The kinetic energy of the deterministic chaos of internal waves in a Newtonian fluid with a constant density and a global velocity field is defined by

where the global velocity field is formed by velocity fields of four wave groups

Substitution of (2) in (1) yields the kinetic energy of the deterministic chaos of the exponential oscillons and pulsons (118) of [3] in terms of the velocity fields of four wave groups

For clarification of summation, we define a matrix of the kinetic energy by

where the braces notation denotes the rectangular summation of all elements of the summation matrix.

Since the velocity fields of four wave groups are expanded in the weighted experimental DVK structures (20) of [3],

matrix (4) takes the following form:

where the summation braces signify the rectangular summation of all elements of a summation matrix with the general term for and, i.e.

since each wave group is composed of M waves.

Primarily, we decompose as follows

Here,

is a diagonal matrix, which includes all diagonal elements of, and

is a complementary matrix, which is composed of the upper and lower triangular matrices of.

The kinetic energy is correspondingly decomposed as

where the first sum is produced by the elements of:

and the second sum by the elements of:

In (12)-(13), the constituents of are

and the constituents of are

Secondly, we expand all sums into internal sums with, which correspond to internal interaction of elementary oscillons of propagation from the th family, and external sums with, which describe external interaction of elementary oscillons of propagation from the th and th families (see Section 2.6).

The summation matrix of the diagonal constituent of

due to the commutative property of the dot products

yields the following reduction of the rectangular summation to the diagonal and triangular summations:

Similar reductions of the remaining diagonal constituents yield

The summation matrices of the non-diagonal constituent of

and

because of the commutative properties of the dot products:

produce the following reduction of the rectangular summation to the diagonal and triangular summations:

If, the asymmetric reduction (23) is converted into the symmetric reduction (18).

Proceeding the same way for the rest of the summation matrices of the non-diagonal constituents returns

Thirdly, we define wave, group, and energy pulsons and oscillons via various combinations of dot products of the DVK structures and express the dot products in terms of the weighted experimental DSD structures (29) of [3].

General terms of the internal sums of (18)-(19) produce wave pulsons of propagation

Since the DVK structures are related with the DSK structures by (20) of [3]

the wave pulsons of propagation in the DSD structures take the following form:

A superposition of a group of the wave pulsons of propagation

is termed a group pulson of propagation. Here, is simplified by the Pythagorean identity for the wave numbers

Eventually, the diagonal summation of all group pulsons of propagation results in an energy pulson of propagation

which shows a cumulative energy of all group pulsons of propagation.

General terms of the external sums of (18)-(19) generate wave oscillons of diagonal external interaction (diagonal wave oscillons for brevity)

Using (26) and Equation (26) with, we obtain the diagonal wave oscillons in the DSD structures

Summation of (32) yields a diagonal group oscillon

where an amplitude produced by wave numbers

The triangular summation of the diagonal group oscillons results in a diagonal energy oscillon

which gives a cumulative energy of all diagonal group oscillons.

So, summation of constituents (12) of the kinetic energy is completed with the following result:

If, then the diagonal wave oscillons (32) are converted into the doubled wave pulsons of propagation (27) and the diagonal group oscillon (33) becomes equal to the doubled group pulson of propagation (28).

General terms of the internal sums of (23)-(24) correspond to wave oscillons of internal interaction (internal wave oscillons for brevity)

In the DSD structures, the internal wave oscillons become

Adding the internal wave oscillons, we get an internal group oscillon

The diagonal summation of the internal group oscillon results in an internal energy oscillon

which returns a cumulative energy of all internal group oscillons.

General terms of the external sums of (23)-(24) correspond to wave oscillons of non-diagonal external interaction (external wave oscillons for brevity)

Using (26) and Equation (26) with, we compute the external wave oscillons via the DSD structures

An external group oscillon takes the following form:

where amplitudes generated by the wave numbers

We then imply the triangular summation of the external group oscillons to find an external energy oscillon

which demonstrates a cumulative energy of all external group oscillons.

Thus, summation of constituents (13) of the kinetic energy is finished as follows:

If, then the external wave oscillons (42) are transformed into the doubled internal wave oscillons (38) and the external group oscillon (43) becomes equal to the doubled internal group oscillon (39).

Members of the th family of elementary oscillons of propagation of the velocity potential

are defined via the weighted experimental DSK structures (1) of [3]

where, are functional amplitudes, 3-d DSK functions are products

of the following 1-d (one-dimensional) DSK functions and an exponential function:

In Equations (47)-(51), are the Cartesian coordinates of a motionless frame of reference, t is time, are the Cartesian coordinates of a frame of reference moving with the th elementary oscillon of propagation, is the celerity of propagation, and are reference values of at, ,. A sign parameter for and for.

A wave oscillon of propagation of the velocity potential

consists of the elementary oscillons of propagation.

Similarly, a group oscillon of propagation of the velocity potential

is composed of the wave oscillons of propagation.

The wave pulsons and oscillons and the group pulsons and oscillons of the kinetic energy are constructed of elementary pulsons and oscillons of the kinetic energy.

