Original Article
J Phys Astr, Volume: 5( 1)

Qualitative Analysis and Asymptotics of Solutions of the Gkp-Class Equations with Variable Dispersion

*Correspondence:
Belashov VYU, Kazan (Volga Region), Federal University, Kazan, Russia, Tel: +86-27-67863760, E-mail: [email protected]

Received: April 4, 2017; Accepted: April 25, 2017; Published: May 2, 2017

Citation: Belashov VYU, Belashova SE. Qualitative Analysis and Asymptotics of Solutions of The Gkp-Class Equations with Variable Dispersion. J Phys Astron. 2017;5(2):108.

Abstract

In this paper we study the dynamical systems associated with the generalized Kadomtsev-Petviashvili (GKP) equation with variable dispersion and consider the structure of possible multidi-mensional solutions and their asymptotics. We also present some considerations on constructing of the phase portraits of the systems in the 8-dimensional phase space for the GKP equation on the basis of the results of qualitative analysis of the generalized equations of the KdV-class.

Keywords

dynamical system; generalized KdV equation; GKP equation; multidimensional solutions; phase space; qualitative analysis; solitons; structure of solutions; variable dispersion

Introduction

Basic equations

Consider the generalized Kadomtsev-Petviashvili (GKP) equation (or so-called Belashov-Karpman (BK) [1-3] equation without the terms describing the dissipation and instability):

physics-astronomy (1)

physics-astronomy

Where and are the transverse coordinates. At Eq. (1) is the classic Kadomtsev-Petviashvili (KP) equation which is the completely integrable Hamiltonian system and has in case the solutions in form of the 1D (for βκ < 0 ) or 2D (for βκ > 0 ) solitons [4,5]. The structure and the dynamics of the solutions of the non-integrable analytically GKP model (1) with β, γ = const has been investigated in detail in Dynamics of two-dimensional solitons in weakly dispersive media [1,2] where it was shown that in dependence on the signs of coefficients β, γ and k the 2D and 3D soliton type solutions with the monotonous or oscil-latory asymptotics can take place. In Radiophysics and quantum electronics [6] the asymptotes of the 1D analogue of Eq. (1) with were studied, and the sufficiently complete classification of its 1D solutions in the phase space was constructed.

Note, that if the dispersion in medium is variable, introducing the high order dispersive correction (term with γ ) into the KP equation has a principal role. Thus, for example, in the problems of the propagation of the 2D gravity and gravity-capillar waves on the surface of "shallow" water [3] when β is defined respectively as physics-astronomy andphysics-astronomy where H is the depth, ρ is the density, and σ is the coefficient of surface tension of fluid, the depth can be variable, H = H(t, x, y) , and in this case β also becomes the function of the coordinates and time. Similar situation takes place on studying of the evolution of the 3D fast magneto-sonic (FMS) waves in a magnetized plasma in case of the inhomogeneous and/or non-stationary plasma and magnetic field [7,8] when β is a function of the Alfvén velocity

physics-astronomy

In these cases the situations, when physics-astronomy or even becomes equal zero are possible, that, however, does not mean that the dispersion in medium is completely absent, simply it is necessary to keep the next order dispersive term in the Taylor expansion of a full dispersion equation with respect to k, at this, a balance between nonlinearity and dispersion, defining the existence of the soliton, will be conserved, and the term proportional to the fifth derivative will appear in the KP equation. Therefore, if the dispersion can vary in medium, it is necessary to consider the BK equation in form (1), namely the GKP equation.

So, our purpose is the generalization of the results obtained in Radiophysics and quantum electronics [6] to the multidimensional cases when β or/and γ are not constants (the dispersion is variable) with due account of the results presented in [3,4].

To avoid unnecessary cumbersome expressions, consider Eq. (1) in the 2D form assuming that physics-astronomy . Further generalization of the technique used (as well as the results obtained) to the full 3D case physics-astronomy is rather trivial, as we demonstrate below. Assume that physics-astronomy and, for clarity,physics-astronomy (the latter can be easily obtained by the scale transformphysics-astronomy in the GKP equation).

