Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtz equations
aa r X i v : . [ n li n . PS ] O c t Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtzequations
K. Tamilselvan a , T. Kanna a, ∗ , Avinash Khare b a Post Graduate and Research Department of Physics, Bishop Heber College, Tiruchirapalli 620 017, Tamil Nadu, India b Raja Ramanna Fellow, Department of Physics, Savitribai Phule Pune University, Pune – 411007, India
Abstract
We obtain a class of elliptic wave solutions of coupled nonlinear Helmholtz (CNLH) equations describing nonparaxialultra-broad beam propagation in nonlinear Kerr-like media, in terms of the Jacobi elliptic functions and also discusstheir limiting forms (hyperbolic solutions). Especially, we show the existence of non-trivial solitary wave profiles inthe CNLH system. The e ff ect of nonparaxiality on the speed, pulse width and amplitude of the nonlinear waves isanalysed in detail. Particularly a mechanism for tuning the speed by altering the nonparaxial parameter is proposed.We also identify a novel phase-unlocking behaviour due to the presence of nonparaxial parameter. Keywords: coupled nonlinear Helmholtz system, Jacobi elliptic function, Lam´e polynomials, solitary waves
1. Introduction
Study of nonlinear waves has time honoured history [1]. In recent years the interest on nonlinear waves is expo-nentially increasing among researchers of various disciplines due to their ubiquitous appearance in di ff erent physicalsystems. There exist several types of nonlinear waves such as elliptic waves, solitons, rogue waves, shock waves,compactons etc. The underlying physical systems in which these nonlinear waves appear are in general described bydi ff erent interesting nonlinear evolution equations. All these systems appear in diverse areas of physics like fluid dy-namics [1], nonlinear optics [2], Bose-Einstein condensates [3], bio-physics [4], plasma physics [5], etc. Especially, innonlinear optics the coupled nonlinear Schr¨odinger (CNLS) system is an important mathematical model with variouspotential applications [2]. From a physical perspective, this CNLS system arises in the context of partially coherentbeam propagation in photorefractive media and also appears as governing equation for pico second pulse propagationin multimode optical fiber [6]-[8]. The CNLS system becomes integrable for specific choice of parameters [9, 10].The integrable CNLS (so called Manakov model) system admits bright soliton solutions for focusing nonlinearity[10, 11] and the dark soliton as well as dark-bright soliton solutions for defocusing nonlinearity [12, 13]. Such typeof multicomponent solitons have also been realized in left handed metamaterials [14].In the framework of nonlinear optics the derivation of CNLS equations stems naturally from the Maxwell’s equa-tions by employing the paraxial wave approximation or slowly varying envelope approximation (SVEA) [2]. Underthis approximation the second order derivative of normalized wave with respect to propagation direction is ignored.As a result of this the CNLS model excludes the di ff raction length of the beam in the transverse direction of the phys-ical system [15]. Here a beam broader than its carrier wavelength, with moderate intensity and is propagating in (or ata negligible angle ) with respect to the reference direction is said to be a paraxial beam. If the beam fails to satisfy atleast any one of the aforementioned properties is said to be a nonparaxial beam. The ultra-narrow beam nonparaxialityviolates only the property of beam broader than its carrier wavelength. Such type of nonparaxiality has been studiedin detail in Ref. [16]. The notion of nonparaxiality was first introduced by Lax et al . [17], in the investigation ofultra-narrow beam, for which the transverse waist w and the carrier wavelength λ are comparable, propagating in ∗ Corresponding author
Email addresses: [email protected] (K. Tamilselvan), [email protected] (T. Kanna), [email protected] (Avinash Khare)
Preprint submitted to Elsevier November 5, 2018 n inhomogeneous isotropic nonlinear media. The corrections to paraxial approximation have been obtained in Ref.[17] by expanding the fields as a power series in the ratio of beam diameter to di ff raction length. Ultimately, it hasbeen shown that the resulting governing equation will be a nonlinear Schr¨odinger (NLS) equation with higher orderdi ff ractive correction terms. After that, the question posed in Ref. [18] on the reliability of standard self-focusingNLS system in describing beam propagation lead to the inclusion of additional longitudinal field oscillations. Goingone step further, the two dimensional NLS equation in self-focusing Kerr medium with nonparaxial term has beennumerically studied in Ref. [19].On the other hand, the nonparaxial ultra-broad beam possesses a broad beam having moderate intensity but ispropagating at an arbitrary angle with respect to the reference direction [20]. To overcome the above inadequacy ofthe paraxial approximation, we modify the CNLS system by including the nonparaxial e ff ect following the nonparaxialtheory developed for beam propagation in Kerr like nonlinear media [21]. The evolution of broad optical beams inKerr like nonlinear media can be well described by the coupled nonlinear Helmholtz (CNLH) type equations. Thegeneral dimensionless CNLH equations describing interaction of two obliquely propagating incoherently coupledoptical fields in Kerr type nonlinear media can be casted as [22]-[26], iq ,ξ + κ q ,ξξ + q ,ττ + ( σ | q | + σ | q | ) q = , (1a) iq ,ξ + κ q ,ξξ + q ,ττ + ( σ | q | + σ | q | ) q = , (1b)where q j , j = , , are the envelope fields of first and second components, subscripts ξ and τ represent the longitudinaland transverse coordinates respectively. In Eq. (1), the second term corresponds to Helmholtz nonparaxiality and theco-e ffi cient κ ( >