Four elementary pulsons of propagation in the DSD structures for are defined by

The wave pulsons of propagation (27) and the group pulson of propagation (28) then become the following superpositions of the elementary pulsons of propagation:

and

There are six elementary oscillons of internal interaction (internal elementary oscillons for brevity) between members of the th family of the elementary oscillons of propagation for

The internal wave oscillons (38) and the internal group oscillon (39) are decomposed via the internal elementary oscillons as follows:

and

There are four elementary oscillons of diagonal external interaction (diagonal elementary oscillons for brevity) between members of the th and th families of the elementary oscillons of propagation from the same wave groups with and

The diagonal wave oscillons (32) and the diagonal group oscillon (33) then are subsequent superpositions of the diagonal elementary oscillons:

and

Eventually, there are six elementary oscillons of non-diagonal external interaction (external elementary oscillons for brevity) between members of the th and th families of the elementary oscillons of propagation from distinct pairs of wave groups with and

The external wave oscillons (42) and the external group oscillon (43) are decomposed in terms of the external elementary oscillons in the following way:

and

If, then the diagonal elementary oscillons (60) are reduced to the doubled elementary pulsons of propagation (54) and then the external elementary oscillons (63) are transformed into the doubled internal elementary oscillons (57).

Finally, we summarize quantization of the kinetic energy of the deterministic chaos of exponential oscillons and pulsons. The cumulative pulson of the kinetic energy (11) (the cumulative energy pulson for brevity) may be decomposed as follows:

where is the energy pulson of propagation (30), is the internal energy oscillon (40), is the diagonal energy oscillon (35), and is the external energy oscillon (45).

The energy pulson and the energy oscillons are composed of the group pulsons and the group oscillons, respectively. The group pulson of propagation (28) is composed of four wave pulsons of propagation (27) that describe a vector self-interaction of velocity fields of th elementary oscillons of propagation (47) from four wave groups, respectively, for.

The internal group oscillon (39) consists of six internal wave oscillons (38) that express a vector internal interaction of velocity fields of th elementary oscillons of propagation from the distinct wave groups, correspondingly, for.

The diagonal group oscillon (33) is constructed of four diagonal wave oscillons (32) that describe a vector external interaction of velocity fields of th and th elementary oscillons of propagation from the selfsame wave groups for,.

The external group oscillon (43) includes six external wave oscillons (42) that express a vector external interaction of velocity fields of th and th elementary oscillons of propagation from the distinct pairs of wave groups &, &, &, &, &, &, respectively, for,.

The wave pulsons of propagation (27) are composed of four elementary pulsons of propagation (54) that describe a scalar self-interaction of th elementary oscillons of propagation from four wave groups with the velocity potentials, respectively, for.

The internal wave oscillons (38) consist of six internal elementary oscillons (57) that express a scalar internal interaction of th elementary oscillons of propagation from the distinct wave groups with the velocity potentials &, &, &, &, &, &, respectively, for.

The diagonal wave oscillons (32) are constructed of four diagonal elementary oscillons (60) that describe a scalar external interaction of th and th elementary oscillons of propagation from the selfsame wave groups with the velocity potentials &, &, &, &, correspondingly, for,.

The external wave oscillons (42) include six external elementary oscillons, (63) that express a scalar external interaction of th and th elementary oscillons of propagation from the distinct pairs of wave groups with the velocity potentials & and &, & and &, & and &, & and &, & and &, & and &, for,.

To clarify topology of the elementary oscillons of propagation in the x-direction, we separate variable x and recast (47)-(51) to the following x-form:

where amplitudes

are expressed via coefficients

are x-shifts in the motionless frame, and

are X_m-shifts in the th moving frame.

For any frozen, the elementary oscillons of propagation (47) are transformed into solo-frequency (1-f) neutral oscillons in x with wavenumber, amplitudes proportional to, and x-shifts

See classification of oscillons and pulsons in Section 3.5.

Let’s consider propagation velocity and period as independent parameters. Then a wave number

and a spatial period along the x-axis (an x-wavelength)

Since integrals of sines and cosines of (67) over x-period vanish,

i.e. the elementary oscillons of propagation are neutral as the average over the x-period vanishes.

To display topology of the elementary oscillons of propagation in the y-direction, we separate variable y and transform (47)-(51) into the following y-form:

where amplitudes

are computed in terms of coefficients

are y-shifts in the motionless frame, and

are Y_m-shifts in the th moving frame.

For any frozen, the elementary oscillons of propagation (47) are reduced to 1-f neutral oscillons in y with wavenumber, amplitudes proportional to, and y-shifts

Since and are independent parameters, a wave number and a spatial period along the y-axis become

The elementary oscillons of propagation are also neutral in the y-direction because

Topology of the elementary oscillons of propagation in time follows from the inhomogeneous Fourier t-form of (47)-(51)

in terms of four trigonometric functions

of two arguments

In Equations (84)-(86),

are amplitudes,

are frequencies,

are t-shifts,

are x- and y-shifts, respectively.

For any frozen, the elementary oscillons of propagation (47) are converted into duet-frequency (2-f) neutral oscillons in t, which are generated by superpositions of sine waves and cosine waves with frequencies, respectively, amplitudes proportional to, and t-shifts depending on

Since temporal periods of and are equal to, respectively,

a temporal period of the elementary oscillons of propagation

where is a least common multiple of and are integers.

