Collision-induced amplitude dynamics of fast 2D solitons in the presence of the generic nonlinear loss

We study the amplitude dynamics of two-dimensional (2D) solitons in a fast collision described by the coupled nonlinear Schrödinger equations with a saturable nonlinearity and weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two onedimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude dynamics in a fast collision of two 2D solitons. Our perturbative approach is based on two major steps. The first step is the standard adiabatic perturbation for the calculations on the energy balance of perturbed solitons and the second step, which is the crucial one, is for the analysis of the collision-induced change in the envelope of the perturbed 2D soliton. Furthermore, we also present the dependence of the collision-induced amplitude shift on the angle of the two 2D colliding-solitons. In addition, we show that the current perturbative technique can be simply applied to study the collision-induced amplitude shift in a fast collision of two perturbed 1D solitons. Our analytic calculations are confirmed by numerical simulations with the corresponding coupled nonlinear Schrödinger equations in the presence of the cubic loss and in the presence of the quintic loss.


I. INTRODUCTION
Solitons are stable shape preserving solitary waves propagating in nonlinear dispersive media. Solitons have attracted considerable attentions in recent years due to the broadened applications of solitons in modern science [1][2][3][4][5][6]. In fact, solitons appear in a variety of fields, including optics, nanophotonics, condensed matter physics [1,4], and plasma physics [7]. In optics, the one-dimensional soliton propagation is stable and can be described by the nonlinear Schrödinger (NLS) model [4]. Due to the stability of the temporal NLS solitons, they can be used as bits of information in the optical fiber transmission technology [4].
However, 2D solitons are generally unstable in nonlinear optical media [5]. In particular, 2D optical solitons do not propagate in uniform cubic (Kerr) nonlinear media because of the catastrophic beam collapse at high powers [8,9]. There have been several investigations to achieve the stabilization of 2D solitons [10][11][12][13]. It was shown that 2D solitons can be stabilized in a layered structure with sign-alternating Kerr nonlinearity [10]. They also can exist and be stabilized in Kerr nonlinear optical media with an external potential [11][12][13].
Recently, the existence and stability of 2D optical solitons in saturable nonlinear media are subjects of continuously renewed interest to achieve the stable transmission of light beams at high velocity. The theoretical analysis of the condition for the existence of 2D and 3D solitons in saturable media were developed in Ref. [14]. Saturable nonlinearities have been observed in many nonlinear materials including photorefractive materials such as LiNbO 3 [3,15]. The 2D solitons can exist in photorefractive crystals due to the relatively slow nonlinear response of these materials. When the light goes through these media, the refractive index changes and the material might force the light to remain confined in its self-generated waveguide. As a result, the light can propagate without changing the shape.
Additionally, it was shown that 2D solitons can be stabilized in the nonlinear media where the cubic domains are embedded into materials with saturable nonlinearities [16]. In such optical media, the soliton propagation can also be described by (2+1)-dimensional ((2+1)D) NLS equation with a saturable nonlinearity [3,15,17].
One of the most fundamental properties of ideal solitons is their shape-preserving property in a soliton collision, that is, a soliton collision is elastic [18]. In optics, the collisions of sequences of solitons are very frequently [1,4]. Therefore, the collisions of two and many 1D solitons have been intensively investigated in several studies, for example, see Refs. [19][20][21][22][23][24][25] and references therein. More specifically, in Refs. [23,24], the authors studied the 1D soliton collision-induced amplitude dynamics in the presence of the cubic loss and the generic nonlinear loss. In optics, the nonlinear loss arises due to multiphoton absorption (MPA) or gain/loss saturation in a silicon media [24,26]. MPA has been received considerable attention in recent years due to the importance of MPA in silicon nanowaveguides, which are expected to play a crucial role in optical processing applications in optoelectronic devices, including pulse switching and compression, wavelength conversion, regeneration, etc. [23][24][25][26][27][28][29]. It has been uncovered that the presence of weak nonlinear loss leads to an additional downshift of the soliton amplitude in a fast collision of two 1D solitons [23,24]. The analytic expressions for the amplitude shift in two-soliton collisions, which is described by the (1+1)D NLS model, in the presence of weak cubic loss, which can be in a result of two-photon absorption (TPA) or gain and loss saturation, were already found in Refs. [23,30] and in the presence of the weak (2m + 1)−order loss, for any m, were found in Ref. [24]. In the previous studies for 1D soliton collision-induced change in the four parameters of solitons [20,21,23,24], the perturbative techniques were based on the projections of the total collision-induced change in the soliton envelope on the four localized eigenmodes of the linear operatorL describing small perturbations about the fundamental NLS soliton, which was derived by Kaup in 1990s [31,32]. However, in this original perturbation theory, the soliton solution of the unperturbed model is used for the calculations on the dynamics of perturbed 1D solitons.