Now, let us introduce the new variables: physics-astronomy Applying firstphysics-astronomy and thenphysics-astronomy to (1) we obtain the pair of one-dimensional equations:

physics-astronomy (2)

written in the reference frame with the axes physics-astronomy andphysics-astronomy rotated through an angle +45? relative to the axes physics-astronomy andphysics-astronomy

Representation (2) means in fact that the starting Eq. (1) admits two types of 1D solutions, physics-astronomy andphysics-astronomy satisfying the first and the second equations of the set (2), respectively. It is necessary, however, to bear in mind that the "onedimensionality" of these solutions nevertheless implicity assumes the linear dependence of each of the new variables physics-astronomy andphysics-astronomy on both coordinates,physics-astronomy andphysics-astronomy

Integrating Eqs. (2) over physics-astronomy andphysics-astronomy , respectively, we obtain equiform generalized KdV equations

physics-astronomy (3)

coupled with each other by the way of the change of the coordinates made above. Now, transferring to the coordinates moving along the corresponding axis with the velocity physics-astronomy applying the changephysics-astronomy in Eqs. (3) and omitting “primes” for simplicity, we write Eqs. (3) in the standard form:

physics-astronomy (4)

So, we can now conduct the analysis for only one generalized equation of the set (3), and then, making the inverse change of the variables, extend the results to the 2D solutions physics-astronomy of the GKP equation (1) withphysics-astronomy

Qualitative analysis for 1D equation

Taking into account the more general case we extend the class of Eqs. (4) by introducing the arbitrary positive exponent p of the nonlinear term:

physics-astronomy (5)

Equation (5) for physics-astronomy is the usual KdV equation when p =1, and it is the modified KdV (MKdV) equation when p=2. The asymptotics of (5) with p =1 were first investigated in [1,9] where it was shown that depending on the signs of the coefficients β and γ the soliton type solutions with monotonous or oscillating asymptotics can take place. But note, that Eq. (5) with physics-astronomy and physics-astronomy is not exactly integrable (i.e., the known analytical methods such as the IST method, are not applicable to this equation). In Radiophysics and quantum electronics [6] Eq. (5) was investigated by the methods of the both asymptotic and qualitative analysis, and, as a result, the sufficiently full classification of its solutions was constructed. In this section we mainly follow the ideas and technique of Radiophysics and quantum electronics [6].

Note, that from the physical point of view, the cases when in Eq. (5) p =1,2 are the most interesting, and applications for p >2 are presently unknown. However, since equations of the family (5) with an arbitrary integer p >0 demonstrate, to a considerable extent, similar mathematical properties, we use here a general approach elucidating, apart from other, the dependence of the characteristics of the solutions on the nonlinearity exponent.

Performing transformation physics-astronomy and integrating (5) inphysics-astronomy we obtain:

physics-astronomy (6)

Assuming without loss of generality that physics-astronomy andphysics-astronomy , after the changephysics-astronomy we convert (6) to

physics-astronomy (7)

where

physics-astronomy

Depending on sign of V in (7) the following two cases can be considered:

physics-astronomy

physics-astronomy

physics-astronomy

physics-astronomy

However, as one can see from (5), the velocity of the wave, V, depends on the equation’s coefficients and it is restricted by:

physics-astronomy

The right-hand sides of inequalities (9) correspond to results obtained in [1} and comparing these relations with expressions (8a) and (8b) leads to contradictions in the cases physics-astronomy andphysics-astronomy respectively. Therefore, below we will limited by the consideration of cases (a) with physics-astronomy and (b) withphysics-astronomy andphysics-astronomy

Qualitative analysis and asymptotes of the solutions

First, we note that every equation of the set (8) is equivalent to the set of the first order ordinary differential equations

physics-astronomy (10)

Where the dots stand for the ?-derivatives, and the signs in the brackets correspond to three cases mentioned above; furthermore,physics-astronomy Solutions of Eqs. (10) are stable if there exist the singular trajectories of the imaging point in the phase space physics-astronomy of the set (10). Each trajectory of its kind is related to the state of equilibrium near the maximum of the soliton-like solution and at the boundaries physics-astronomy Assuming as the boundary conditions that

physics-astronomy (11)

we can find from (10) the number as well as the coordinates of the singular points:

physics-astronomy (12)