0) is the nonparaxial parameter. The nonparaxial parameter κ value has been estimated of the order of10 − to 10 − [22, 26]. Normally, in the paraxial approximation (the beam wavelength is much smaller than the widthof the beam) the terms κ q j ,ξξ , j = , , are neglected. However, these terms will come into picture when the width ofthe beam is of the order of wavelength [19] and influence the dynamics of the nonlinear waves. Physically, here wehave included the di ff raction in both transverse and longitudinal directions.In Eq. (1), we have explicitly introduced the real nonlinearity coe ffi cients σ l , l = ,
2, that can be absorbedinto the equation itself with a simple transformation, in order to deal with a broader class of nonlinearities, namelyfocusing nonlinearity ( σ > σ > σ < σ < σ > σ < σ < σ >
0) within a single framework. For σ = σ = ±
1, the system(1) reduces to the Helmholtz-Manakov system discussed in Ref. [22]. In real situations one require the cross-phasemodulation coe ffi cients (coe ffi cients of | q | in Eq.(1a) and | q | in Eq.(1b)) to be the same. This can be achievedfor the nonlinearity σ > σ <
0, by considering the complex conjugate of Eq. (1a) along with Eq. (1b) as it isand making the transformation ψ → √ σ q ∗ , ψ → √| σ | q will result in a coupled equation in which ψ is in theanomalous dispersion regime and ψ is in the normal dispersion region. Note that now in spite of having opposite signsfor σ and σ , the cross-phase modulation coe ffi cients are equal to 1. Similar analysis can be carried out for the otherchoice ( σ < σ >
0) too. The mixed (focusing-defocusing) nonlinearity has been considered earlier in the absenceof nonparaxial terms ( κ q j ,ξξ , j = ,
2) in Ref. [27]. Here we include the nonparaxial term and discuss the mixed typenonlinearity. This type of mixed (focusing-defocusing) nonlinearity can be realized in left-handed materials, couplerswith left-handed and right-handed composite materials, optical materials with quadratic nonlinearities and also in twocomponent Bose-Einstein condensates using Feshbach mechanism.There exist several analytical studies on the scalar (single component) nonlinear Helmholtz (NLH) system. Partic-ularly, it has been shown that the one dimensional scalar NLH system with focusing nonlinearity admits bright solitonsolutions [23] and scalar NLH system with defocusing nonlinearity supports dark soliton solutions [24] while the twodimensional scalar NLH system supports the vortex solitons [25]. However, results are scarce for the above CNLH(vector NLH) system. Recently, Eq. (1) with σ = σ = ± ff ect of nonparaxial parameter on theelliptic as well as solitary waves in the CNLH equations.This paper is organised as follows: In Sec. 2, we briefly outline the mathematical procedure to construct theelliptic solutions of the CNLH system. In Secs. 3 and 4, respectively we obtain the elliptic wave solutions in terms ofLam´e polynomials of order one and two, and also present a complete discussion on the e ff ect of the nonparaxial term.We conclude our results in Sec. 5.
2. The Method
In this section, we briefly outline our procedure to construct a rich variety of doubly periodic elliptic waves as wellas localized solitary wave solutions of the CNLH system (1). This will also provide the criteria for the existence ofnonlinear elliptic waves in the CNLH system. For constructing the Jacobi elliptic solutions of system (1), we introducethe traveling wave ansatz q j = f j ( u ) e i α j , (2a)where u = β ( τ − v ξ + δ ) , α j = k j ξ − ω j τ + δ j , j = , , (2b)in Eq. (1). Here, f j , j = , , are real functions of ξ and τ while β , δ and δ j j = , ω j sare frequencies of the two components of the CNLH system, v is the velocity and k j is the wave number of the q thj component. On equating the real and imaginary parts of the resulting equations, we obtain the following equations: d f j du − + κ v ) β h k j (1 + κ k j ) + ω j − σ f + σ f ) i f j = , j = , , (2c)where v = − ω + κ k ! , ω = ω + κ k + κ k ! . (2d)Note that Eq. (2c) is di ff erent from the earlier systems studied in Refs. [30] and [32] due to the explicit appearance ofnonparaxial term κ in Eq. (2c). We assume the Lam´e function ansatz for f j , that is, f j = ρ j ψ ( l ) j , l , j = , , (3)where ψ ( l ) j can be any one of the three first order Lam´e polynomials for l = l = d ψ ( l ) j du + [ λ ( l ) j − l ( l + m sn ( u , m )] ψ ( l ) j = , (4)where m (0 ≤ m ≤
