωjζ1kand Φ1XTωjζ. The solution of this equation system enables one to define Φ1XTζfrom (71).

By knowing Φ1XTζ, we can take extra information into account, namely, that the expansion of Φ1XTζas an asymptotic series in λ11ζconnects with WXTas follows (cf. Eq. (2.7) in [33]):

Φ1XTζ=113λ1ζWXTW+Oλ12ζ.E72

Consequently, the solution WXTand the matrix BXTζcan be reconstructed from the spectral data.

6. The interaction of the loop-like solitons

We will discuss the exact N-soliton solution of the VPE via the inverse scattering method [24]. To do this we consider (71) with Q1jζ0. Then there is only the bound state spectrum which is associated with the soliton solutions.

Let the bound state spectrum be defined by K poles. The relation (71) is reduced to the form

Φ1XTζ=1k=1Kj=23γ1jkTexpλjζ1kλ1ζ1kXλ1ζ1kλ1ζΦ1XTωjζ1k.E73

Eq. (73) involves the spectral data, namely, the poles ζ1kand the quantities γ1jk. First we will prove that Reλ=0for compact support. From Eq. (47) we have

ψXXXX+UψXXλψX=0,E74

and together with Eq. (47), this enables us to write

X2X2ψXψ3ψXXψX+UψXψ2ReλψXψ=0.E75

Integrating Eq. (75) over all values of X, we obtain that, for compact support, Reλ=0since, in the general case, ψXψdX0.

As follows from Eqs. (2.12), (2.13), (2.36) and (2.37) of [33], ψXζis related to the adjoint states ψXAζ. In the usual manner, using the adjoint states and Eq. (14) from [35] and Eq. (2.37) from [33], one can obtain

ϕ1XXζ=i3ϕ1XXω2ζϕ1(Xω3ζ)ϕ1X(Xω3ζ)ϕ1(Xω2ζ).E76

It is easily seen that if ζ11is a pole of ϕ1Xζ, then there is a pole either at ζ12=ω2ζ11(if ϕ1Xω2ζhas a pole) or at ζ12=ω3ζ11(if ϕ1Xω3ζhas a pole). For definiteness let ζ12=ω2ζ11. Then, as follows from (76), ω3ζ12should be a pole. However, this pole coincides with pole ζ11, since ω3ζ12=ω3ω2ζ11=ζ11. Hence, the poles appear in pairs, ζ12n1and ζ12n, under the condition ζ12n/ζ12n1=ω2, where nis the pair number.

Let us consider Npairs of poles, i.e., in all there are K=2Npoles over which the sum is taken in (76). For the pair nn=12Nwe have the properties

iζ12n1=iω2ξn,iiζ12n=iω3ξn.E77

Since Uis real and λis imaginary, ξkis real. The relationships (77) are in line with the condition (2.33) from [33]. These relationships are also similar to Eqs. (6.24) and (6.25) in [34], whilst γ1jkturns out to be different from γ˜1jkfor the Boussinesq equation (see Eqs. (6.24) and (6.25) in [34]). Indeed, by considering (76) in the vicinity of the first pole ζ12n1of the pair nand using the relation (73), one can obtain a relation between γ12kand γ13k. In this case the functions ϕ1,XXζ, ϕ1Xω2ζand ϕ1,XXω2ζalso have poles here, whilst the functions ϕ1Xω3ζand ϕ1,XXω3ζdo not have poles here. Substituting ϕ1Xζin the form (73) into Eq. (76) and letting X, we have the ratio γ132n/γ122n1=ω2and γ122n=γ132n1=0. Therefore, the properties of γijkshould be defined by the relationships

iγ122n1=ω2βk,γ132n1=0,iiγ122n=0,γ132n=ω3βk,E78

where, as it will be proved below, βkis real when U=WXis real.

By defining

ΨkXT=j=23γ1jkTexpλjζ1kXΦ1XTωjζ1k,E79

we may rewrite the relationship (73) as (see, for instance, Eqs. (6.33) and (6.34) in [34])

Φ1XTζ=1k=12Nexpλ1ζ1kXλ1ζ1kλ1ζΨkXT.E80

From (72) and (80), it may be shown that (cf. Eq. (6.38) in [34])

WXTW=3k=12Nexpλ1ζ1kXΨkXT=3XlndetMXT.E81

The 2N×2Nmatrix MXTis defined as in relationship (6.36) in [34] by

MklXT=δklj=23γ1jk0exp3λjζ1k1+3λ1ζ1k1T+λjζ1kλ1ζ1lXλjζ1kλ1ζ1l,E82

and

n=1,2,,N,λ1ζ12n1=iω2ξn,λ2ζ12n1=iω3ξn,γ122n1=ω2βn,γ132n1=0,λ1ζ12n=iω3ξn,λ3ζ12n=iω2ξn,γ122n=0,γ132n=ω3βn.

