# The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Exact Two-Soliton Solution of the Kadomtsev–Petviashvili Equation

_{1,2}are thicknesses of the upper and lower layers. In the general case of internal waves in a continuously stratified fluid with a shear flow, the coefficients were derived in [13] and then, in many other papers.

^{2}/α, whereas its speed V depends both on k and l:

^{−1}(l/k) is the angle between the direction of soliton propagation normal to its front and the x-axis. Note that in the Korteweg–de Vries (KdV) equation, which is the particular case of the KP2 equation when φ = 0, the relationship between the soliton speed and amplitude is V = αA/3, so that the angle correction to the soliton speed is relatively small and proportional to φ

^{2}within the KP approximation. The components of soliton velocity V = (V

_{x}, V

_{y}) are:

_{1,2}and l

_{1,2},

_{1}= k

_{2}= k > 0, l

_{1}= −l

_{2}= −l > 0 and obtain:

^{2}/2α (the same restriction was obtained by Miles [16]). If this restriction is not met, then the stationary two-soliton pattern cannot exist; the solitons, apparently, experience a more complicated nonstationary interaction. This interesting issue has not been studied yet.

^{2}/α, whereas at the point of front intersection, the pattern amplitude increases and attains the value:

_{c}, and l remains finite, we obtain a triad solution (see Figure 1b) with the amplitude of the third soliton ${A}_{3}=2\left|l\right|\sqrt{6\beta c}/\alpha $, whereas the amplitudes of both original solitons equal to $A=\left|l\right|\sqrt{6\beta c}/2\alpha $, so that we obtain the well-known result A

_{3}= 4A. This case can be physically acceptable, if |l| << k

_{c}. Figure 2 illustrates the examples of two symmetric soliton solutions for the cases when: (a) k = 0.05; l

_{2}= −l

_{1}= 0.005, l

_{2}/k = 0.1, B = 4; then the phase shift along the x-axis is Φ

_{x}= ln B/k = 27.73 and along the y-axis is Φ

_{y}= ln B/l

_{2}= 277.3; (b) k = 0.05; l

_{2}= −l

_{1}= 0.00433012702, l

_{2}/k = 0.0866, B = 2 × 10

^{9}, Φ

_{x}= ln B/k = 27.73, Φ

_{y}= ln B/l

_{2}= 428.4.

_{c}, the parameter B increases, the bridge between the soliton fronts also increases, and in the limit when k = k

_{c}, the bridge reduces to the third soliton; then the upper pair of soliton fronts disappear at the infinity. As follows from the formula for k

_{c}and A, in the triad solution of this type, we have:

## 3. The Approximate Asymptotic Approach to the Description of Two-Soliton Interaction

#### 3.1. Symmetric Configuration of Two Plane Solitons in the KP2 Equation

_{pat}as per Equation (17). Then, the KP2 Equation (1) can be presented in the form:

_{0}

^{2}= 2V

_{pat}/c plays a role of the squared speed of long linear waves. Thus, Equation (20) can be considered as the one-dimensional equation in space and “time y” with the following soliton solution:

^{2}= 12β/αA, and W

^{2}= (2/c)(V

_{pat}− αA/3); the solution exists if V

_{pat}> α A/3. Soliton fronts shown, for example, in Figure 2 can be treated as the space-time diagrams for the soliton solutions of the Boussinesq Equation (20).

_{1}+ η

_{2}where η

_{1,2}represent two separate soliton solutions described by Equation (21). Then the soliton of amplitude A

_{1}being at a big distance from another soliton of amplitude A

_{2}(see Figure 3) experiences its influence only through the exponentially small asymptotic (a “tail”) of the second soliton field. The tail part of the soliton field follows from the asymptotic solution of Equation (21), η = 4 A

_{2}exp(–2 d/∆), where d is the distance between the soliton maxima along the x-axis. The external field produced by the second soliton and represented by its “tail” can be considered as slowly varying in the space pedestal; at the place of the first soliton, the tail field is almost constant because it is assumed that the characteristic scale of pedestal variation in space is much greater than the characteristic width of the first soliton; this is shown schematically in Figure 3. A soliton solution of Equation (20) on the constant pedestal p is the same as in Equation (21), but with the modified speed, $W=\sqrt{\frac{2}{c}\left({V}_{pat}-\alpha p-\frac{\alpha A}{3}\right)}$; the solution exists if V

_{pat}> α A/3 + α p.