1) is the modulus parameter of the Jacobi elliptic function sn( u , m ), l ( = ,
2) represents theorder of the Lam´e polynomial ψ ( l ) j and λ ( l ) j is the corresponding eigenvalue. Thus we will have two distinct families ofsolutions corresponding to the Lam´e polynomials of order 1 ( l =
1) and of order 2 ( l = . First order elliptic and solitary wave solutions The first order solutions of the CNLH system (1) consist of six distinct solutions. Additionally, one novel su-perposed elliptic solution can be constructed in terms of dn and cn elliptic functions. These elliptic solutions reduceto two broad types of hyperbolic solitary wave solutions namely bright solitary wave and dark solitary wave in thehyperbolic limit. Each of the six elliptic solutions is characterized by twelve real parameters with five conditions,thereby admitting seven arbitrary real parameters. In the following, we present these first order solutions one by oneand clearly demonstrate how the amplitude, speed and width of the nonlinear waves are influenced by the nonparaxialparameter κ . Solution 1(a) Cnoidal waves
The first elliptic solution of the CNLH system (1) is given below. q q ! = A e i α B e i α ! √ m cn (u,m) , (5a)where v = − ω + κ k ! , ω = ω δ , δ = + κ k + κ k ! , β = k (1 + κ k ) + ω δ (2 m − κ v + , A = σ h (2 κ v + β − σ B i , ω = (cid:2) δ − (cid:3) [ k (1 + κ k ) − k (1 + κ k )] . (5b)Note that all the first order and second order solutions discussed in this paper admit the same constraint conditions(2d) for v and ω . The above solution 1(a) is possible for the focusing nonlinearity ( σ and σ are positive) butfor the defocusing nonlinearity ( σ and σ are negative) the above solution does not exist. However for the mixednonlinearity physically admissible solution is possible for appropriate choice of the solution parameters. (b) Bright solitary waves The solution 1(a) reduces to the following hyperbolic bright solitary wave solution in the limit m = q q ! = A e i α B e i α ! sech( u ) , (5c)provided β = k (1 + κ k ) + ω δ (2 κ v + . (5d)Here all other constraint parameters are same as given in Eq. (5b). Solution 2(a) Snoidal waves
The second elliptic solution of the CNLH system (1) can be expressed as. q q ! = A e i α B e i α ! √ m sn (u,m) , (6a)where β = − k (1 + κ k ) + ω δ (1 + m )(2 κ v + , A = − σ h (2 κ v + β + σ B i , ω = (cid:2) δ − (cid:3) [ k (1 + κ k ) − k (1 + κ k )] . (6b)4he solution parameters and the system parameters have to be properly chosen to ensure that β , A and ω are indeedpositive. It can be inferred that for the focusing nonlinearity ( σ > σ > A becomes negative and the abovesolution does not exist. However, the solution exists for the defocusing and mixed nonlinearities with appropriatechoice of parameters. (b) Dark solitary waves In the limit m = q q ! = Ae i α Be i α ! tanh( u ) . (6c)Now, all the constraints parameters are same as given in Eq. (6b) except for β which now becomes as β = − k (1 + κ k ) + ω δ κ v + . (6d) Solution 3 : Dnoidal waves
Another solution for the CNLH system is given below: q q ! = A e i α B e i α ! dn (u,m) . (7a)The constraint conditions turn out to be the same as given in Eq. (3b) except for β which is now given as β = k (1 + κ k ) + ω δ (2 − m )(2 κ v + . (7b)This solution also exists for focusing and mixed nonlinearities. The solution 3 reduces to the hyperbolic solution 1(b)for the limiting case m = Solution 4(a) Dnoidal-Snoidal waves
Yet another elliptic solution of the CNLH system (1) is, q q ! = A dn (u,m) e i α B √ m sn (u,m) e i α ! , (8a)provided A = σ h (2 κ v + β + σ B i , B = σ [2 k (1 + κ k ) + ω δ + ( m − κ v + β ] ,ω = h δ − (cid:16) + κβ (1 + κ k ) (cid:17)i h k (1 + κ k ) − k (1 + κ k ) + β i . (8b)The above solution exists in case σ > , σ < σ < , σ >
0. On theother hand, for focusing and defocusing nonlinearities, the existence of the solution requires the parameters to satisfysome conditions. Particularly, for the defocusing case we require (2 κ v + β < | σ B | and | ( m − κ v + β | > k (1 + κ k ) + ω δ . One should also choose the solution as well as the system parameters appropriately to ensure that ω is indeed positive definite. 5 b) Bright-Dark solitary waves The above solution 4(a) reduces to the following hyperbolic solution in the limiting case m = q q ! = A sech( u ) e i α B tanh( u ) e i α ! , (8c)where B = σ [2 k (1 + κ k ) + ω δ − (2 κ v + s ) β ] . (8d)Here the first component q and the second component q comprise of bright and dark solitary waves respectively.Such type of coexistence of bright and dark solitary waves even for the focusing nonlinearity is a special feature ofthe present system. Solution 5: Cnoidal-Snoidal waves
Next elliptic wave solution of the CNLH system (1) is obtained as q q ! = A √ m cn (u,m) e i α B √ m sn (u,m) e i α ! . (9a)The constraint relations for this solution are same as those of solution 4(a) (see Eq.(8b)) except for B and ω whichare now given by mB = σ h k (1 + κ k ) + ω δ − (2 m − κ v + β i ,ω = h δ − (1 + m κβ (1 + κ k ) ) i h k (1 + κ k ) − k (1 + κ k ) + m β i . (9b)The above solution 5 reduces to the hyperbolic solution 4(b) in the limit m = Solution 6: Dnoidal-Cnoidal waves