Because integrals of sine waves and cosine waves over t-periods vanish, the elementary oscillons of propagation are neutral in time, as well, since

The elementary oscillon of propagation with is visualized in Figure 1 for the following independent parameters:

In Figures 1-18, two adjacent x-, y- and t-periods are shown by dotted vertical lines.

The elementary oscillons of propagation are displayed via 1-f neutral oscillons in x with periods, in terms of 1-f neutral oscillons in y with periods, and by 2-f neutral oscillons in t with periods, five local t-maximums, and five local t-minimums, the numbers of which are specified by.

In agreement with (67) and (74),

So, x-extrema of the elementary oscillon, which are equal to, correspond to x-zeroes of the x-complementary oscillon and x-zeroes of to x-extrema of, which are also equal to. Similarly, x-extrema of, which are equal to, correspond to x-zeroes of the x-complementary oscillon and x-zeroes of to x-extrema of, which are equal to, as well.

Because of (76) and (82),

Thus, y-extrema of the elementary oscillon, which are equal to, correspond to y-zeroes of the y-complementary oscillon and y-zeroes of to y-extrema of, which are also equal to. Analogously, y-extrema of, which are equal to, correspond to y-zeroes of the y-complementary oscillon and y-zeroes of to y-extrema of, which are equal to, as well.

Consider the th and th families of inhomogeneous x-eigenfunctions of propagation

of four arguments

which are parametrized by two wavenumbers and with and.

An x-matrix of external interaction between x-eigenfunctions of propagation of the th and th families takes the following form:

Because of the product rules for sine and cosine waves, there are 16 inhomogeneous x-eigenfunctions of external interaction

of eight arguments

which are parametrized by two wavenumbers

with,. Thus, the x-eigenfunctions of external interaction include sine and cosine waves of all possible combinations of x-wave-numbers and x-shifts.

Computing products (100) of x-eigenfunctions of propagation yields the following expansions via the x-eigenfunctions of external interaction (101):

If, then two wavenumbers of the x-eigenfunctions of external interaction (103) are reduced to a single degenerated wavenumber since

Consequently, eight arguments (102) of the x-eigenfunctions of external interaction are transformed into four arguments as

Therefore, we define the following four arguments of x-eigenfunctions of internal interaction:

If, x-eigenfunctions of external interaction are converted into eight x-eigenfunctions because

Thus, six x-eigenfunctions of internal interaction and two vertical shifts are defined as follows:

The x-matrix of internal interaction becomes a symmetrical one, namely,

Computing elements of the x-matrix of internal interaction via the x-eigen-functions of internal interaction (109) returns

Computation of eigenfunctions of the inhomogeneous Fourier series in y coincides with the computation of the eigenfunctions of the inhomogeneous Fourier series in x up to the following substitutions:

Consider the th (85) and th families of inhomogeneous t-eigenfunctions of propagation

of four arguments

which are parametrized by four frequencies, and with,.

A t-matrix of external interaction between t-eigenfunctions of propagation of the th and th families takes the following form:

Due to the product rules for sine and cosine waves, there are eight arguments of t-eigenfunctions of external interaction

which include all possible sums and differences of the arguments of t-eigenfunctions of propagation (114).

Consequently, there are 16 inhomogeneous t-eigenfunctions of external interaction

of eight arguments

which are parametrized by eight frequencies

with,.

Computation of elements of the t-matrix of external interaction (115) returns the following expansions in terms of the t-eigenfunctions of external interaction (117):

If, then eight frequencies of the t-eigenfunctions of external interaction are reduced to four degenerated frequencies since

So, we define four frequencies of t-eigenfunctions of internal interaction

Consequently, eight arguments of the t-eigenfunctions of external interaction are transformed into four independent arguments as

Thus, we define the following four arguments of t-eigenfunctions of internal interaction:

If, 16 t-eigenfunctions of external interaction are converted into eight functions because

Therefore, eight t-eigenfunctions of internal interaction are defined as follows:

Substitution of (126) in (125) yields asymptotics of t-eigenfunctions of external interaction as approaches

The t-matrix of internal interaction is a symmetrical one, viz.

Computing elements of the t-matrix of internal interaction via the t-eigenfunctions of internal interaction (126) gives

In eigenfunctions (98) with, the elementary oscillons of propagation of the velocity potential (47), (67) are represented as follows:

Substitution of (130) in (54) gives

Computing squares of by (111) in terms of the x-eigenfunctions of internal interaction (109) yields

For any frozen, the elementary pulsons of propagation are reduced to 1-f critical positive pulsons in x with wavenumber, amplitudes proportional to

x-shifts (72), and energy shifts proportional to (133). The elementary pulsons of propagation are critical since they are produced by the complete squares of.

The x-period of the elementary pulsons of propagation

Because integrals of and over vanish,

Integrals (135) show that two pairs of the elementary pulsons of propagation and on average transfer equal positive amounts of the kinetic energy along the x-axis.

