**Vasily Yu Belashov ^{1}*, Elena S Belashova^{2} and Oleg A Kharshiladze^{3}**

*
^{1}Kazan Federal University, Russia
^{2}Kazan National Research Technical University named after A.N. Tupolev - KAI,
Russia
^{3}Iv. Javaxishvilis Tbilisi State University, Georgia
*

***Corresponding Author:** Vasily Yu Belashov, Kazan Federal University, Russia.

**Received:
** January 21, 2021; **Published:** February 27, 2021

The results of numerical study of evolution of the solitons of gravity and gravity-capillary waves on surface of shallow fluid when the characteristic wavelength is essentially greater then depth, λ > H , are presented for the cases when dispersive parameter is a function of time and spatial coordinates, β = β(t, x, y) . This corresponds to the problems when the relief of bottom is changed in time and space. We use both one-dimensional approach (the equations of the KdV-class) and also two-dimensional description (the equations of the KP-class) where it is necessary.

**Keywords: **Structure; Shallow Fluid; KdV-class

Let us consider the models of the Korteweg – de Vries (KdV)
and Kadomtsev – Petviashvili (KP) equations in their application
to hydrodynamics, namely, to description of the gravity waves on
the surface of an ideal incompressible fluid of small (compared to
wavelength) depth. In this case, the generalized density and velocity
of “sound” in the general set of the hydrodynamic equations
[1] acquire the sense of fluid depth, H, and velocity c=√gH,the
term gH^{2}/2 plays the role of the pressure, this corresponds to
the effective adiabatic index γ = 2 [2]. Then the Boussinesq equations
take the form

(h=const is the depth of the fluid). It is easy to add into these equations the terms associated with the capillary effects. Assuming that the curvature of the surface is not too large and the additional pressure to the fluid caused by the surface tension is defined by the Laplace formula

Where σ is the surface tension coefficient, R_{1}
and R_{2} are the main
curvature radii, we can write δp = −σΔη where η(x, y,t ) is
the surface function (the value of η is sufficiently small). Changing *ρgh*
in (1) by *ρgH*+ δp (ρ is the fluid density) we obtain

Equations (2), (3) are the Boussinesq equations taking into accounting the capillary effects [2]. Change of the factor at the dispersive term in the dispersion relation in its standard form [2] leads to the change of the dispersion equation, and, instead of we have

Where C_{o} = √gH. In this case the dispersive factor is defined by

Using furthermore the results of [1] we transform (1) and (2) to the form

i.e. obtain the KP equation for the gravity-capillary waves on
shallow fluid. Note that for sufficiently large σ>1/3δpH^{2}
the dispersive
parameter changes its sign that involves the qualitative change
of the character of the evolution and the form of the solutions [3].

Consider now in more detail the following interesting case. Often
there are the cases when the factor β is unusually small. As it
follows from (5) β = 0 at H = (3σ / ρg )^{1/ 2} ≈
0.48cm( for pure water). However β = 0 does not mean that there is no dispersion
in medium. It simply means that in this case the next term in the
Taylor expansion in k of the full dispersion relation must be taking
into account. In this case, the corresponding additional term proportional
to the next odd derivative appears in the equation. This
generalization leads to the KP equation which can be written as

Where the coefficients are

Numerical integration of (6) using the methods based on implicit
and explicit difference schemes [1,3] enables us to investigate
the structure of the one-dimensional (1D) and two-dimensional
(2D) solitons on shallow fluid in the case of anomalously weak dispersion. We have found that the qualitative form of the solutions
depends significantly on the value of parameter ε = (β /V )
(−V /γ ) ^{ 1/2}<< 2 where V is the soliton’s velocity in the reference
frame moving along the x-axis with the phase velocity ε = 0. In 1D
case at the structure of propagating solitons does not differ
qualitatively from the structure of solitons of the usual KdV
equation (see [2]), and in 2D case – from the structure of the algebraic
KP-solitons [1,3]. Such solitons on the surface of fluid have
negative polarity (the hollow solitons). When ε >0, for example,
in the case of the increasing fluid depth starting from the depth
H = (3σ/ ρg )^{1/ 2} , the structure of the solitons radically changes:
by remaining to decay from their maximum to zero in the transverse
direction as before, now their sign varies along the direction
of their propagation (at this, the amplitude of the 2D solitons falls
from maximum to zero in the transverse direction as before). At
ε → 2 the number of the oscillations in the tails increases and
the solitons become similar to 1D and 2D high-frequency trains,
respectively, i.e. envelope solitons1. Note that a similar structure is
typical also for solitons of internal gravity waves, which were considered
in detail in [1,4,5].