Consequently, it is very hard to apply a similar technique for studying the effects of small perturbations on the interactions of solitons in higher dimensions, in which the unperturbed equations are nonintegrable. One needs to develop a new approach for studying the soliton collision-induced dynamics in the presence of nonlinear dissipation in higher dimensions instead of using the Kaup's perturbation theory. It is worthy to note that the collisioninduced corrections to solitons amplitudes were investigated in Ref. [33] in the framework of unperturbed nonintegrable wave models. However, the study for the soliton amplitude dynamics in the nonintegrable wave models with nonlinear dissipation has not been explored.
So far, to the best of our knowledge, the studies for the collision-induced amplitude dynamics of 2D solitons in the presence of nonlinear dissipation in two or higher dimensional nonlinear optical media is a long standing open problem.
In this work, this important and challenging problem will be addressed. We study fast collisions between two 2D solitons in weakly perturbed nonlinear optical media. The dy-namics of the collision is described by the systems of coupled (2+1)D NLS equations with the saturable nonlinearity, which are nonintegrable models, in the presence of the generic weak (2m + 1)-order of the nonlinear loss. We derive the analytic expression for the amplitude dynamics of a 2D single-soliton and, particularly, the collision-induced amplitude dynamics in a collision of two fast 2D solitons in the presence of nonlinear loss. For the above purposes, we develop a perturbative method for perturbed 2D solitons. Our perturbative method significantly extends the perturbative technique in Refs. [20,21,23,24] for calculating the effects of weak perturbations on fast collisions between two 1D solitons of the NLS equation and the recent perturbative method presented in Ref. [34] for calculating the collision-induced amplitude dynamics of two 1D pulses for the perturbed linear waves.
The crucial points in the current perturbative approach are the uses of the solution of the perturbed NLS model instead of the unperturbed NLS model and the single soliton dynamics in calculating the total collision-induced change in the soliton envelope. These are the key improvements compared to the perturbation techniques for studying the perturbed 1D solitons presented in Refs. [20,21,23,24]. More specifically, our perturbative approach is based on a procedure of two steps. The first step is for calculations on the energy balance of perturbed solitons based on the perturbed solution and a standard adiabatic perturbation theory for solitons. The second step, which plays a crucial role for our perturbative approach, is for calculations on the collision-induced change in the soliton envelope and a technical approximation of integrals based on the assumption of a fast and complete collision. We verify the analytic expressions by the numerical simulations with the corresponding (2+1)D NLS models with the cubic loss (m = 1) and with the quintic loss (m = 2). Additionally, we also demonstrate that the current perturbative approach can be simply applied to calculate the collision-induced amplitude shift in a fast collision of two 1D solitons for a wider class of perturbed (1+1)D NLS equations in a straightforward manner. As a concrete example, we use the current perturbative technique to derive the expression for the collision-induced amplitude shift in a fast collision of two 1D solitons of (1+1)D cubic NLS model in the presence of the delayed Raman response.
The rest of the paper is organized as follows. In sections II A, II B, and II C, we first study the dynamics of a single-soliton propagation in saturable nonlinear optical media in the presence of the generic weak nonlinear loss. Then, we use the perturbative technique to calculate the collision-induced amplitude dynamics in a fast collision of two 2D solitons. The analytic predictions will be validated by simulations in section III. Section IV is reserved for conclusions. In Appendix A, we demonstrate the robustness and the simplicity of the current perturbative method for other perturbed soliton equations. We consider fast collisions between two 2D solitons propagating in saturable nonlinear optical media in the presence of the generic weak (2m+1)-order of the nonlinear loss, for any m ≥ 1. The dynamics of the collision is described by the system of coupled NLS equations as follows [3,17]: y is the transverse Laplace operator, ψ j is the envelope of soliton j, x and y are the spatial coordinates, z is the propagation distance, α is the strength of the nonlinearity, I 0 is the saturation parameter, and 2m+1 , which satisfies 0 < 2m+1 1, is the (2m+1)-order of the nonlinear loss coefficient [3,35]. On the left-hand side of equation (1), the second term corresponds to the secondorder dispersion and the third term represents the effects of the saturable nonlinearity. On the right-hand side of equation (1), the first and second terms describe the effects of intrabeam and inter-beam interaction due to the (2m+1)-order of the nonlinear loss, respectively.