The explicit form of the sixth solution of the CNLH system (1) is given below. q q ! = A dn (u,m) e i α B √ m cn (u,m) e i α ! . (10a)The constraint conditions associated with the above solution are A = σ h (2 κ v + β − σ B i , ( m − B = σ h k (1 + κ k ) + ω δ − (2 − m )(2 κ v + β i ,ω = h δ − (cid:16) − κβ (1 + κ k ) (cid:17)i h k (1 + κ k ) − ( β + k (1 + κ k )) i . (10b)The above mentioned solution 6 can exist for focusing nonlinearity with the condition | κ v + β | > | σ B | but doesnot exist for the defocusing nonlinearity. For the mixed nonlinearity case, it demands the parameters to satisfy someconditions: | (2 − m )(2 κ v + β | > k (1 + κ k ) + ω δ for the mixed case with σ > , σ <
0. Also the other choiceof the mixed case σ < , σ >
0, requires (2 κ v + β < σ B . To make ω positive, we should appropriately choosethe solution and system parameters.As expected, in the limit m =
1, this solution goes over to the hyperbolic solution 1(b).6 olution 7: Superposed first order elliptic waves
Finally, we find that the linear superposition of two di ff erent elliptic solutions is also a solution of the CNLHsystem (1) [33]. The corresponding superposed solution is q q ! = (cid:16) A dn( u , m ) + D √ m cn( u , m ) (cid:17) e i α (cid:16) B dn( u , m ) + E √ m cn( u , m ) (cid:17) e i α , (11a)provided A = σ h (2 κ v + β − σ B i ,β = h k (1 + κ k ) + ω δ i (1 + m )(2 κ v + , ω = (cid:2) δ − (cid:3) [ k (1 + κ k ) − k (1 + κ k )] . (11b)Here the signs of D = ± A and E = ± B are correlated. This solution exists for the focusing nonlinearity but for thedefocusing nonlinearity solution 7 can not exist. The mixed type nonlinearity also allows this solution only for thechoice (2 κ v + β < | σ B | . One can note that the solution 7 reduces to the hyperbolic solution 1(b) in the limit m = ff erent elliptic waves we have pointed out the type of nonlinearity admitting thosesolutions. We tabulate below the solutions along with nonlinearities which support the solutions for a better under-standing. Table 1: Types of nonlinearities and their corresponding solutions
Solutions Types of nonlinearity supporting the solutions1, 3, 6, 7 Focusing nonlinearity ( σ > σ > σ > σ < σ < σ > σ < σ < σ > σ < σ < σ > σ > σ > σ < σ < σ > σ < ff erent nonlinear elliptic wave solutions. This is an important aspect of the present study. For illustrative purpose,we consider solution 1(a), superposed solution [solution 7] and the bright soliton / solitary wave solution 1(b). Onecan infer from the first order solutions in the absence of nonparaxial term ( κ =
0) (note that for this choice theCNLH system reduces to the standard integrable CNLS system) the frequencies as well as the wave numbers inboth components become equal. Ultimately the solutions become phase-locked [34]. This phase-locking propertywill be removed when the nonparaxial parameter is introduced and now the two components will possess distinctphases. Further in order to interpret the role of the nonparaxial parameter, one should carefully analyse the constraintconditions on the physical parameters like speed (modulus of velocity), pulse width (1 /β ) and the amplitudes.Figs. 1(a-c) show the variation of speed, pulse width and amplitude of the first component of solutions 1(a), 7and the hyperbolic solution 1(b) respectively, with respect to κ . The nature of the curve depends upon the sign of k in particular the denominator term (1 + κ k ) in the expression for v (see Eq. (5b)). In Fig. 1(a) the speed decreasesgradually as κ increases and reaches a saturation for the parametric choice ω = . , B = . , k = . κ = . A of the first component behave in an opposite manner as κ increases. It can beinferred from the above three solutions (1(a), 1(b), 7) that the amplitude of the second component is independent of κ .In Figs. 1(b-c), also we notice the speed decreases rapidly as κ increases and reaches a saturation. But the pulse width7nitially increases as κ increases and after some nonparaxial value κ , it decreases. Finally, we observe that the firstcomponent amplitude A also initially increases as κ increases and after some nonparaxial value, it attains a constantvalue. In all the three figures we notice that the speed is decreasing as κ is increased. This shows that by tuning thenonparaxial parameter we can decelerate the elliptic waves as well as the solitary waves.Another important observation from the Figs. 1(b-c) corresponding to superposed solution and hyperbolic solutionis that the amplitude of the first component attains a saturation after a particular value of κ . However the speedcontinues to change. Thus in this saturation region by altering κ we can change the speed without a ff ecting theamplitude of the first component. (a) Κ H È v È , (cid:144) Β , A L (b) Κ H È v È , (cid:144) Β , A L (c) Κ H È v È , (cid:144) Β , A L Figure 1: Plots of speed (solid dark green), pulse width (dotted blue) and amplitude of the first component (dotdashed red) versus nonparaxialparameter κ . (a) solution 1(a) with ω = . k = . B = .