Similarly, the elementary oscillons of propagation of the velocity potential may be written in the y-direction as

Therefore, the elementary pulsons of propagation of the kinetic energy via become

and in terms of take the following form:

For any frozen, the elementary pulsons of propagation are transformed into 1-f critical positive pulsons in y with wavenumber, amplitudes proportional to

y-shifts (81), and energy shifts proportional to (139). The elementary pulsons of propagation are also critical since they are formed by the complete squares of.

Since amplitudes and shifts do not effect periods, the y-period

Because integrals of and over vanish,

Therefore, two pairs of the elementary pulsons of propagation and on average transfer equal positive amounts of the kinetic energy along the y-axis.

With the help of (54), (47), and (84), the elementary pulsons of propagation become the complete squares of trigonometric polynomials

Expanding the trigonometric polynomials and substituting elements of the t-matrix of internal interaction (129) via the t-eigenfunctions of internal interaction (126) returns

where

For any frozen, the elementary pulsons of propagation are converted into at most quartet-frequency (4-f) critical positive pulsons in t, which are formed by superpositions of cosine waves with frequencies, correspondingly, amplitudes proportional to

t-shifts depending on (91), and energy shifts proportional to

The elementary pulsons of propagation are critical as they are described by the complete squares of trigonometric polynomials (142).

Temporal periods of, respectively, are

Therefore, a temporal period of the elementary pulsons of propagation

where are integers.

Because integrals of cosine waves over vanish, averages of the elementary pulsons of propagation over become

Therefore, integral (149) validates that the elementary pulsons of propagation on average transfer in time the same positive amount of the kinetic energy.

The elementary pulson of propagation with, and is shown in Figure 2 for independent parameters (95). The elementary pulsons are visualized by 1-f critical pulsons in x with periods and in terms of 1-f critical pulsons in y with periods.

The elementary pulsons are displayed as 4-f critical pulsons in t with periods, local t-maximums, and local t-minimums, the numbers of which do not exceed or equal to due to merged t-maximums and t-minimums. The elementary pulsons, as 4-f critical pulsons in t with periods, local t-maximums, and local t-minimums, the numbers of which do not exceed or equal to because of merged t-maximums and t-minimums.

In the view of (131), (132), and (134),

So, x-maximums of the elementary pulson, which are equal to, correspond to x-zeroes of the x-complementary pulson and x-zeroes of to x-maximums of, which are also equal to. Similarly, x-maximums of the elementary pulson, which are equal to, correspond to x-zeroes of the x-complementary pulson and x-zeroes of to x-maximums of, which are equal to, as well.

Due to (137), (138), and (140),

Thus, y-maximums of the elementary pulson, which are equal to, correspond to y-zeroes of the y-complementary pulson and y-zeroes of to y-maximums of, which are also equal to. Analogously, y-maximums of the elementary pulson, which are equal to, correspond to y-zeroes of the y-complementary pulson and y-zeroes of to y-maximums of, which are also equal to.

Expressing definitions (57) in the x-eigenfunctions (98) with and using the x-matrix of internal interaction (111) yields the internal elementary oscillons via the x-eigenfunctions of internal interaction (109) in the following form:

For any frozen, the first pair of internal elementary oscillons is reduced to 1-f neutral oscillons in x, which are formed by sine waves with wavenumber. The second pair and the third pair are presented by 1-f nonneutral oscillons in x, which are generated by cosine wave and sine wave with wavenumber and energy shifts proportional to

For the first pair, amplitudes are proportional to (133) and x-shifts are specified by (72), for the second and third pairs amplitudes are proportional to and x-shifts depend on (72). The x-period of the internal elementary oscillons coincides with the x-period of the elementary pulsons of propagation (134).

Since integrals of for over vanish,

It is at tedious but a straightforward procedure to show that

where trigonometric functions of the double argument

Integrals (154) show that the first pair of internal elementary oscillons does not transfer on average any kinetic energy, the internal elementary oscillons of the second pair transfer on average equal amounts of the kinetic energy of the same sign, and the internal elementary oscillons of the third pair transfer on average equal amounts of the kinetic energy of opposite signs along the x-axis.

For two vector fields and, a scalar field of the kinetic energy of interaction is given by

By the definition of the dot product

where is an angle between vector fields and at a point.

So, the maximal magnitude of

does not depend on, but the sign of is determined by. If the vector fields and are unidirectional at, then. If the vector fields have opposite directions, then. The kinetic energy of interaction vanishes, if the vector fields and are orthogonal at.

Thus, becomes negative if on the contrary to the kinetic energy of propagation of the vector field

that is always positively defined. To summarize, the scalar field of the kinetic energy of propagation has a pulsatory nature and the scalar field of the kinetic energy of interaction an oscillatory one.

Let a nonlinear wave with an energy shift and amplitudes be decomposed into the inhomogeneous Fourier expansion via M cosine waves and sine waves with wavenumbers and -shifts proportional to

such that

Then a period of in a variable

and the average of over period

An oscillon is termed a neutral oscillon, if the average of over vanishes, i.e.. So, the neutral oscillon doesn’t transfer the kinetic energy along the -direction.

An oscillon is called a positive (nonneutral) oscillon, if the average of over is positive since and the nonneutral oscillon transfers the positive amount of the kinetic energy along the -direction.