For the N-soliton solution, there are Narbitrary constants ξnand Narbitrary constants βn.

The final result for the N-soliton solution of the VPE is defined by relationship (81) with (82).

6.1 Examples of one- and two-soliton solutions of the VPE

In order to obtain the one-soliton solution of the VPE (27)

WXXT+1+WTWX=0,

we need first to calculate the 2×2matrix MXTaccording to (82) with N=1. We find that the matrix is

1ω2β13ξ1exp3ξ1X3ξ11Tiω3β12ξ1exp2iω3ξ1X3ξ11Tiω2β12ξ1exp2iω2ξ1X3ξ11T1ω3β13ξ1exp3ξ1X3ξ11TE83

and its determinant is

detMXT=1+β123ξ1exp3ξ1XT3ξ122.E84

Consequently, from Eq. (81) we have the one-soliton solution of the VPE

UXT=WXXT=92ξ12sech232ξ1XT3ξ12+α1,E85

where α1=12lnβ1/23ξ1is an arbitrary constant. Since Uis real, it follows from (85) that β1is real. Note that with β1/ξ1<0we have the real solution in the form of the singular soliton [41].

UXT=92ξ12sinh2η,η=32ξ1XT3ξ12+α1.E86

Let us now consider the two-soliton solution of the VPE. In this case MXTis a 4×4matrix. We will not give the explicit form here, but we find that

detMXT=1+q12+q22+b2q12q222,E87

where

qi=exp32ξiXT3ξi2+αi,b2=ξ2ξ1ξ2+ξ12ξ12+ξ22ξ1ξ2ξ12+ξ22+ξ1ξ2,E88

and αi=12lnβi/23ξiare arbitrary constants. The two-soliton solution to the VPE as found by the IST method is given by (81) together with (87).

Finally we note that comparison of (81) with W=6lnfXfrom (42) shows that

lndetMXT=2lnf.E89

so that detMXTis a perfect square for arbitrary N.

6.2 The two-loop-like solitons of the VE

We discuss the two-loop soliton solution of the VE in more detail. Let us consider what happens in x-tspace. The relations (20), (25) and (29) determine the solutions in x-tthroughout the solutions in X-T. In these coordinates x-t, we have the loop-like solitons.

The shifts, δi, of the two-loop solitons u1and u2in the positive x-direction due to the interaction may be computed as follows. The larger loop soliton is always shifted forwards by the interaction. However, for smaller u2with r=ξ1/ξ2, there is a value rc=0.88867in that we have a different form of the phase shift:

  1. For rc<r<1, δ1<0so the smaller loop soliton is shifted backwards.

  2. For r=rc, where rc=0.88867is the root of lnb+3/r=0, δ1=0, so the smaller loop soliton is not shifted by the interaction.

  3. For 0<r<rc, δ1>0so the smaller loop soliton is shifted forwards.

At first sight it might seem that the behaviour in (b) and (c) contradicts conservation of ‘momentum’. That this is not so is justified as follows. By integrating (9) with respect to x, we find that udx=0; also, by multiplying (9) by xand integrating with respect to x, we obtain xudx=0. Thus, in x-tspace, the ‘mass’ of each soliton is zero, and ‘momentum’ is conserved whatever δ1and δ2may be. In particular δ1and δ2may 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 68, respectively; in these figures uis plotted against xfor various values of t. For convenience in the figures, the interactions of solitons are shown in coordinates moving with speed v1+v2/2.

Figure 6.

The interaction process for two-loop solitons with ξ 1 = 0.99 and ξ 2 = 1 so that r = 0.99 and δ 1 < 0.

Figure 7.

The interaction process for two-loop solitons with ξ 1 = 0.88867 and ξ 2 = 1 so that r = 0.88867 and δ1 = 0.

Figure 8.

The interaction process for two-loop solitons with ξ1 = 0.5 and ξ2 = 1 so that r = 0.5 and δ1 > 0.

7. Discussion on the loop-like solutions

We have already mentioned the important question on stability of loop-like solutions (Section 2).

