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European Journal of Applied Sciences – Vol. 12, No. 1

Publication Date: February 25, 2024

DOI:10.14738/aivp.121.16384

Volobuev, A. N., Adyshirin-Zade, K. A., Antipova, T. A., Novikov, V. A., & Aleksandrova, N. N. (2024). Occurrence and Evolution of

Gaussian-Type Soliton-Like Waves in An Open Water Channel. European Journal of Applied Sciences, Vol - 12(1). 544-560.

Services for Science and Education – United Kingdom

Occurrence and Evolution of Gaussian-Type Soliton-Like Waves

in An Open Water Channel

A. N. Volobuev

K. A. Adyshirin-Zade

T. A. Antipova

V. A. Novikov

N. N. Aleksandrova

ABSTRACT

The possibility of a transition from the nonlinear equations of impulse, continuity

and the complex velocity potential of an inviscid liquid to the nonlinear Schrodinger

equation with logarithmic nonlinearity is shown. The solution to this equation in an

open water channel is a solitary wave, the shape of which is described by Gauss's

law. This wave is not a soliton due to the significant excess of the effects of the

nonlinearity of the equations of hydrodynamics over the phenomenon of

dispersion. The further evolution of the solitary wave under the examined

conditions leads to an increase in the steepness of forward front of the wave, the

appearance of a rupture surface and the overturning of the solitary wave. The

calculation of these phenomena is presented.

Keywords: solitary wave, open water channel, nonlinear Schrodinger equation, rupture

surface, overturning wave.

INTRODUCTION

The propagation of a solitary wave in an open water channel, which was observed and

accompanied by J.S. Russell on horseback in 1834, has marked the beginning of the research of

solitary waves and the discovery of solitons - special stable wave formations. Having arisen, a

solitary wave can evolve in two directions.

The first direction is characterized by a gradual decrease in the height of the solitary wave

during its movement and finally its disappearance. This development of a solitary wave is

observed in media and waves with high dispersion. The harmonic components of the solitary

wave (modes) move at different speeds [1]. This leads to the dispersed of the wave packet and

the disappearance of the solitary wave. Often, before disappearing, a solitary wave turns into a

long oscillating wave packet – a sequence of waves. This kind of a solitary wave evolution was

observed by J.S. Russell [2].

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545

Volobuev, A. N., Adyshirin-Zade, K. A., Antipova, T. A., Novikov, V. A., & Aleksandrova, N. N. (2024). Occurrence and Evolution of Gaussian-Type

Soliton-Like Waves in An Open Water Channel. European Journal of Applied Sciences, Vol - 12(1). 544-560.

URL: http://dx.doi.org/10.14738/aivp.121.16384

Another direction of wave evolution is observed when the dispersion effects are small, and the

equation describing a solitary wave has a significantly nonlinear character. In this case, the

steepness of the wave front gradually increases and the solitary wave overturns. The vivid

example of such evolution of waves there are surf waves. But even on the free surface of the

water, during the formation of waves, the overturning of wave crests is often observed.

In which direction the evolution of a wave on water will go is determined, first of all, by the

ratio of the amplitude of the wave to its length. In a high wave, the second direction of wave

development is most often realized.

The case of the balance of dispersion and nonlinear effects with the appearance of solitons, in

practice, cannot have a long time. Although from the point of view of theory research of solitons

turned out to be very productive for hydrodynamics.

In this paper, we consider the conditions for the occurrence of a solitary wave with weak

dispersion described by the Gauss function of the so-called soliton-like wave. Its nonlinear

equation and evolution are investigated. A comparison is made with other types of waves.

HYDRODYNAMIC EQUATIONS FOR A SOLITARY WAVE IN AN OPEN WATER CHANNEL

In [3] the possibility of the occurrence of Korteweg - de Vries solitons in an open water channel

was shown, Fig. 1.

Fig. 1: A solitary wave in an open water channel

However, the occurrence of a stable soliton is possible only at a small height

 max

. The soliton

solution of the equations of hydrodynamics arises under the condition of taking into account

the zero and first order of expansion into series of the equations of impulse and continuity by a

small parameter

1

max

= 

h





, where h is the depth of the channel.

Let's consider another approximate solution. We will consider the fluid to be inviscid and the

transverse velocity of the fluid to be small compared to the longitudinal velocity

W V

. But

we will make the restriction on the height of the solitary wave less stringent. Let's consider the

evolution of a solitary wave under these conditions.

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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 1, February-2024

The impulse equation is used in a form similar to [3] for the pressure term:

0

1 ( )

+ =





+





S X

PS

X

V

V

t

V







, (1)

where



is the density of the liquid, V is the longitudinal velocity of the fluid, X is the

longitudinal coordinate, t is the time,

S = S1 + S0

is the cross section of the flow,

S1 = b

is the

cross section of the wave, b is the channel width,

S0 = bh

is the cross section of the flow in the

channel outside the wave, Fig. 1. Pressure in the fluid is

P = Pa + Pg

, where

P gh const a

=  =

2

1

is the average gravitational pressure in the flow outside the wave, g is the acceleration of

gravity,

Pg

is the gravitational pressure in the wave at the level of undisturbed liquid at

Z = 0

. Thus, the vertical movement of the fluid in the equations of hydrodynamics is taken into

account only due to the changing area

S1

.

The continuity equation is used in the form [4]:

= 0





+





X

VS

t

S

. (2)

We assume that solitary waves on the water surface are determined only by gravitational

forces. In this case, the pressure at the level of the undisturbed fluid at

Z = 0

is equal to:

b

S

P g g

1 =  . (3)

THE VELOCITY OF A SOLITARY WAVE IN AN OPEN WATER CHANNEL

Let's copy the impulse equation (1) in the form:

0

1 ( )

=





















+ +





X

V

S V

PS V

t

V







. (4)

If to consider (4) a nonlinear equation of a wave of the first degree, then the velocity of this

wave is equal to:

S V

PS

с







1 ( )

= . (5)

Carrying out similar transformations in the continuity equation (2), we find:

= 0





















+ +





X

S

S

V

V S

t

S

. (6)