When, the positive oscillon is transformed into a critical positive pulson, which also transfers the positive amount of the kinetic energy along the -direction. If, the critical positive pulson is converted into a supercritical pulson, which transfers the positive amount of the kinetic energy along the -direction, as well.

Similarly, an oscillon is named a negative (nonneutral) oscillon, if the average of over is negative because and the nonneutral oscillon transfers the negative amount of the kinetic energy along the -direction.

When, the negative oscillon is transformed into a critical negative pulson, which also transfers the negative amount of the kinetic energy along the -direction. If, the critical negative pulson is converted into a subcritical pulson, which transfers the negative amount of the kinetic energy along the -direction, as well.

We then represent definitions (57) in the y-eigenfunctions with and use the y-matrix of internal interaction to compute the internal elementary oscillons in terms of the y-eigenfunctions of internal interaction as follows

For any frozen, the first pair and the third pair of internal elementary oscillons are represented by 1-f nonneutral oscillons in y, which are generated by cosine wave and sine wave with wavenumber and energy shifts proportional to

The second pair is transformed into 1-f neutral oscillons in y, which are formed by sine waves with wavenumber. For the first and third pairs, amplitudes are proportional to and y-shifts depend on (81). For the second pair, amplitudes are proportional to (139) and y-shifts are specified by (81). The y-period of the internal elementary oscillons equals to the y-period of the elementary pulsons of propagation (140).

Because integrals of for over the period vanish,

It may be shown that

where trigonometric functions of the double argument

Thus, the internal elementary oscillons of the first pair transfer on average equal amounts of the kinetic energy of the same sign, the second pair of internal elementary oscillons on average does not transfer on average any kinetic energy, and the internal elementary oscillons of the third pair transfer on average equal amounts of the kinetic energy of opposite signs along the y-axis.

Usage of definitions (57) together with trigonometric polynomials (84) in t-eigenfunctions (113) and application of the t-matrix of internal interaction (129) yields the internal elementary oscillons in terms of the t-eigenfunctions of internal interaction (126) in the following t-form:

where

For any frozen, the first pair of internal elementary oscillons is presented by at most trio-frequency (3-f) neutral oscillons in t, which are produced by superpositions of sine waves with frequencies, correspondingly. The second pair is represented by at most 3-f neutral oscillons in t, which are generated by superpositions of sine waves with frequencies, respectively. The third pair is formed by 2-f nonneutral oscillons in t, which are created by superpositions of cosine waves with frequencies, correspondingly, and energy shifts proportional to. For all pairs, amplitudes are proportional to (145) and t-shifts depend on (91).

A temporal period of the first pair

where are integers and are given by (147).

Similarly, a temporal period of the second pair

where are integers.

Analogously, a temporal period of the third pair

where are integers.

Because integrals of over vanish, averages of the internal elementary oscillons over the relevant t-periods become

Therefore, the first and second pairs of internal elementary oscillons do not transfer on average any kinetic energy in time, the internal elementary oscillons of the third pair on average transfer in time equal amounts of the kinetic energy of the opposite signs.

The internal elementary oscillon with is visualized in Figure 3 for independent parameters (95). The first pair is displayed by 1-f neutral oscillons in x with periods, the second and third pairs via 1-f nonneutral oscillons in x with periods.

The first and third pairs are visualized in terms of 1-f nonneutral oscillons in y with periods and the second pair by 1-f neutral oscillons in y with periods.

The first pair is shown by 3-f neutral oscillons in t with periods, local t-maximums, and local t-minimums, the numbers of which do not exceed or equal to due to merged t-maximums and t-minimums, respectively. The second pair is visualized via 3-f neutral oscillons in t with periods, local t-maximums, and local t-minimums, the numbers of which do not exceed or equal to due to merged t-maximums and t-minimums. The third pair is displayed in terms of 2-f neutral oscillons in t with periods, local t-maximums, and local t-minimums, the numbers of which are indicated by.

In agreement with (152),

Locations of x-maximums of coincide with locations of x-minimums of and locations of x-minimums of coincide with locations of x-maximums of. Locations of x-maximums and x-minimums of coincide with locations of x-maximums and x-minimums of.

Because of (165),

Locations of y-maximums of coincide with locations of y-minimums of and locations of y-minimums of coincide with locations of y-maximums of. Locations of y-maximums and y-minimums of coincide with locations of y-maximums and y-minimums of.

Due to (170),

Locations of t-maximums and t-minimums of coincide with locations of t-maximums and t-minimums of.

Using (130), we recast definitions (60) of the diagonal elementary oscillons in terms of x-eigenfunctions (98) and convert them with the help of the x-matrix of external interaction (104) into the following form:

For any frozen, the diagonal elementary oscillons are transformed into 2-f neutral oscillons in x, which are produced by superpositions of two pairs of cosine waves and with two wavenumbers, respectively, amplitudes proportional to

and x-shifts depending on

If, the diagonal elementary oscillons (179) are reduced to the doubled elementary pulsons of propagation (132) in the view of (108)-(109).

A spatial period of the diagonal elementary oscillons along the x-axis

where

are x-periods of and, respectively, and are integers.

