The study of soliton collisions is based on the fact that the collision of N solitons is given by the N soliton solution (NSS) [75]. Soliton solutions of a second dynamical system coupled to optical solitons - higher-order coupled nonlinear Schr¨odinger equations Sasa-Satsume.
Introduction to the Lax Formalism
The integrability aspect of any dynamical system using ISM depends purely on the discovery of these operators for that dynamical system.
Application of ISM: An example
The direct problem
Time evolution of the scattering data
The inverse problem
The transmission coefficient T(k) does not appear in the GLME, as it has been eliminated in its derivation. The eigenvalues of the bound state κn appear as simple poles of T(k) on the positive imaginary k-axis. 2.22) The eigenvalues of the bound state can therefore be derived from the denominator of the transmission coefficient.
Soliton Solutions
The solitary wave solutions or the one soliton solution for the KdV equation at different times are shown graphically in Figure 2.2. This is the initial profileφ(x,0) = −6 sech2xthat is used to obtain the two soliton solution to the KdV equation using N = 2.
AKNS Scheme
AKNS formalism
AKNS SCHEMES 21 Thus, a number of dynamical equations can be formulated with the same time-evolving Lax operator L. Such a set of evolution equations is said to belong to the same successor, in this case, the ANKS successor.
Inverse Scattering in the AKNS Scheme
The N simple poles arise from the fact that α22(λ)−1 is analytic in the lower half of the λ plane, except at these N points. To calculate the time evolution of the scattering data, the second Lax equation (2.51) is evaluated at the asymptotes, where it has the form 2.65) Since soliton solutions are associated with reflectionless potentials, the second term in r.h.s (2.62) is absent.
Summary
This multicomponent version of the AKNS scheme is introduced and used in the next chapter to obtain N soliton solutions for several coupled integrable models related to optical solitons. In the above equation q=q(z, t) denotes the normalized complex envelope of the slowly varying optical pulse. The dynamic fields q(x, t) represent the envelope function of the current pulses of the electric field.
The small dimensionless parameter ε represents the ratio of the spectral width to the carrier frequency [104].
Lax Operators
Since the CHNLS equation has higher order effects added to the CNLS equation, these effects are ignored for the broader picosecond pulses of the CNLS. On the other hand, a direct set of Lax pairs for the CSS equation (3.5c) has not yet been discovered. Thus, using we obtain the explicit forms of ¯Σ and ¯A from (3.16) give n equations and their complex conjugates of n-CCmKdV and each equation appears twice in the marix implementation of the zero curvature condition.
Thus, equations (3.14) and (3.5c) remain unchanged and the dynamical fields ui CCmKdV have n components as before.
Direct Problem
Scattering data for CNLS and Coupled Hirota Equations 33
Using (3.22), the elements of the scattering matrix can be expressed by Jost functions. The orthonormality of the Jost functions (3.22), the scattering equation (3.23) and the scattering data (3.24) remain the same for CCmKdV, except for the fact that in these equations the indices i, j = 1,2,.
Gel’fand Levitan Marchenko Equation
GEL'FAND LEVITAN MARCHENKO EQUATION 35 So (3.23) yields. λ)φ(j)(x;λ) (3.29) Equation (3.28) implies the fact that the scattering matrix is unitary and gives the following relation for each of its elements. 3.31b). Unless explicitly stated, the same convention will be followed in the remainder of this section.
Time Evolution of the Scattering Data
CNLS Case
Coupled Hirota Case
CCmKdV Case
N Soliton Solutions
CNLS Case
Coupled Hirota Case
CCmKdV Case
Summary
The N soliton solutions of the three dynamical systems derived in the previous chapter are compact and contain all information about the propagating solitons of these dynamical systems. The two-soliton (2SS) solution is the next level of sophistication and represents the simplest form of composite solitons. All these studies are an intermediate step to the asymptotic study of the compound solitons.
The 3SS study continues with the restructuring of the NSC (3.54) necessary for its asymptotic study.
One Soliton Solution
CNLS Case
Equation (4.5) describes the complex wave-envelope function for a soliton of the CNLS system described by (3.5a). SOLUTION OF A SOLITON 45 Considering xandt to represent position and time coordinates, from the above expression, it is seen that 1SS has these characteristics. From (4.11a) it is evident that Cn+1,i(1) = 0 reduces that particular component of solitons which does not exist.
Form (4.11d), the condition for 1SS to have zero initial phase is. 4.12) It can also be from equation (4.11) that the different components of 1SS can differ from each other only with respect to their amplitude, all the other functions - group speed, pulse width and initial phase remain the same for all the components of the soliton.
Coupled Hirota Case
Two Soliton Solution
The parameters κi,j and Λi,j appearing in the above expression are defined in equations (3.57) and (3.58) respectively. 2SS denotes two peaked pulses, where the two peaks move independently in a character similar to figure (2.3), but with n components. Thus, 2SS denotes the next level of complexity after 1SS, and further investigation of all these characteristics of 2SS will be carried out in the next chapter.
Three Soliton Solution
N Soliton Solution
Summary
By asymptotically studying the nature of the soliton solutions, one can observe the behavior of the soliton interactions. In this chapter, the study of soliton collisions is carried out using such an asymptotic analysis of the soliton solutions. In the work of Radhakrishnan et.al., it was found that for coupled systems each component of the soliton can interact in an inelastic manner.
Reduction of the terms forn = 2 describes the Manakov system and its extension to include higher order effects, respectively.