6, (b) solution 7 with ω = . k = . B = E = ω = . k = . B = . m =
1. In all the figures σ = σ = δ = δ = δ = ξ = . τ = .
5. Except in Fig. 1(c) m = . Next we consider the solutions 4(a), 5 and 6. A careful look at these solutions and their constraint relations showsthat these solutions behave distinctly from the solutions 1(a), 7 and 1(b). For these solutions, the pulse width becomesarbitrary while the amplitude of second component is not arbitrary. Also the pulse width a ff ects the frequency andultimately influences the speed. To get further insight we present the plots of amplitude of the first and secondcomponents and speed for these three solutions 4(a), 5 and 6 in Fig. 2.In Figs. 2(a-c) we display the plots of speed (modulus of velocity), amplitude of the first and second componentsof solutions 4(a), 5 and 6 respectively, with respect to κ . In Figs. 2(a-c) the speed decreases gradually as κ increases.But the amplitudes of the first (A) and second (B) components are increasing as κ increases. One can infer from theabove three solutions that the pulse width of the solution is independent of κ . Fig. 2 also shows that by changing thenonparaxial parameter we can decelerate the elliptic waves as well as the solitary waves. (a) Κ H È v È , A , B L (b) Κ H È v È , A , B L (c) Κ H È v È , A , B L Figure 2: Plots of speed (solid dark green), amplitude of the first component (dotted blue) and amplitude of the second component (dotdashedblack) versus nonparaxial parameter. (a) solution 4(a) with ω = . k = . β = .
9, (b) solution 5 with ω = k = . β = . ω = . k = . β = . σ = −
1. In all the figures σ = σ = δ = δ = δ = τ = . m = . We show the propagation of the elliptic waves supported by the solutions 1(a), 2(a) and 4(a) respectively inFigs. 3(a-c). Here, the top and middle panels of Fig. 3(a) show the intensity profiles of the solution 1(a). The intensityprofiles of the first and second components are in-phase and this is demonstrated in the bottom panel of Fig. 3(a).Next, the top and middle panels in Figs . 3(b-c) show the intensity profiles for the solutions 2(a) and 4(a) respectively.The bottom panel of Fig. 3(b) shows that both components are in-phase while that of Fig. 3(c) shows both componentsare out-of-phase. 8 igure 3: The intensity plots of solution 1(a) with ω = . k = . B =
1, solution 2(a) with ω = . k = − . B = σ = − ω = k = . β = . q (dark green) and q (dashed blue) components with τ = .
5. In all the figures σ = κ = . σ = δ = δ = δ = m = . Further, we illustrate all first order hyperbolic solutions 1(b), 2(b) and 4(b) in Fig. 4. Particularly, Fig. 4(a) showsthe bright soliton / solitary wave profiles exhibited by solution 1(b). Next, we show in Fig. 4(b) that in solution 2(b)both components admit dark soliton / solitary wave profiles. Finally, we depict solution 4(b) in Fig. 4(c) showing theco-existence of bright and dark solitary waves respectively in the first and second components.9 igure 4: The intensity plots of solution 1(b) with ω = . k = . B =
1, solution 2(b) with ω = . k = − . B = σ = − ω = k = . β = . q (dark green) and q (dashed blue) components with τ = .