Because integrals of for over vanish and integrals of for over also vanish, averages of the diagonal elementary oscillons over vanish since

in the view of (179), (182) and the additive interval property. The diagonal elementary oscillons are neutral since on average they do not transfer any kinetic energy along the x-axis.

Similarly, representation of definitions (60) in y-eigenfunctions with the help of (136) and transformation into y-eigenfunctions of external interaction yield

For any frozen, the diagonal elementary oscillons are transformed into 2-f neutral oscillons in y, which are formed by superpositions of two pairs of cosine waves and with wavenumbers, respectively, amplitudes proportional to

and y-shifts depending on

If, the diagonal elementary oscillons (185) are reduced to the doubled elementary pulsons of propagation (138).

A y-period of the diagonal elementary oscillons

where

are y-periods of and, respectively, and are integers.

In the view of vanishing integrals of for over and vanishing integrals of for over,

The diagonal elementary oscillons are also neutral along the y-axis since on average they do not transfer any kinetic energy.

In agreement with (60), (47), (84), we use the t-matrix of external interaction (120) to compute the inhomogeneous Fourier expansion of the diagonal elementary oscillons via the t-eigenfunctions of external interaction (117) as follows:

For any frozen, the diagonal elementary oscillons are transformed into at most octet-frequency (8-f) neutral oscillons in t, which are produced by superpositions of cosine waves with frequencies for, amplitudes proportional to

and t-shifts depending on

If, the diagonal elementary oscillons (191) are reduced to the doubled elementary pulsons of propagation (143) due to (127).

Since a t-period of for is

a t-period of the diagonal elementary oscillons

where are integers.

In the view of vanishing integrals of over,

Integral (196) demonstrates that the diagonal elementary oscillons are neutral in time, as well, as they do not transfer on average any kinetic energy.

The diagonal elementary oscillon with is shown Figure 4 for independent parameters (95).

Along the x-axis, the diagonal elementary oscillons are visualized by 2-f neutral oscillons in x with periods local x-maximums, and x-local minimums, the numbers of which are specified by.

Along the y-axis, the diagonal elementary oscillons are displayed via 2-f neutral oscillons in y with periods local y-maximums, and local y-minimums, the numbers of which are indicated by.

The diagonal elementary oscillons are [7-f, 7-f, 6-f] neutral oscillons in t with periods, local t-maximums, and local t-minimums, the numbers of which do not exceed or equal to due to merged t-maximums and t-minimums. The diagonal elementary oscillons are [7-f, 7-f, 6-f] neutral oscillons in t with periods, local t-maximums, and local t-minimums, the numbers of which do not exceed due to merged t-maximums and t-minimums.

With the help of (130), we transform definitions (63) of the external elementary oscillons via x-eigenfunctions (98) and express them using the x-matrix of external interaction (104) as follows:

For any frozen, the first pair of external elementary oscillons is reduced to 1-f neutral oscillons in x, which are formed by a pair of sine waves with wavenumber. The second pair is transformed into 2-f neutral oscillons in x, which are generated by superpositions of two pairs of cosine waves and with wavenumbers, respectively. The third pair is converted into 2-f neutral oscillons in x, which are produced by superpositions of two pairs of sine waves and with wavenumbers, correspondingly.

For all three pairs, amplitudes are proportional to parameters (180) and x-shifts depend on parameters (181). The x-period of the first elementary pair is and the x-period of the second and third elementary pairs is that are defined by (182)-(183). If, the external elementary oscillons (197) are reduced to the doubled internal elementary oscillons (152) due to (108)-(109).

Averages over the x-period of six external elementary oscillons vanish since

because of the same reasons as for the diagonal elementary oscillons (184). The external elementary oscillons are also neutral because on average they do not transfer any kinetic energy along the x-axis.

Analogously, definitions (63) written in y-eigenfunctions with the help of (136) and the y-matrix of external interaction give

For any frozen, the first pair of external elementary oscillons is transformed into 2-f neutral oscillons in y, which are generated by superpositions of two pairs of cosine waves and with wavenumbers, correspondingly. The second pair is reduced to 1-f neutral oscillons in y, which are formed by a pair of sine waves with wavenumber. The third pair is converted into 2-f neutral oscillons in y, which are produced by superpositions of two pairs of sine waves and with wavenumbers, respectively.

For all pairs, amplitudes are proportional to parameters (186) and y-shifts depend on parameters (187). The y-period of the first and third elementary pairs is and the y-period of the second elementary pair is that are specified by (188)-(189). If, the external elementary oscillons (199) are transformed into the doubled internal elementary oscillons (165).

The external elementary oscillons are also neutral along the y-axis since on average they do not transfer any kinetic energy as

With the help of (63), (47), and (84), we employ the t-matrix of external interaction (120) to compute the inhomogeneous Fourier expansion of the external elementary oscillons via the t-eigenfunctions of external interaction (117) in the following t-form:

For any frozen, the first pair of external elementary oscillons is reduced to at most 4-f neutral oscillons in t, which are formed by superpositions of four sine waves with frequencies, respectively. The second pair is transformed into at most 4-f neutral oscillons in t, which are generated by superpositions of four sine waves with frequencies, correspondingly. The third pair is converted into at most 4-f neutral oscillons in t, which are produced by superpositions of four cosine waves with frequencies, respectively.