We first discuss the form of the single ideal 2D soliton j which is the fundamental solution of the following unperturbed model [3,16]: The soliton solution of equation (2) with the velocity vector d j = (d j1 , d j2 ) can be found in the form:ψ where is the initial position of soliton j, α j is related to the phase, d j1 and d j2 correspond to the velocity components in the x and y directions, respectively, µ j is the propagation constant, and U j is the amplitude function.
From equations (2) and (3), it can be shown that the localized function U j satisfies the following elliptic equation [3,36]: B. The 2D soliton dynamics of the single-soliton propagation Next, we investigate the effects of the (2m + 1)-order of the nonlinear loss on the singlesoliton propagation described by the following perturbed equation: By using an energy balance calculation for equation (5), it implies: We assume that the initial envelopes of the 2D solitons can be expressed in the general form where A j (0) is the initial amplitude parameter,ψ j0 (x, y, 0) is the fundamental soliton solution of equation (2), that is,ψ j0 (x, y, 0) is given by equation (3), and j = 1, 2. We note that for an initial envelope of the unperturbed soliton solution, one can define A j (0) = 1, that is ψ j0 (x, y, 0) =ψ j0 (x, y, 0). In the presence of the nonlinear loss, we look for the solution of equation (5) in the form of where , is the amplitude parameter taking into account of the effects of nonlinear loss for z > 0, andψ j0 (x, y, z) is given by equation (3). We substitute the relation for ψ j0 (x, y, z) into the equation (6) and apply the standard adiabatic perturbation theory for the NLS soliton [37]. It then yields: where By the definition of U j , one can obtain that I 2,j (z) and I 2m+2,j (z) are constants. Solving equation (9) on the interval [0, z], it implies the equation for the amplitude dynamics of a single soliton as follows: where I 2,j0 = I 2,j (0) and I 2m+2,j0 = I 2m+2,j (0).
Equation (10) describes the effects of the nonlinear loss on the amplitude parameter of a single 2D soliton.

C. The collision-induced amplitude dynamics of two 2D solitons
We now study the collision-induced amplitude dynamics in a fast two-soliton collision described by equation (1). For this purpose, we assume two solitons are well-separated at the initial propagation distance z = z 0 and at the final propagation distance z = z f for a complete collision. By deriving the energy balance of equation (1), one then obtains: Based on the perturbative calculation approach in Refs. [21,23,30], it is useful to look for the solution of equation (1) in the form: where ψ j0 is the single-soliton propagation solution of equation (5) and φ j describes a small correction to ψ j0 , i.e., the correction is solely due to collision effects. We substitute the relation (12) into equation (11) and take into account only leading-order effects, that is, the effects of order of 2m+1 . Therefore, based on the standard adiabatic perturbation theory for the NLS soliton [37], the terms containing φ j on the right-hand side of the resulting equation can be neglected. It then leads to the following differential equation for soliton 1: (13) represents the energy balance for soliton 1. The last term on the right-hand side in equation (13) is responsible for the contribution of the interaction term during the collision. We note that when 2m+1 = 0 then equation (13) becomes a conservation law for energy and the calculations to obtain the equation for soliton 2 are the same. From equations (6) and (13) and noting that ψ j0 satisfies equation (6), one then obtains the energy balance equation for soliton 1 via the use of the perturbed single-soliton propagation solution as follows: In a fast collision, the collision takes place in a small interval where z c is the collision distance, which is the distance at which the maxima of |ψ j (x, y, z)| coincide at the same point (x 0 , y 0 ), and ∆z c is the distance along which the envelopes of the colliding solitons overlap. Integrating over z of equation (14), it implies: where  (15), respectively. That is, The expression for ∆ 1 can be expressed in the term of the change in the soliton envelope: . We introduce the following approximation: where ∆A (c) 1 is the total collision-induced amplitude shift of soliton 1, A j (z − c ) is the limit from the left of A j (z) at z c , and ∆A (s) 1 (z c ) is the amplitude shift of soliton 1 which is due to the single-soliton propagation from z − c to z + c . By the definition of ψ 1 , ψ 10 , and U j : Substituting the relations (17) and (18) into equation (16), it yields the following key approximation for the total collision-induced change in the soliton envelope: We note that 10 (x, y, z)dxdy is a conserved quantity of the propagation equation (5) when 2m+1 = 0. Therefore, the following relation holds for all z. Substituting the relation (20) into equation (19) and taking into account only leading order terms, it implies On the other hand, ∆ 10 can be expressed as By the definition of ψ 10 , one can use the approximations |ψ 10 . Substituting these relations into equation (22) and then expanding the first integrand on the right-hand side while keeping only leading terms, it implies We substitute equations (21) and (23) into equation (15). It arrives at the equation for the collision-induced amplitude dynamics of soliton 1: where Finally, we simplify equation (24) by integrating M k,m using the decompose approximation of the integrand based on the assumption of a fast soliton collision. We note that only functions on the right-hand side of equation (24) that contain fast variations in z, which are the factors X j and Y j , are U 1 and U 2 . The slow varying amplitudes A 1 (z) and A 2 (z) can be approximated by A 1 (z − c ) and A 2 (z − c ), respectively. Therefore, equation (24) can be re-written: where N k,m = zc+∆zc zc−∆zc dxdydz. Since the integrand on the right-hand side of equation (25) is sharply peaked at a small interval about z c , we can extend the limits of this integral to 0 and z f . Therefore, it yields ∆A (c) where dxdydz. Equation (26)  exchange of 1D soliton studied in Ref. [38], where the phase difference between the colliding solitons strongly affects the amplitude of colliding solitons in a slow collision.
We note that equation (19) plays an important role in our analysis. In previous studies for the collision-induced dynamics of two 1D solitons [20,21,23,24], the perturbative method derived by Kaup Therefore, there will be a collision at the origin at the propagation distance z c = −x 10 /d 11 .
Second, we define the relative error in the approximation of ∆A is defined by: In equation (28), A 1 (z − c ) is measured from equation (10) and A 1 (z + c ) is calculated by solving equation (9) with j = 1 on the interval [z c , z f ]: where A 1 (z f ) is measured by simulations of equation (1). From equation (26) with cubic loss (m = 1) and it is ∆A (c)(th) 1 with quintic loss (m = 2). Additionally, we define the relative error in measuring the soliton patterns at the propagation distance z by |ψ where ∆A To validate equation (26), we carry out the simulations with equation (1) using the split-step method with the second-order accuracy [11]. As an example, we present the simulation results of equation (1) for α = 1 and I 0 = 1 with m = 1 and m = 2. For simplicity, we use the dimensionless parameters. The initial conditions of equation (1) are defined from equation (3) at z = 0. The amplitude functions U j in equation (3) at z = z 0 are measured by simulations of equation (4) using the Accelerated Imaginary-Time Evolution Method [11,39]. To implement simulations with equation (4), we use the input Also we use the final propagation distance z f = 10. Figure 1 represents the initial soliton profile and the evolution of its profiles |ψ 1 (x, y, z)| obtained by the simulation of equation presented using the level colormap. In addition, the amplitude parameters A

(z) is measured by the simulation of equation (5) and
A (th) 1 (z) is calculated from the theoretical prediction with equation (10). As can be seen, the agreement between the analytic calculations and the simulation results for m = 1 is very good. In fact, the relative error in measuring A 1 (z) for z ∈ [0, z f ], which is defined by |A with z c = 10/d 11 . One can measure cos θ = −0.1111 and |d| = 1.9026d 11 . We observe that   Figure 5 shows the dependence of ∆A (c) 1 on θ with 2m+1 = 0.01 and 2m+1 = 0.02 for m = 1 (a) and m = 2 (b). One can observe that the magnitude of ∆A (c) 1 is smaller for a larger value of θ, i.e., for a faster collision. Figure 6 shows the dependence of the relative change p on θ with 2m+1 = 0.01 and 0.02 for m = 1 (a) and for m = 2 (b). As can be seen, the relative difference p is independent of the choices of 2m+1 and m. The values of p are decreasing from p max = 0.25 at θ = 0 to p min = 0 at θ = π. The maximal relative error in calculations of p over [0, π] is 0.016 for m = 1 and it is 0.026 for m = 2.