5. In all the figures σ = κ = . σ = δ = δ = δ = m =
4. Second order Elliptic and solitary wave solutions
In this section, we construct seven distinct elliptic solutions of the CNLH system (1) in terms of Lam´e polynomialsof order 2 and also obtain a novel superposed elliptic solution. We also discuss the associated hyperbolic solutionswhen the modulus parameter m becomes unity. Each of the elliptic solutions given below has six arbitrary parameters.We refer to these solutions as second order elliptic solutions as they are stemming from the second order Lam´epolynomials. Solution 8(a) cn dn − sn dn waves The first second order elliptic solution of the CNLH system (1) is obtained as, q q ! = A √ m cn (u,m) dn (u,m) e i α B √ m sn (u,m) dn (u,m) e i α ! , (12a)10here α , and u are as defined in Eq. (2).The above solution requires, σ A = σ B , mA = σ h κ v + β i , β = k (1 + κ k ) + ω δ (2 κ v + − m ) ,ω = (cid:0) (5 m − δ − (5 − m ) (cid:1) [ k (5 − m )(1 + κ k ) − k (5 m − + κ k )] . (12b)A careful look at the expression for A shows that the choice σ < A = σ σ B , wefind that the solution will exist only for the focusing nonlinearity. Also, the parameters k , k and m have to be chosenappropriately so that ω is indeed positive. (b) Blue-White solitary waves In the hyperbolic limit ( m =
1) the above solution results in a specially coupled solitary wave solution. Here thefirst component admits symmetric profile whereas that of the second component is antisymmetric. Such type of q and q components can also be viewed as blue and white solitary waves respectively [35] and can be expressed as q q ! = A sech ( u ) e i α B sech( u ) tanh( u ) e i α ! . (12c)The solution parameters satisfy the following conditions: A = σ h κ v + β i , β = k (1 + κ k ) + ω δ κ v + , ω = (cid:0) δ − (cid:1) [4 k (1 + κ k ) − k (1 + κ k )] (12d)and all other constraints parameters are as given in Eq. (12b). Solution 9: cn dn − sn cn waves Another second order elliptic solution of the CNLH system (1) is q q ! = A √ m cn (u,m) dn (u,m) e i α B m sn (u,m) cn (u,m) e i α ! , (13a)in which the solution parameters satisfy the relations, σ A = σ B , A = σ h κ v + β i , β = k (1 + κ k ) + ω δ (2 κ v + m − ,ω = (cid:0) (5 m − δ − (5 m − (cid:1) [ k (5 m − + κ k ) − k (5 m − + κ k )] . (13b)As in the previous solution, here too the solution is ruled out for the choice σ < q and q , respectively in the hyperbolic limit m = Solution 10: sn dn − sn cn waves Yet another second order elliptic solution for the CNLH system (1) is q q ! = A √ m sn (u,m) dn (u,m) e i α B m sn (u,m) cn (u,m) e i α ! , (14a)11ith the conditions σ A = − σ B , (1 − m ) A = − σ h κ v + β i , β = − + m ) k (1 + κ k ) + ω δ (2 κ v + ,ω = (cid:0) (4 + m ) δ − (4 m + (cid:1) [ k (4 m + + κ k ) − k (4 + m )(1 + κ k )] . (14b)One can infer from the expression for A and the condition σ A = − σ B that the above solution is possible only forthe mixed type nonlinearity. This solution becomes singular in the hyperbolic limit ( m = Solution 11(a) dn − sn cn waves We also identify the following second order solution for the CNLH system (1) q q ! = (cid:16) A dn (u,m) + D (cid:17) e i α B m sn (u,m) cn (u,m) e i α ! , (15a)provided σ A = σ B , p = DA , p = − (cid:16) (2 − m ) ± √ − m + m (cid:17) , (2 + p − m ) A = σ [3(2 κ v + β ] , β = k (1 + κ k ) + ω δ (2 κ v + b ,ω = a δ − b ) [ b k (1 + κ k ) − a k (1 + κ k )] , (2 + p − m ) a = [6(1 + p ) − (4 + m )(2 + p − m )] , (2 + p − m ) b = + p ) − (2 + p − m )((1 + m ) + √ − m + m ] . (15b)It can be inferred from the expression for p that it admits two values corresponding to the ± sign before the term √ − m + m . Ultimately, for the positive sign the above solution 11(a) exists only for the defocusing nonlinearity butfor the negative sign this solution exists only for the focusing nonlinearity. (b) Red-White solitary waves In the hyperbolic limit this solution 11(a) with p = − (cid:16) (2 − m ) − √ − m + m (cid:17) reduces to the solution 8(b). Thesame solution 11(a) with p = − (cid:16) (2 − m ) + √ − m + m (cid:17) reduces to the following special hyperbolic solitary wavesolution in the limit m = q q ! = A (cid:16) sech ( u ) − (cid:17) e i α B sech( u ) tanh( u ) e i α ! , (15c)provided σ A = σ B , A = − σ h κ v + β i ,β = − k (1 + κ k ) + ω δ κ v + , ω = δ −
8) [8 k (1 + κ k ) − k (1 + κ k )] . (15d)Here all other constraints parameters are same as that of solution 11(a). One can also view the first component q andthe second component q as red and white solitary waves respectively [35]. In fact, q admits like W-shape solitarywave profile. 12 olution 12(a) dn − sn dn waves One more second order elliptic solution for the CNLH system (1) is q q ! = (cid:16) A dn (u,m) + D (cid:17) e i α B √ m sn (u,m) dn (u,m) e i α ! , (16a)where the constraint conditions read as σ A = σ B , p = DA , p = − (cid:16) (2 − m ) ± √ − m + m (cid:17) , A = p + σ h κ v + β i , β = k (1 + κ k ) + ω δ κ v + b ,ω = a δ − b ) [ b k (1 + κ k ) − a k (1 + κ k )] , a = + p ) p + − (1 + m ) ! , b = + p ) p + − (cid:16) (1 + m ) + √ − m + m (cid:17)! . (16b)The ± sign appearing in the expression for p results in two di ff erent values for 2 p +
1. Particularly, for all valuesof m , the + ( − ) sign yields negative (positive) values for 2 p +
1. For 2 p + >
0, the solution is allowed only forfocusing nonlinearity while for 2 p + <