For all pairs, amplitudes are proportional to parameters (192) and t-shifts depend on parameters (193). If, the external elementary oscillons (201) are converted to the doubled internal elementary oscillons (170) because of (127).

Temporal periods of the first pair

the second pair

and the third pair

where, are integers, for are specified by (194).

Because integrals of sine waves over the relevant t-periods vanish for, averages of the external elementary oscillons also vanish for the reason that

due to (201)-(204) and the additive interval property. In agreement with (205), the external elementary oscillons are also neutral because on average they do not transfer in time any kinetic energy.

The diagonal elementary oscillon with is shown in Figure 5 for independent parameters (95).

Along the x-axis, the first pair is visualized by 1-f neutral oscillons in x with periods. The second and third pairs are displayed by 2-f neutral oscillons in x with periods, local x-maximums, and local x-minimums, the numbers of which are specified by.

Along the y-axis, the first and third pairs are depicted by 2-f neutral oscillons in y with periods, local y-maximums, and local y-minimums, the numbers of which are indicated by. The second pair is shown by 1-f neutral oscillons in y with periods.

The first pair is presented by 4-f neutral oscillons in t with periods, local t-maximums, local t-minimums, the numbers of which do not exceed or equal to due to merged t-maximums and t-minimums, respectively. The second pair is represented via [3-f, 4-f, 4-f] neutral oscillons in t with periods, local t-maximums, and local t-minimums, the numbers of which do not exceed or equal to because of merged t-maximums and t-minimums. The third pair is visualized by [4-f, 4-f, 3-f] neutral oscillons in t with periods, local t-maximums, and local t-minimums, the numbers of which are provided by.

Separation of variable x in (52) gives

where

is an amplitude, which is expressed via coefficients

In Equation (206),

is an x-shift in the motionless frame and

is an X_m-shift in the th moving frame.

For any frozen, the wave oscillon of propagation is transformed into a 1-f neutral oscillon in x with the wavenumber, an amplitude proportional to and the x-shift. The x-period of the wave oscillon of propagation coincides with (74).

The wave oscillon of propagation of the velocity potential is neutral in the x-direction since

Proceeding with the same approach, we separate variable y in (52) to find

where an amplitude

is computed via coefficients

In Equation (212),

is a y-shift in the motionless frame and

is an Y_m-shift in the th moving frame.

For any frozen, the wave oscillon of propagation is converted into a 1-f neutral oscillon in y with the wavenumber, an amplitude proportional to and the y-shift. The y-period of the wave oscillon of propagation is specified by (82).

The wave oscillon of propagation of the velocity potential is also neutral along the y-axis because

Topology of the wave oscillon of propagation in time is described by the inhomogeneous Fourier t-expansion of (52)

expressed via two trigonometric functions

of two arguments

In Equations (218), (220),

are amplitudes and

are t-shifts.

For any frozen, the wave oscillon of propagation is reduced to a 2-f neutral oscillon in t, which is formed by superpositions of sine waves with frequencies, correspondingly, amplitudes proportional to, and t-shifts depending on,. The t-period of the wave oscillon of propagation is given by (92)-(93).

Because integrals of sine waves over vanish, respectively, average of the wave oscillon of propagation over t-period vanishes, as well, since

The wave oscillon of propagation with is visualized in Figure 6 for independent parameters (95). The wave oscillons of propagation are displayed by 1-f neutral oscillons in x with periods, via 1-f neutral oscillons in y with periods, and through 2-f neutral oscillons in t with periods, 5 local t-maximums, and 5 local t-minimums, the numbers of which are specified by.

Combining (55) and (132) yields that the wave pulsons of propagation (27) in terms of the x-eigenfunctions of internal interaction (109) become

For any frozen, the wave pulsons of propagation are reduced to 1-f supercritical pulsons in x, which are formed by a superposition of cosine waves with wavenumber, amplitudes proportional to and (133), x-shifts (72), and energy shifts depending on, and (133). Therefore, the x-period of the wave pulsons of propagation is specified by (134).

Since integrals of and over vanish,

Two pairs of the wave pulsons of propagation and, on average transfer equal positive amounts of the kinetic energy along the x-axis.

Similarly, we combine (55) and (138) to compute the inhomogeneous Fourier expansion for the wave pulsons of propagation (27) via the y-eigenfunctions of internal interaction in the following form:

For any frozen, the wave pulsons of propagation are converted into 1-f supercritical pulsons in y, which are produced by a superposition of cosine waves with wavenumber, amplitudes proportional to and (139), y-shifts (81), and energy shifts depending on, and (139). Therefore, the y-period of the wave pulsons of propagation is specified by (140).

Because integrals of and over vanish,

So, two pairs of the wave pulsons of propagation and on average transfer equal positive amounts of the kinetic energy along the y-axis.

Substitution of (143) in (55) and collection of like terms results in the inhomogeneous Fourier expansion of the wave pulsons of propagation (27) through the t-eigenfunctions of internal interaction (126)

For any frozen, the wave pulsons of propagation are transformed into 2-f supercritical pulsons in t, which are generated by superpositions of cosine waves with frequencies, respectively, amplitudes proportional to, and (145), t-shifts depending on (91), and energy shifts proportional to and (146).