7.1 Remarks on the existence and uniqueness theorem

In [42], the existence and uniqueness theorem is formulated for system (one) differential equations. The loop-like solutions take place on travelling waves. In this case, the initial equation is reduced to an ordinary differential equation (ODE) (see Section 2). It has been this equation which we are exploring. Now we note some important remarks. In particular, in order to investigate the ODE (the solution on travelling waves), it is still necessary to reconcile this solution with the initial problem, which is described by the differential equation in partial derivatives (evolution equation). Consequently, the ambiguous solutions for the ODE during their reconstruction into the initial coordinates should be checked by means of some restrictive conditions (see 7.2).

It is necessary to note that if the conditions of the existence and uniqueness theorem break down, then nevertheless, this does not restrict the existence of solutions. Hence, the solutions can exist, for example, the multivalued solutions. Here we point out an example: the exact solutions for the Camassa-Holm equation (CHE) and the Degasperis-Procesi equation (DPE) can be constructed as the component solutions, through separate parts (branches) of solutions (see [43]).

The selection of possible multivalued solutions will be discussed in 7.2.

7.2 Selection for the loop-like solutions

Solutions must satisfy the following conditions:

  1. At the point η=0, the solution must pass over the ellipse z2+2vη2ηη=0(see Eq. (4.3) in [3]);

  2. According to the conservation law uxtdx=const0for t>0. The ‘mass’ of individual soliton equals to zero. This condition will be satisfied if point 1 takes place.

  3. As you know [3], taking into account dissipation in the physical process allows one to select a solution from an array of possible solutions that are inherent to the equation without dissipation. This condition also selects a solution as in a point 1 if α0.

  4. During the interaction of the solitons [24, 29], you must take into account all the parts of loop-like soliton (see end of Section 7.3). The soliton has a form satisfying point 2.

Thus, we cannot arbitrarily combine the solutions at η=0. The solutions, in particular, solitons should be specific loop-like form.

7.3 Physical interpretation of the multivalued solutions

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 αuxtends to infinity. The question arises: Are there solutions of Eq. (8) in a loop-like form? That the dissipation is likely to destroy the loop-like solutions can be associated with the following well-known fact [5]. For the simplest nonlinear equation without dispersion and without dissipation, namely,

ut+uux=0,E90

any initial smooth solution with boundary conditions

ux+=0ux=u0=const.>0

becomes ambiguous in the final analysis. When dissipation is considered, we have the Burgers equation [47]:

ut+uux+μ2ux2=0.

The dissipative term in this equation and in Eq. (6) for low frequency is 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 αu(see Eq. (18) in [29])? Will the inclusion of dissipation give rise to unambiguous solutions?

By direct integration of Eq. (8) (written in terms of the variables (11)) within the neighbourhood of singular points z=0where zη±and zτzη, it can be derived (see [3]) that the dissipative term, with dissipation parameter less than some limit value α, 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 (15) and (16), 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), for high frequencies, a dissipative term arises which is similar to the corresponding term in Eq. (7) 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 interpenetratable. A similar situation, when several components with different hydrodynamic parameters occupy one microvolume, has been assumed in mixture theory (see, for instance, [48]). Such a fundamental assumption in the theory of mixtures is physically impossible (see [48], p.7), but it is appropriate in the sense that separated components are multi-velocity interpenetratable 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 and not just the separate parts.

7.4 Conclusion

Loop-like solitons are a class of interesting wave phenomena, which take place in some nonlinear systems. This interest consisted not only in the interpretation of the solutions obtained but also in the explanation of the experimental results. The ambiguous structure of the loop-like solutions is similar to the loop soliton solution to an equation that models a stretched rope [44]. Loop-like solitons on a vortex filament were investigated by Hasimoto [45] and Lamb, Jr. [46]. The loop-like solutions appear in description of physical phenomena, in particular, electromagnetic terahertz pulses in asymmetric molecules [49], high-frequency perturbations in a relaxation medium [3, 50, 51] and soliton in ferrites [52]. As a typical multivalued structure, loop soliton has been discussed in some possible physical fields including particle physics [53] and quantum field theory [54].

It must be admitted that we are a long way still from complete awareness of physical processes which can be described by loop-like solutions. However, the approach, considered here, will hopefully be interesting and useful in understanding the birth and death process for particles, since the mass and momentum of individual loop-like soliton are zero. Furthermore, the investigations in optics, magnetism and hydrodynamics clearly indicate the acceptability of the approach on loop-like solitons. Indeed, the phase shifts observed at interaction of solitons can be explained by means of loop-like solutions.

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Vyacheslav O. Vakhnenko, E. John Parkes and Dmitri B. Vengrovich (September 27th 2019). Loop-like Solitons, Research Advances in Chaos Theory, Paul Bracken, IntechOpen, DOI: 10.5772/intechopen.86583. Available from:

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