0, the solution will exist if the nonlinearity is of defocusing type. Howeverto get regular solutions the parameters have to be chosen appropriately. Here too the solution 12 in the limit m = p = m = p = − it reduces to solution 11(b). Solution 13(a) dn − cn dn waves We have obtained the following elliptic solution for the CNLH system (1): q q ! = (cid:16) A dn (u,m) + D (cid:17) e i α B √ m cn (u,m) dn (u,m) e i α ! , (17a)along with the constraints σ A = − σ B , p = DA , p = − (cid:16) (2 − m ) ± √ − m + m (cid:17) , (2 p + − m ) A = σ [3(2 κ v + β ] , β = k (1 + κ k ) + ω δ κ v + b ,ω = a δ − b ) [ b k (1 + κ k ) − a k (1 + κ k )] , (2 p + − m ) a = ((1 + m ) − + p ) − m )) , (2 p + − m ) b = (cid:16) + p ) − m ) − (2 p + − m ) (cid:16) (1 + m ) + √ − m + m (cid:17)(cid:17) . (17b)The condition σ A = − σ B restricts the solution to be allowed only for the mixed type nonlinearities. For thecondition p = − (cid:16) (2 − m ) + √ − m + m (cid:17) , we should take the value of σ as negative to make A as positive definite.But the same solution with p = − (cid:16) (2 − m ) − √ − m + m (cid:17) , requires σ to be positive for getting positive definitevalue for A . 13 b) Red-Blue solitary waves Here, in the limit m = p = p = − ,yields the following hyperbolic solution comprising is W-shape bright solitary wave and a standard solitary wave. q q ! = (cid:0) A sech ( u ) − (cid:1) e i α B sech ( u ) e i α ! . (17c)Here the solution parameters satisfy the relations σ A = − σ B , σ < , σ > , A = − σ h (2 κ v + β i , p = − ,β = − k (1 + κ k ) + ω δ κ v + , ω = − δ +
1) [ k (1 + κ k ) + k (1 + κ k )] . (17d)Here the first component q and the second component q are comprised of red and blue solitary waves respectively[35]. Solution 14(a) dn − dn waves A distinct second order elliptic solution for the CNLH system (1) is q q ! = A (dn (u,m) + D ) e i α B (dn (u,m) + E ) e i α ! , (18a)provided σ A = − σ B , p , q = − (cid:16) (2 − m ) ± √ − m + m (cid:17) , A = p − q ) σ h (2 κ v + β i , p = DA , q = EB ,β = − k (1 + κ k ) + ω δ √ − m + m (2 κ v + , ω = − δ +
1) [ k (1 + κ k ) + k (1 + κ k )] . (18b)As in the previous solution, here also the condition σ A = − σ B confines the system to admit the solution only formixed type nonlinearities. (b) Blue-Red solitary waves In the hyperbolic limit the solution 14(a) with p = − and q = p = q = − reduces to the following hyperbolic solitary wave solution in the limit m = q q ! = A sech ( u ) e i α B (cid:16) sech ( u ) − (cid:17) e i α , (19a)where the solution parameters satisfy the relation A = σ h (2 κ v + β i . (19b)Here all other constraints parameters are same as that of solution 13(b). Then the first component q and secondcomponent q are comprised of blue and red solitary waves respectively [35].14 olution 15: Superposed second order elliptic waves Finally, we have also constructed a novel second order superposed elliptic solution that can be expressed as acombination of dn and cndn elliptic functions. This special superposed elliptic solution is given below. q q ! = (cid:18) A dn ( u , m ) + D + F √ m cn ( u , m ) dn ( u , m ) (cid:19) e i α (cid:18) B dn ( u , m ) + E + G √ m cn ( u , m )dn ( u , m ) (cid:19) e i α . (20a)Here the constraint conditions read as σ A = − σ B , A = σ ( p − q ) (cid:16) (2 κ v + β (cid:17) ,β = − h k (1 + κ k ) + ω δ i √ + m + m (2 κ v + , p = DA , q = EB , p , q = − (cid:16) (5 − m ) ± √ + m + m (cid:17) ,ω = − δ +
1) [ k (1 + κ k ) + k (1 + κ k )] . (20b)This solution exists exclusively for the mixed nonlinearity as follows from the constraint relation σ A = − σ B andalso from the expression for A . In solution 15, the signs of F = ± A and G = ± B are correlated. The hyperbolicsolution is only possible for the case F = A and G = B . Note that p and q admit two values corresponding to the ± sign before the term √ + m + m in their corresponding expression. Here, in the limit m = p = − and q = m = p = q = − reduces to the solution 14(b).In the following table, we tabulate the solutions and the types of nonlinearities admitting that solution. Table 2: Types of nonlinearities and their corresponding second order solutions
Solutions Types of nonlinearity supporting the solutions8, 9 Focusing nonlinearity ( σ > σ > σ > σ < σ < σ > σ < σ <
0) when p = − (cid:16) (2 − m ) + √ − m + m (cid:17) Focusing nonlinearity ( σ > σ >
0) when p = − (cid:16) (2 − m ) − √ − m + m (cid:17) For illustrative purpose, we present the variation of speed (modulus of velocity), pulse width and amplitude of thefirst component of solutions 8(a), 10 and 13(a) respectively, with respect to κ in Figs. 5(a-c). In Fig. 5(a) the speed andthe pulse width increases slightly as κ increases. Then the amplitude A of the first component behaves opposite to thespeed and pulse width i.e., the amplitude A of the first component decreases slightly as κ increases. In Figs. 5(b-c),also we notice the speed and pulse width increases as κ increases. But the amplitude A decreases as κ increases. (a) Κ H È v È , (cid:144) Β , A L (b) Κ H È v È , (cid:144) Β , A L (c) Κ H È v È , (cid:144) Β , A L Figure 5: Plots of speed (solid dark green), pulse width (dotted blue) and amplitude of the first component (dotdashed red) versus nonparaxialparameter κ . (a) solution 8(a) with ω = . k = − . m = . σ =
1, (b) solution 10 with ω = . k = − . ω = . k = − .