A temporal period of the wave pulsons of propagation

where are integers and given by (147).

Average of the wave pulsons of propagation over takes the following form:

Integral (230) validates that the wave pulsons of propagation on average transfer the same positive amount of the kinetic energy in time.

The wave pulson of propagation with is shown in Figure 7 for independent parameters (95). The wave pulsons of propagation are visualized by 1-f supercritical pulsons in x with periods, through 1-f supercritical pulsons in y with periods, and via 2-f supercritical pulsons in t with periods, local t-maximums, and local t-minimums, the numbers of which are specified by.

In agreement with (224),

Therefore, x-maximums of correspond to x-minimums of and x-minimums of to x-maximums of. Similarly, x-maximums of correspond to x-minimums of and x-minimums of to x-maximums of. Locations of x-maximums and x-minimums of coincide with locations of x-maximums and x-minimums of. Analogously, locations of x-maximums and x-minimums of match locations of x-maximums and x-minimums of.

Due to (226),

Consequently, y-maximums of correspond to y-minimums of and y-minimums of to y-maximums of. Analogously, y-maximums of correspond to y-minimums of and y-minimums of to y-maximums of. Locations of y-maximums and y-minimums of match locations of y-maximums and y-minimums of. Similarly, locations of y-maximums and y-minimums of coincide with locations of y-maximums and y-minimums of.

Because of (228),

So, locations of t-maximums of and correspond to locations of t-minimums of and, respectively. Analogously, locations of t-minimums of and to locations of t-maximums of and, correspondingly.

Substituting (152) in (58) results in the inhomogeneous Fourier expansion of the internal wave oscillons (38) along the x-axis

For any frozen, the first pair of internal wave oscillons is transformed into 1-f neutral oscillons in x, which is produced by a superposition of sine waves with wavenumber, amplitudes proportional to and (133), and x-shifts (72). The second pair and the third pair are reduced to 0-f (uniform) noncritical pulsons in x with energy shifts proportional to, and (153). The x-wavelength of the internal wave oscillons is given by (134).

Averages over the x-wavelength of six internal wave oscillons

prove that the first pair of internal wave oscillons on average does not transfer any kinetic energy along the x-axis, the internal wave oscillons of the second pair on average transfer along the x-axis equal amounts of the kinetic energy, and the internal wave oscillons of the third pair on average transfer along the x-axis equal amounts of the kinetic energy of opposite signs.

Analogously, we substitute (165) in (58) to find the inhomogeneous Fourier expansion of the internal wave oscillons along the y-axis

For any frozen, the first pair and the third pair of internal wave oscillons are converted into 0-f noncritical pulsons in y with energy shifts proportional to, and (166). The second pair is reduced to 1-f neutral oscillons in y, which is formed by a superposition of sine waves with wavenumber, amplitudes proportional to and (139), and y-shifts (81). The y-wavelength of the internal wave oscillons is specified by (140).

The internal wave oscillons of the first pair on average transfer along the y-axis equal amounts of the kinetic energy, the second pair of internal wave oscillons on average does not transfer any kinetic energy along the y-axis, and the internal wave oscillons of the third pair on average transfer along the y-axis equal amounts of the kinetic energy of opposite signs because

Substitution of (170) in (58) yields the inhomogeneous Fourier expansion of the internal wave oscillons in time

For any frozen, the first pair and the second pair of internal wave oscillons are transformed into 1-f neutral oscillons in t, which are formed by sine waves with frequencies, respectively, amplitudes proportional to, and (145), and t-shifts depending on (91). The temporal period of the first and second pairs are and, which are specified by (147). The third pair is transformed into 0-f noncritical pulsons with energy shifts depending on.

For the same reasoning as for the internal elementary oscillons,

Similar to the internal elementary oscillons (175), the first and second pairs of internal wave oscillons on average do not transfer in time any kinetic energy and the internal wave oscillons of the third pair transfer in time equal amounts of the kinetic energy of the opposite signs.

The internal wave oscillon with is visualized in Figure 8 for independent parameters (95). The first pair is displayed by 1-f neutral oscillons in x with periods, the second and third pairs via 0-f noncritical pulsons in x with periods.

In the y-direction, the first and third pairs are visualized by 0-f noncritical pulsons in y with periods and the second pair via 1-f neutral oscillons in y with periods.

The first pair is represented by 1-f neutral oscillons in t with periods, the second pair via 1-f neutral oscillons in t with periods, and the third pair in terms of 0-f noncritical pulsons in t with periods.

Substituting (179) in (61), we convert definitions (32) into the inhomogeneous Fourier expansion of the diagonal wave oscillons along the x-axis

where

For any frozen, the diagonal wave oscillons are presented by 2-f neutral oscillons in x, which are formed by superpositions of two pairs of cosine waves and with two wavenumbers, respectively, amplitudes depending on, and (180), and x-shifts, which depend on (181). The x-wavelength of the diagonal wave oscillons is given by (182)-(183). If, the diagonal wave oscillons (240) are transformed by (108)-(109) into the doubled wave pulsons of propagation (224).