Let us consider now, separately for 1D and 2D cases, some our results of numerical simulation of the soliton dynamics on surface of shallow fluid which describes by the standard KdV and KP equations (equation (6) with γ = 0 ) when factor β is a function of space coordinates and time.

At first, let us consider the evolution of the 1D solitons in the framework of model (6) with γ = 0 and right-hand side being equal to zero (the KdV equation):

on surface of a fluid with varying in time and space dispersive
parameter β = β(t ,x). Such situation can take place, for example,
in the problems on propagation of the gravity and gravity-capillary
waves on surface of shallow fluid [3] when β = c_{0} H^{2}/6
and
β = (c_{0}/6 )[H^{2} − 3σ / ρg] , respectively (see above). In these
cases, if H =H (t ,x), the dispersive parameter becomes also the
function of the x coordinate and time.

In [1,4] it was shown that the solutions of the KDV equation at
β=const in dependence on value of β are divided into two classes: at
β < u_{0} (0, x) l / 12 (where l is the characteristic wavelength
of the initial disturbance) they have soliton character, in the opposite
case the solutions are the wave packets with asymptotes being
proportional to the derivative of the Airy function (see also [2]). In
these cases, the KdV equation can be integrated analytically by the
inverse scattering transform (IST) method. But, even in 1D case, if
β = β(t ,x), this approach is impossible principally, it is necessary
to resort to a numerical simulation in the conforming problems.

Let us formulate the problem of numerical simulation of the KdV equation with β = β(t ,x) and consider some results of our numerical experiments on study of structure and evolution of the solitary waves on surface of shallow fluid.

To solve the initial problem for the KdV equation (7) with variable
dispersion we used an implicit difference scheme [1] with
O(τ^{2} , h^{4} ) approximation. Initial conditions were chosen in form
of the solitary disturbance,

and in form of a “smoothed step”:

With different values of parameters u_{0}, l and с, which were defined
by the convenience of numerical calculation for specific sizes
of the numerical integration area. The zero conditions on boundaries
of the computation region were imposed, and simulation has
been conducted for a few types of model types of function β (see
figure 1 and 2) when for t < t_{cr} β = β_{0} = const, and for t ≥ t_{cr}

Where a and c are constants. In terms of the problem of the
wave propagation on surface of the shallow water that accordingly
means that on reaching t_{cr} we have: 1) sudden "breaking up of the
bottom", 2) gradual "changing of height" of the bottom area, and 3)
"bottom oscillation" with time.

**Figure 1: *** Dependence β = β(t ,x) of type of “step”, models (10)
and (11).*

**Figure 2: *** Dependence β = β(t ,x) of type of “bottom
oscillations”, model (12).*

Consider briefly some results of numerical simulation for two types initial conditions and different kinds of model function β = β(t ,x).

In the first series of numerical experiments we investigated of the evolution of the initial disturbance in form of the solitary soliton-like pulse (8) for the models with spasmodic change of dispersion [models of bottom of type of “step” (10) and (11)] with values of parameter а corresponding (at t = 0 ) to the position of the “break” behind and ahead soliton, and values с < 0 (“negative" step) and с > 0 (“positive” step). The obtained results showed that in all cases the deformation of initial pulse occurs with time. If the step is located behind the soliton, that in both cases, c<0 and c>0, the waving tail which is not associated with the main maximum of the outgoing forward main pulse is formed, and its evolution is entirely determined by the value β in its location. In case when at t = 0 the “step” is located ahead front of initial pulse, at с > 0 in the model (11) a steep front is formed quite quickly, that leads to the overturning of the wave with time. At с > 0, the destruction of the soliton can be observed (Figure 3), which occurs due to the fact that in the region of localization of its front, the relative role of nonlinear effects falls due to the increase of the dispersive parameter here, and dispersive effects prevail.

**Figure 3: *** Evolution of the KdV soliton in the model (11)
with с< 0.*

The second series of numerical experiments was devoted to study of the evolution of the initial disturbance of type (9) for the models of “bottom” (10), (11) at different values of parameters а and с.

Figure 4 shows the result of numerical simulation of the evolution of initial disturbance (9) for the model of “bottom” in the form of positive step in the case when “break” is located directly under the region of the disturbance front of the fluid surface. It is seen that due to the fact that the development of perturbations occurs mainly in the region where the value of the dispersion parameter corresponds to multisoliton solution of the KdV equation [1,2], solitary disturbance propagates with the development of highfrequency oscillatory structure behind the shock front, and in the region of the soliton "tail", where dispersive effects dominate over the nonlinear ones, the high-frequency train of oscillations decays rather quickly to zero and it is limited in the region x < 0 .