In summary, the very good agreement between the analytic calculations for ∆A Our current perturbative approach can be applied for studying the soliton collisioninduced amplitude dynamics for a larger class of soliton equations, even in a nonintegrable system, with other dissipative perturbations in a similar manner. Furthermore, we showed that the current perturbative approach can be applied to simply calculate the collision-

Conflicts of interest
The authors declare that they have no conflict of interest. In this Appendix, we illustrate that the current perturbation method is robust and simple to study the collision-induced amplitude dynamics in fast collisions of solitons of perturbed NLS equations. More specifically, one can apply the current perturbative approach to simply derive the expression for the collision-induced amplitude shift in a fast collision of two 1D NLS solitons with delayed Raman response in a straightforward manner. This expression has been derived in Ref. [40] by the Taylor expansion and in Ref. [21] by the traditional perturbation technique developed by Kaup [31,32] for 1D NLS solitons.
For the above purpose, we consider fast collisions between two solitons in nonlinear optical waveguides described by the system of coupled (1+1)D cubic NLS equations in the presence of the delayed Raman response as follows [21,41]: where R is the Raman coefficient, 0 < R 1, 1 ≤ j, l ≤ 2, and j = l. The first term on the right-hand side of equation (A1) describes the Raman-induced intra-pulse interaction while the second and third terms describe the Raman-induced inter-pulse interaction. We note that the unperturbed NLS equation i∂ z ψ j + ∂ 2 t ψ j + 2|ψ j | 2 ψ j = 0 has the fundamental soliton solution ψ cs,j (t, z) = Ψ cs,j exp(iχ j ), where x j = η j (t − y j − 2β j z), χ j = α j + β j (t − y j ) + (η 2 j − β 2 j )z, and parameters η j , β j , α j , and y j are related to the amplitude, frequency, phase, and position of the soliton j, respectively.
Similarly to Ref. [21], we assume that 1/|β| 1 with β = β 2 − β 1 and that two solitons are well separated at the initial propagation distance z = z 0 and at the final distance z = z f .
We now apply the current perturbation technique to calculate the collision-induced amplitude shift in a fast collision of two solitons described by equation (A1). We first perform the calculations for the energy balance of equation (A1). It implies: Equation (A4) can be written as: where C k = ψ k ∂ t (ψ * k ) with k = l, j. By the definition of ψ c,k , it implies C k = Ψ c,k ∂ t Ψ c,k − iβ k Ψ 2 c,k , where Ψ c,k = Ψ c,k0 + Φ c,k . Substituting the relation for C k into equation (A5) and using a standard adiabatic perturbation theory for the NLS soliton with concentrating only on the leading-order effects of the collision, it implies the energy balance equation for soliton 1: We note that ∞ −∞ Ψ 2 c,j0 (t, z)dt = 2η j0 (z), where η j0 (z) is the amplitude parameter of ψ c,j0 in the presence of the delayed Raman respone. By using the standard adiabatic perturbation theory, one can obtain η j0 (z) = η j0 (0). On the other hand, Ψ c,1 (t, z + c ) can be expressed in the manner where η 1 (z + c ) η 1 (z − c ) + ∆η 1 is the total collision-induced amplitude shift of soliton 1. Next, we calculate the integral M by using the algebra approximations which was used to calculate the integral M k,m in equation (24). That is, one can take into account only the fast dependence of Ψ c,j0 on z, which is the factor v j = t−y j −2β j z.
This approximation of Ψ c,j0 (t, z) can be denoted byΨ c,j0 (v j , z c ). Moreover, since the integrand of M is sharply peaked at a small interval [z c − ∆z c , z c + ∆z c ] about z c , the limits of the integral M can be extended to −∞ and ∞. By changing the integration variable with v j = t − y j − 2β j z, it implies:  (20) in Ref. [21] which was originally based on the perturbation technique developed by Kaup for 1D NLS solitons [31,32]. In addition, we emphasize that one can apply the current perturbative approach to simply derive the expression for the collision-induced amplitude shift in a fast collision of two 1D NLS solitons with nonlinear loss in a straightforward manner.