_{1,2}the positions of soliton centres, then the kinematic condition of soliton motion requires that Ṡ

_{1,2}= ±W

_{1,2}, where Ṡ stands for the derivative of S with respect to the “effective time” y, and the signs ± pertain to solitons moving to the right or left. Considering solitons moving towards each other, we can set Ṡ

_{1}= −W

_{1}, and Ṡ

_{2}= W

_{2}. In the expressions for W

_{1,2}, we can substitute instead of the constant pedestals d

_{1,2}the expressions of the tails of reciprocally external solitons or simply the external soliton fields, because we assume that the distance between the solitons is much greater than their characteristic width, |S

_{1}− S

_{2}| >> ∆

_{1,2}. Thus, we obtain the set of two equations for S

_{1,2}:

_{2}, and the lower sign—to the soliton at S

_{1}. Here the slowly varying pedestal p

_{1.2}(y) is determined by Equation (21).

_{1,2}(y). However, soliton amplitudes experience variations only within a relatively short time when the “effective particles” experience a head-on collision. Therefore, in the first approximation, we can assume that soliton amplitudes remain unchanged and equal to their values at the infinity, i.e., far from the collision zone where the soliton fronts intersect (see Figure 2a). Substituting in Equation (22) A

_{1,2}= 3β k

^{2}/α and V

_{pat}from Equation (17), we obtain:

_{1}= A

_{2}= A, and a real solution to Equation (23) exists provided that $\left|l\right|\ge {l}_{cr}\equiv \alpha A\sqrt{2/3\beta c}$ (cf. Equations (14) and (15)). Equation (23) can be readily solved. To this end, we note that S

_{1}+ S

_{2}= 0 in the symmetric case; subtracting then one equation from another and denoting X = S

_{2}− S

_{1}, we obtain:

^{1/2}. After the separation of variables in Equation (24), we find:

_{2}– S

_{1}and S

_{1}+ S

_{2}= 0, we find solutions for S

_{1}and S

_{2}in the form:

_{1}= A

_{2}= 7.5 × 10

^{−3}, k = 0.05; l

_{2}= −l

_{1}= 0.005, l

_{2}/k = 0.1, B = 4. The dependences S

_{1,2}(y), which represent the soliton fronts in the KP2 equation, are shown in the figure in terms of inverse functions y

_{1,2}(S

_{1,2}). The directions of the effective (imaginary) particle motion within the approximate approach are shown by small arrows.

_{2}− S

_{1}| >> ∆

_{1,2}. The lower dashed line in Figure 3 shows the initial trajectory of an imaginary particle 1 before the collision with the imaginary particle 2, and the upper dashed line shows the trajectory of particle 1 after the collision with particle 2. The phase shift between the soliton fronts due to interaction with another soliton can be determined as the difference in the “particle trajectory” experiencing a collision with another particle, S

_{2}(y), and the trajectory of freely moving particle. The former trajectory is given by the inversion of Equation (26), and the latter follows from Equation (23) with A

_{1}= 0: S

_{20}(y) = (3β l

^{2}/αA)

^{1/2}y. The “particle trajectory” experiencing collision with another particle can be easily calculated in the asymptotic when y → ∞; then from Equation (26) we obtain:

_{x}and Φ

_{y}. These expressions for the phase shifts are exactly the same as those which follow from the analytical solution (11), (13) and (16). The phase shift between the fronts of one of the interacting solitons is shown in Figure 4. The phase shift becomes infinite when the condition (19) is fulfilled, and the triad configuration is formed. Figure 5 shows the example of soliton interaction at the near-critical condition when the bridge between them becomes long. This case corresponds to the exact solution shown in Figure 2b with the same parameters: k = 0.05; l

_{2}= −l

_{1}= 0.00433012702, l

_{2}/k = 0.0866, B = 2 × 10

^{9}, Φ

_{x}= 428.3, and Φ

_{y}= 4947. One can see again that there is a very good agreement between the asymptotic theory and exact solution even in the case of the big phase shift.

#### 3.2. A symmetric Pattern in the 2dBO Equation Formed by Two Plane Solitons

**V**= (V

_{x}, V

_{y}), the formulae are the same as in Equation (10).

_{2}= −l

_{1}, then they form a symmetric pattern which moves stationary along the x-axis with the speed (cf. Equation (16)):

_{0}

^{2}= 2 V

_{pat}/c plays a role of a squared speed of long linear waves. Therefore, Equation (34) can be treated as the one-dimensional equation in space and “time” with the following soliton solution:

^{2}= (2/c)(V

_{pat}− αA/4); the solution exists if V

_{pat}> α A/4.