**Figure 4: *** Evolution of “step” (9) in the model (10) with с>0: a − t
≈ 0.25; b − t ≈ 0.5.*

Figure 5 shows the example of the results of simulation of the evolution of initial disturbance in the form of the “smoothed step” (9) in case when the break of the “bottom” is negative and located in front of the localization region of the fluid surface disturbance. It can be seen that in this case, the front of the disturbance becomes more gentle with time, the oscillatory soliton structure in the front region is not formed, but the development of low-frequency oscillations behind the main maximum occurs. This result is easily explained within the framework of the similarity principle for the KdV equation [2]: the evolution of the "tail" of the initial disturbance occurs in the region of small values of the dispersive parameter, whereas in the front region, where the dispersion is relatively large, the formation of a shock wave does not occur.

As for the third law of change of β (harmonic oscillations of the
parameter β with time on all x-axis), a series of numerical experiments
for various k_{0} = const at variable frequency ω [see law of change (12)] showed that at some values of ω the stationary (locally)
standing waves can be formed, in other cases it is possible
formation of the stationary periodic wave structures, in intermediate
cases a chaotic regime is usually realized.

**Figure 5: *** Evolution of “step” (9) in the model (10) with с<0: a − t
≈ 0.25; b − t ≈ 0.75.*

Let us now consider the problem of evolution of the 2D solitons in the framework of the standard KP equation

With varying in time and space dispersive parameter β = β (t , x, y) . This situation can take place in the problems on propagation of gravity and gravity-capillary waves on surface of shallow fluid [1] when the fluid depth is the function of the spatial coordinates and time H = H(t , x, y) .

Here, situation is the same as for 1D model of the KdV equation described above take place: if the analytical solutions of the KP equation are known that in case β = β (t,r) the dispersion term of equation becomes quasi-linear that the model being not exactly integrable (the IST method is not applicable) [1].

The problem of numerical simulation of the KP equation with
O(τ^{2} , h^{4} ) is formulated analogously the problem for the KdV
equation (see previous section). To solve the initial problem for
the KP equation (13) with variable dispersion (varying relief of the
bottom) we used an implicit difference scheme [3] with O(τ^{2} , h^{4} )
approximation. Initial conditions was chosen in form of the exact 2D one-soliton solution of the KP equation [1], the complete absorption
conditions on boundaries of the computation region [1,3]
were imposed, and simulation has been conducted for the same
types of model function β as for the KdV equation [see formulae
(10)-(12)].

Consider basic results of the numerical experiments on the investigation of the structure and the evolution of 2D solitary waves on the fluid surface with variable dispersion.

The first series of numerical experiments have been aimed
at study of the soliton dynamics under spasmodic character of
the dispersion change (function β = β(t, x, y) has form of the
"step"). At first, we investigated the evolution of initial pulse in case
when at t_{cr} the spasmodic change of β has a place behind soliton
["negative" step when с < 0 in formulae (10), (11)]. At this, the dependence
of the spatial structure of solution on the parameter a
value in models (10) and (11) was studied. The obtained results
(see an example in figure 6) showed that in all cases the evolution
leads to the formation of waving
tail which is not connected with
soliton going away and caused only by local influence of sudden
change
of the “relief” β = β(t, x) . Consequently,
the formation of
oscillatory structure is connected not so much with decreasing of a
role of the dispersion
effects behind soliton as with the spasmodic
changing of β in space.

**Figure 6: *** Solution of eq. (13) for the dispersion law (10) with
a=5.0, c=−0.0038 at t=0.6.*

In the next series of numerical simulation, we considered the
evolution of a 2D soliton in case when the sudden change of the dispersion parameter has a place directly under or in front of an
initial pulse ("negative" step). An example of the results of that series
is shown in figure 7. From the analysis of the results of whole
of the series one can see that for such character of the relief of function
β the disturbance
caused by sudden change of the dispersive
parameter has also a local character, i.e. it doesn't propagate
together with the going away soliton. But, unlike the cases considered
in the first series of simulation, the asymptotes of leaving
soliton become oscillating (in any case in the time limits of
numerical experiment),
besides, against a background of the longwave
oscillations of the waving
tail we can also see the appearance
of the wave fluctuations. The effects noted may be interpreted as
a result of those that for the areas of the wave surface with different
values of local wave number k_{x} the value of the dispersive effects
is different. As a result, the intensity of the phase mixing of the
Fourier-harmonics within (x, y)-region varies with the coordinates
and, therefore, it reacts differently to the nonlinear generation of
the harmonics with various (in particular, large) wavenumbers
k_{x}.