3. In all the figures σ = − σ = δ = δ = δ = ξ = . τ = . m = . κ = . q ) component the width of thepulse train is compressed in the second ( q ) component. Finally, intensity profiles of the superposed solution 15, areshown in the first and middle panels of Fig. 6(c) while the associated two dimensional plot is presented in the bottompanel. We also note that in the superposed solution the amplitude of the second ( q ) component gets enhanced andadmit simply periodic pulse trains whereas the first ( q ) components admits doubly periodic pulse trains with slightlylesser amplitude. The above type of distinct behaviours of second order elliptic waves for di ff erent choice of solutionparameters suggest the possibility of pulse shaping, a desirable property in nonlinear optics, in the CNLH system. Figure 6: The intensity plots of solution 8(a) with ω = . k = − σ =
1, solution 10 with ω = . k = − . ω = . k = . q (dark green) and q (dashed blue) components with τ = .
5. In all the figures σ = − κ = . σ = δ = δ = δ = m = . The hyperbolic second order solutions display a rich variety of intensity profiles. To give further impetus in thisaspect we have plotted all the second order hyperbolic solutions given above. In Fig. 7(a) we show the intensity plotsof solution 8(b). Here the first component admits single-hump bright soliton / solitary wave and the second componentbears a double-hump bright soliton / solitary wave intensity profile. Another interesting type of soliton / solitary wave isshown in Fig. 7(b), namely double-hump dark (W-shape) as well as bright soliton / solitary wave corresponds to solution161(b). The last interesting co-existing bright-dark-structure comprising double-hump dark (W-shape) solitary wave in q component and bright solitary wave in q component are displayed in Fig. 7(c) (see the hyperbolic solution 13(b)).An important observation from these plots is that even though the parameters k and ω are changed the intensityprofiles remain unaltered. This property is a reminiscent of the degenerate property of solitary waves reported forcoupled Sasa-Satsuma system in Ref.[36]. Figure 7: The intensity plots of solution 8(b) with ω = . k = . σ = σ =
1, solution 11(b) with ω = − . k = − σ = − σ =
1, solution 13(b) for the choice ω = . k = σ = − σ =
1, are shown in first two rows of Figs. 7(a-c) respectively. The third rowshows the corresponding two dimensional plots of the q (dark green) and q (dashed blue) components with τ = .
5. All these figures κ = . δ = δ = δ = m =
5. Conclusion
In this paper, we have obtained two distinct sets of elliptic wave solutions of the CNLH system which are expressedin terms of Lam´e polynomials of order one and two. The resulting solitary waves in the limit m = κ increases. We have demonstrated that the speed can be altered by tuning the nonparaxialparameter. This is the most significant e ff ect of the nonparaxial parameter. Importantly, it has been shown that thee ff ect of nonparaxiality is to remove the phase-locking behaviour of the solutions. In the second order solutions, wehave presented novel single and double-hump bright solitary waves, double-hump (W-shape) dark and bright solitary17aves and coupled double-hump dark (W-shape) and standard bright solitary waves. These solitary waves with distinctprofiles were found to admit the degenerate property in which the shape of the solution profile remains unaltered bythe solution parameters even though these parameters influence the amplitude, pulse width and velocity. These resultswill give further impetus into the dynamics of nonlinear travelling waves in the CNLH system and will find applicationin nonlinear pulse shaping. This procedure can be extended to variants of CNLH system and other CNLS systems byincluding quintic and power-law nonlinearities. The elliptic solutions as well as hyperbolic solitons obtained here willserve as good platform for further numerical investigation of the CNLH system. Their experimental realization willgive further insight into the the nonparaxial e ff ects in optical waveguides. Further investigation of nonparaxial e ff ectsin graded index medium is a separate issue and yet to be studied. Acknowledgments
The authors T.K. and K.T. acknowledge the Principal and Management of Bishop Heber College, Tiruchirapallifor the constant support and encouragement. The author A.K. acknowledges DAE, Govt. of India for the financialsupport through Raja Ramanna Fellowship.
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