where the points physics-astronomy andphysics-astronomy correspond, respectively, tophysics-astronomy and the bending points of the functionphysics-astronomy for the odd and j = 2, 3 for the even p, in the last case physics-astronomy Considering only real roots of (12) we immediately conclude (using the Sturm’s theorem) that for the odd p there are two singular points, and for any even p there are three singular points. The distance between the singular points defines the amplitude of the soliton-like solution of (6). Besides, the value of the nonlinearity exponent p defines a character of dependence physics-astronomy for p >1 this dependence becomes nonlinear ( Figure 1 ) unlike the known linear one for p =1 (e.g., in the case of the KdV equation). As one can see in Figure 1 , for the even p the solutions of Eq. (6) can have the positive as well as negative polarity physics-astronomy for any sign of V .

To investigate the types of the singular points, it is necessary to linearize the set (10) in the neighborhood of every point. Using the Taylor’s formula, we obtain from Eqs. (10) [6]:

physics-astronomy-curves-corresponds

Figure 1: Dependence V = f (u) for Eq. (6) for different values of p. Numbers of curves corresponds p =1, 2, ..., 6.

1) for the singular point physics-astronomy that corresponds tophysics-astronomyin (6), taking into account the conditions (11),

physics-astronomy

2) for the singular point physics-astronomy that corresponds tophysics-astronomy

physics-astronomy

Since the sets (13) are essentially four-dimensional, we investigate them by expanding of the corresponding canonical systems into the subsystems [4]. In this case, it is possible to consider the phase portraits of the linear sets (13) as projections of the singular points and trajectories onto two planes. For singular point physics-astronomy we obtain that the eigenvalues of the matrices of subsets of set (13.1) [4] corresponding the phase planes P1(w, x1) and P2(x2,x3) are defined by:

physics-astronomy (14)

In the case (a) with physics-astronomy are real on the phase plane P1 and pure imaginary on the phase plane P2, besides,physics-astronomy on both planes. In the case (b) withphysics-astronomy taking into account the conditions (9), the characteristic roots physics-astronomy and physics-astronomy are complex with positive and negative real part on the planes P1 and P2, respectively, andphysics-astronomy In the case (b) with physics-astronomy taking into account condition (9.2), all four roots are real and physics-astronomy on both planes. Therefore, the singular points w1= 0 of three types exist in the phase space, namely: the “saddle–center” point, the “stable focus–unstable focus” point, and the “saddle-saddle” point in the cases mentioned above, respectively.

Considering by analogy the matrix of subsets corresponding to set (13.2) we obtain eigenvalues for the singular points wj defined by (12) in three cases considered for the subsets corresponding to the projections to the phase spaces P1(w, x1) and P2(x2, x3) [6]:

physics-astronomy (15)

One can see from (15) that the character of the singular point depends on the nonlinearity exponent p defining the wave velocity physics-astronomy . Nevertheless, conditions (9) remain valid and in these cases as well.

Analysis of (15) enables to conclude the following. The eigenvalues physics-astronomy andphysics-astronomy are complex (moreoverphysics-astronomy with the positive real parts on the plane P1 and the negative ones on the plane P2 in case (a) physics-astronomy taking into account the condition (9.1). In case (b) with physics-astronomy andphysics-astronomy are real on the plane P1 and pure imaginary on plane P2, andphysics-astronomy in the both cases. In the case (b) withphysics-astronomy the situation is analogous to that of the case (b) withphysics-astronomy Therefore, only singular points wj of the type “stable focus–unstable focus” take place in phase space in the case (a), and of the type “saddle-center” take place in the case (b) for both signs of β.

To study the global phase portraits including singular trajectories corresponding the stable solutions of the Eqs. (10), in Solitary waves in dispersive complex media [4,6] the Bendixon and Dulac criterions have been used, and also the first and the second Lyapunov quantities have been calculated. Omitting here cumbersome mathematical calculations, we note that in all three cases considered the closed trajectories have place in the phase space. At this, formulae (14) and (15) enable us to obtain such parameters of the curves as their directions and, consequently, the angles with respect to the coordinate axes on both planes, and, therefore, to construct the global phase portraits.