**Figure 7: *** Solution of eq. (13) for the dispersion law (11) with
a=4.0, c=−0.0038 at t=0.6.*

In the third series of the experiments with dispersive parameter
changing with the laws (10) and (11) we considered the cases of
"positive" step [ c > 0 in formulae (10) and (11)] being both in
front of and behind of initial pulse for the wide diapason of values
of parameter а. The examples of the most interesting results are
shown in figure 8. One can see that when "positive" step is far in
front of maximum of the function u (0, x, y) the soliton evolution on the initial stage does not practically differ qualitatively from that
for β = const (Figure 8,a), but in the future the evolution character
is defined by presence of the step, namely the processes, caused
by the same causes which have been noted for the results of the
second series of numerical simulation, begin to be developed (figure
8,b). As we can see in the figure, the appreciable change of the
soliton structure which can lead to wave falling is observed owing
to intensive generation of the harmonics with big k_{x} in the soliton
front region, even for rather small height of the step (i.e. even if the
value of parameter c in formulae (10), (11) is rather small). Thus,
as it follows from the results of this series, the disturbance of the
propagating 2D soliton caused by sudden change in time and space
of the dispersive parameter with с > 0 has also local character.

**Figure 8: *** Evolution of soliton of eq. (13) for the dispersion law
(11) with a=5.0, c=0.0038: (a) t=0.6, (b) t=0.8.*

As to the second law of the β change (model (12) − harmonic oscillation
of the parameter β with time on the whole (x, y)-plane),
the series of numerical simulation for different k_{0} = const and
variable frequency ω [see law (12)] showed that for some values of
ω the stationary (locally) standing waves can be formed, in other
cases the formation of the stationary periodical wave structures is
possible, and in the intermediate cases a chaotic regime is usually
realized.

In the experiments made for different values of the parameter
k_{0} and ω = const , we found that the stable (in any case in the
limits of the numerical computation time) solutions can be formed
only for k_{0} ≤ β_{0} in formula (12), and the solutions are unstable
in another cases. An example of evolution of the 2D soliton when
its structure along the x and y axes acquires the wave character
and the amplitude of its maximum decreases with time is shown
in figure 9.

**Figure 9: *** Evolution of 2D soliton of eq. (13) at t =0.4, 1.2, 2.0.*

Summing up the above, one can note that the numerical simulation of evolution of the 2D solitons describing by the model of the KP equation with β = β(t, x, y) enabled us to found different types stable and unstable solutions including the solution of the mixed "soliton non-soliton" type for various character of the dispersion change in time and space.

Obtained results open the new perspectives in the investigation of a number applied problems of the dynamics of the nonone- dimensional nonlinear waves in the specific physical media, including upper atmosphere (ionosphere), magnetosphere and in a plasma [1,3-5].

The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. This work was also supported by the Shota Rustaveli National Science Foundation (SRNF), grant no. FR17 252.

In the paper the results of numerical study of evolution of the solitons of gravity and gravity-capillary waves on surface of shallow fluid when the characteristic wavelength is essentially greater then depth, β = β(t, x, y) , were presented for the cases when dispersive parameter is a function of time and spatial coordinates, β = β(t, x, y) . This corresponds to the problems when the relief of bottom is changed in time and space. We have considered three cases of variable dispersion when the sudden "breaking up of the bottom", the gradual "changing of height" of the bottom area, and the "bottom oscillation" with time take place. To solve the problem, we used both 1D approach (the equations of the KdV-class) and also 2D description (the equations of the KP-class). For all cases the numerical solutions of the problem in 1D and 2D geometry were presented. It was noted that the approach realized can be useful also in other applications of nonlinear wave theory such as dynamics of 1D and multidimensional solitary waves in other specific physical media, including upper atmosphere (ionosphere), magnetosphere and in a plasma (see, for example, the papers [6-8]).

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- Belashov VYu. “The KP equation and its generalizations”. Theory, applications. NEISRI FEB RAS, Magadan (1997): 162.
- Belashov VYu and Belashova ES. “Solitons: theory, simulation, applications”. Kazan Federal University, Kazan (2016): 270.
- Belashov VYu.,
*et al*. “Nonlinear wave structures of the soliton and vortex types in complex continuous media: theory, simulation, applications”.*Lecture Notes of TICMI*18 (2017): 90. - Belashov VYu.,
*et al*. “Structure and evolution of IGW and TID in regions with sharp gradients of the ionospheric parameters”.*Journal of Geophysical Research*112 (2007): A07302. - Belashov VYu and Belashova ES. “Dynamics of IGW and traveling ionospheric disturbances in regions with sharp gradients of the ionospheric parameters”.
*Advances in Space Research*2 (2015): 333-340. - Popel SI.,
*et al*. “Evolution of perturbations in a dusty plasma with a variable charge”.*Plasma Physics*5 (2001): 1497-1504.

**
Citation: ** Vasily Yu Belashov.,

**Copyright: ** © 2021 Vasily Yu Belashov., *et al*. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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