_{1,2}:

_{1,2}= 4β k/α and V

_{pat}from Equation (33) and assuming that each soliton plays a role of external slowly varying pedestal p

_{1.2}(y) to another soliton, we obtain:

_{1}= A

_{2}= A, a real solution to Equation (37) exists provided that $l\ge {l}_{cr}\equiv \left(\alpha A/2\beta \right)\sqrt{\alpha A/2c}$. Equation (37) can be readily solved using the fact that S

_{1}+ S

_{2}= 0 in the symmetrical case. Then, we have:

^{2}= (αA)

^{3}/8cβ

^{2}l

^{2}= 8β k

^{3}/cl

^{2}. Graphically, this solution is similar to what is shown in Figure 4 and Figure 5.

_{1,2}→ ±∞. Then, for the phase shift we obtain:

_{1,2}= ± kΩ, where Ω = tan φ = 2, 1, 0.5 in three different runs. In all these runs, the angle φ is not small as it is required in the 2dBO model. Nevertheless, it is of interest to compare the results of the asymptotic theory developed here with the numerical data presented in [24]. In the first run with Ω = 2, the authors observed a stationary moving pattern formed by two solitons; “it seems that there exist very small phase shifts” in the soliton fronts. With the chosen set of parameters, we can estimate the phase shift using Equation (40); this gives Φ

_{y}= 0.8485. Such a small phase shift is indeed difficult to detect in Figure 3(a) of paper [24]. In two other runs with Ω = 1 and 0.5, the initial conditions used by the authors formally lead to too big values of the parameter γ (see above), which becomes greater than 1. In such cases, according to the developed here asymptotic theory, stationary patterns do not exist. The authors of [24] indeed observed rather complicated nonstationary dynamics of initial perturbations, especially in the latter case of Ω = 0.5 (see Figure 3(b,c) in [24]).

_{y}= 0.978.

^{2}/2α. The same restriction was obtained for the soliton amplitudes by Matsuno [8] in the case of the generalized two-dimensional BO-type equation. The solitons, apparently, experience a more complicated nonstationary interaction if this restriction is not met. This interesting issue deserves further study.

## 4. Discussion and Conclusions

## Funding

## Conflicts of Interest

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**Figure 1.**(colour online). Photographs of the observed wave patterns on shallow water demonstrating an almost symmetric X-type soliton interaction (frame (

**a**)) and a Y-type triad pattern (frame (

**b**)) [3].

**Figure 2.**(Colour online). Contour-plots of soliton fronts as per solution (15): (

**a**) k = 0.05; l

_{2}= −l

_{1}= 0.005, l

_{2}/k = 0.1, B = 4; (

**b**) k = 0.05; l

_{2}= −l

_{1}= 0.00433012702, l

_{2}/k = 0.0866, B = 2 × 10

^{9}. Each plot was generated for α = β = 1 and c = 2 in the domain (−10

^{3}, 10

^{3}) × (–10

^{4}, 10

^{4}).

**Figure 3.**(Colour online). The sketch of two solitons of different amplitudes being at a distance d = |S

_{1}− S

_{2}| from each other and interacting through their tails.

**Figure 4.**(colour online). Solutions y

_{1,2}(S

_{1,2}) as per Equation (26) of the derived set of equations for A

_{1}= A

_{2}= 0.0285 (k

_{1}= k

_{2}= 0.097468), l

_{1}= −0.01771, l

_{2}= −l

_{1}. The yellow circle shows the domain where the asymptotic theory is formally inapplicable because the distance between the solitons is not big here.

**Figure 5.**(Colour online). Solutions y

_{1,2}(S

_{1,2}) as per Equation (26) of the derived set of equations for A

_{1}= A

_{2}= 0.0285 (k

_{1}= k

_{2}= 0.05), l

_{1}= −0.00433012702, l

_{2}= −l

_{1}. The yellow circle shows the domain where the asymptotic theory is formally inapplicable.

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Stepanyants, Y. The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns. *Symmetry* **2020**, *12*, 1586.
https://doi.org/10.3390/sym12101586

**AMA Style**

Stepanyants Y. The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns. *Symmetry*. 2020; 12(10):1586.
https://doi.org/10.3390/sym12101586

**Chicago/Turabian Style**

Stepanyants, Yury. 2020. "The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns" *Symmetry* 12, no. 10: 1586.
https://doi.org/10.3390/sym12101586