The examples of such phase portraits for the cases (a) and (b) for p =1, 2 are shown in Figure 2a and 2b and Figure 3a and 3b.

physics-astronomy-phase-portraits

Figure 2: The phase portraits of the solutions of (10a) with β=1 for p=1 (a) and p=2 (b) (solid and broken lines correspond the phase trajectories respectively in the planes P1 and P2) and numerical solution of Eq. (5) with γ=1, β=-1 for p=1

physics-astronomy-phase-portraits

Figure 3:Phase portraits of the solutions of (10b) with β=1 for the same values of parameters as in FIG. 2a and 2b – respectively the positions (a), (b), and numerical solution of (5) with γ=1,β=3.16, p=1 (c)

Using the values of the characteristic roots physics-astronomy andphysics-astronomy (14) for singular points w1=0, taking into account the conditions (9) and (11) we obtain the asymptotics of the solutions of Eq. (5) for considered cases [4], namely:

1) for the cases (a) and (b) with physics-astronomy

physics-astronomy (16.1)

(The upper/lower sign corresponds to the cases (a)/(b), respectively);

2) For the case (b) with physics-astronomy

physics-astronomy

Where A1, A2 and Θ are arbitrary constants. As we can see from (16), in the solutions of (5) in dependence of the signs of V and β the solitons with both monotonous and oscillating asymptotics can take place. At this, the phase portraits shown in Figure 2a and 2b correspond to the solitons with monotonous asymptotics, and the phase portraits shown in Figure 3a and 3b correspond to solitons with oscillating asymptotics. Figure 2c and 3c show the results of numerical integration of Eq. (5) for the initial condition physics-astronomy that corresponds the results of the asymptotic analysis.

Generalization of the results to the BK-class equations

Making the inverse change of the variables, extend the obtained results to the 2D solutions physics-astronomy of the GKP equation (1) with physics-astronomy The asymptotics of the solutions are defined by relations [3,4]:

physics-astronomy

physics-astronomy (17)

physics-astronomy

physics-astronomy

where A1, A2 and physics-astronomy are the arbitrary constants,physics-astronomy andphysics-astronomy (here the signs "plus" and "minus" relate to the first and second equations of the set (2), respectively). As we can see from (17), (18), solitons with monotonous as well as oscillating asymptotics can take a place depending on the signs of V and β as the solutions physics-astronomy of Eq. (1). (Note, that forphysics-astronomy and anyphysics-astronomy solutions of Eqs. (2) have formphysics-astronomy and, consequently, also describe a soliton with monotonous asymptotics [3,4].) Figure 4 shows the results of numerical integration of Eq. (1) with the initial condition that confirms the results of our asymptotic analysis.

physics-astronomy-General-view

Figure 4: General view of a 2D soliton of Eq. (1) with physics-astronomy for p =1: (a) γ=1, β=-0.8 (t = 0.2); (b) γ=1, β=3.16(t=0.5).

physics-astronomy

Other correlations of signs of V and β are not realized [3,4].

As to the proper transform of the phase portraits of the system and "bind" them for the 2D equation, owing to the phase space is 8-dimensional in this case, we can employ the results obtained in [5,6] coupling the characteristics of every singular point of each equation of the set (2) in the 8-dimensional phase space, at this the type of the singularities in the 4-dimensional subspaces [4] under the inverse transform of the coordinates, physics-astronomy , is not changed, and only parameters of the phase portraits that correspond to solutions of the same class change (leading to the respective changes such parameters as the amplitude, the fronts steepness, frequency of the oscillations etc) [7-9].

Concluding remarks

To conclude, we note that for the BK equation we considered only the particular cases when it is possible to neglect the effects of dissipation and different types of instability here. For other cases when it is necessary to account dissipation and instabilities, more complex wave structures resulting from the simultaneous presence of all the effects discussed, for example, in Solitary waves in dispersive complex media [4] can be observed. Indeed, results obtained numerically in [10-12], demonstrate that in general BK model [4] with physics-astronomy in the presence of the Gaussian random fluctuations of the wave field (for harmonic initial conditions and initial conditions in the form of a solitary pulse) stable wave structures of the soliton-like type can be formed too, with the time evolution. Furthermore, stable soliton structures can be formed also for physics-astronomy An analytical study of such cases is highly complicated, however, although the approach considered above can be used for them as well. Note also that the results presented here for the GKP equation can be very useful when studying solutions and interpreting the multidimensional phase portraits of more complicated multidimensional model equations [3,4,13].

References