Exclusive Phase Change in 1SS
This chapter begins with the development of a scaling that results in a change in the initial phase of the 1SS without affecting any other characteristic of the soliton. This is due to the fact that a collision sequence of N soliton collisions cannot be studied from two soliton collisions. The studies presented in this chapter were developed for the CNLS system and the Hirota-type CHNLS system for n-component fields.
It should also be noted that there is an exchange of time and position coordinates between the Manakov system and the dynamical systems presented here.
Asymptotic Nature of the Two Soliton Solution
0 and (x, t) represent a position away from the central maxima of the soliton, or where the soliton is absent. Again, simultaneously the two one-solitons of the 2SS, represented by (5.5) for the CNLS system. Furthermore, the two solitons will meet, or the intersection of their paths based on their initial conditions at the same time.
Similarly, the separation of the two peaks of the en-solitons in 2SS for the Hirota-type CHILLS system is given by.
Collision of Two Solitons
4λ1I are described by the transformation of the parameter Cn+1,i(1). 5.18) The changes that the S(1) soliton undergoes contain the parameters of the S(2) soliton and can be considered to have affected S(2) non-trivially. However, the collision does not affect the group velocity and width of the soliton, as can be seen from (5.5b). The condition for inelastic collision at the component level for the collision of two solitons is,. 5.31), where γ > 0 is a real quantity, then (5.1) ensures that the collision will be elastic for each of the components of the two solitons.
Thus, the exchange of energy between the components during the elastic band is only between the components of the same soliton and not from one soliton to another colliding soliton.
Collision of Three Solitons
This asymptotic reduction of the 3SS at t → +∞ can easily be extended to the other solitons. As in the case of two soliton interactions, the soliton energy is exchanged between the components of the same soliton, the sum total of the energies of each soliton remains unaffected. In fact, the amplitude and relative phase difference as well as the velocity and width maintain their invariance under any simple or combinations of γ-scaling of the three solitons.
Hence the amplitude change that a soliton undergoes in this three-soliton interaction is independent of the sequence of soliton collisions.
Collision of N Solitons
Cn+1,i(p)‡?Cn+1,i(p)‡ (5.46) So the entire transformation from the p-th soliton before a collision took place to the situation after all collisions took place can be represented by a transformation of the parameter Cn+1,i(p). During the collision of the soliton S(p) with the other N−1 solitons, the group velocity and phase remain unchanged. As in the case of three-soliton interactions, the order of interactions between the N one-solitons has no asymptotic influence on the soliton characteristics.
Summary
On the other hand, the transfer of energy between the components of a multicomponent soliton can be used in the implementation of binary logic operations. This is due to the fact that the output of colliding multicomponent soliton depends on the other soliton(s) participating in the collision. This transactive nature of solitons was presented in the previous chapter, especially in the context of two and three soliton collisions.
Contrary to some of the techniques used in the aforementioned works, here different data solitons move at different speeds.
Model for Soliton Based Logic Gates
In addition, there may be other solitons, which can be called as actuators, moving in the same x-direction and colliding with the data solitons in a non-trivial manner causing inelastic collisions between the components. On the other hand, the functions evaluated at the "after collision" are actually evaluated at the t → +∞ asymptotes. Thus, in terms of the 2-CNLS and 2-CHNLS, the binary levels are represented by the 2-component solitons polarized only in one specific direction, for example, the soliton may have its field in the y direction and its field in the z direction is completely absent.
This means that the soliton with its field only in the y-direction will initially now have its field in the z-direction and its field in the z-direction is now zero.
Logic Gates from Two Soliton Collisions
Fractional transformation as shown in Figure:6.2, then the corresponding transformation derived from (4.20) is the same. 2)IN ), so the COPY gate is impossible to realize with two soliton collisions using the linear fractional transform (LFT) given in (6.2).
Logic Gates from Three Soliton Collisions
The corresponding output represented by (6.8) is no longer a linear fractional transform in terms of the input soliton, i.e. %(2)IN (or %(3)IN), but the truth table for the copy port may still be. In this form of COPY gate, the output data bit undergoes a rate change relative to the input data bit. In the case of a NOT gate, loading the conditions from the truth table of the NOT gate, i.e.
If the NAND gate is to be represented by the collision of three solitons, then a possible implementation of the truth table requires that %(2)IN and %(3)IN be the .
NAND Gate with four soliton collisions
In the context of inputs, which are now two in number, (6.17) can no longer be considered to be a linear fractional transformation of both inputs at the same time. The imposition of the NAND gate truth table yields the set of constraints given by (6.15). On the other hand, if the scheme is different, and therefore is represented by a figure different from (Fig:6.5), and the set of constraints can be (6.16) or any other depending on the situation.
Logic Gates with Higher Soliton Collisions
The thesis is mainly concerned with the study of the nature of Hirota-type n-CNLS and n-CHNLS soliton collisions. N soliton solutions of another dynamical system belonging to the same hierarchy, the Sasa Satsuma-type n-CHNLS, were also obtained. Furthermore, the outcome of these inelastic soliton collisions at the component level is also found to be independent of the sequence of interactions.
By dividing both sides of the above equation by if(x, t), we get back the KdV equation.
Elastic collisions of three solitons (N = 3) for a 2-coupled
Elastic collisions of three solitons for the same system as in
Contour plot for Fig:5.15
Contour plot for Fig:5.16
Inelastic collisions of three solitons (N = 3) for a 2-coupled
Inelastic collisions of three solitons for the same system as in
Contour plot for Fig:5.19
Contour plot for Fig:5.20
Soliton Gates
