Spatiotemporal Similaritons in (3+1)-Dimensional Inhomogeneous Nonlinear Medium with Cubic-Quintic Nonlinearity

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1 Commun. Theor. Phys. 55 ( Vol. 55, No. 5, May 15, 211 Spatiotemporal Similaritons in (3+1-Dimensional Inhomogeneous Nonlinear Medium with Cubic-Quintic Nonlinearity CHEN Yi-Xiang (í 1,2, and LU Xuan-Hui (öï 1 Department of Physics, Institute of Optics, Zhejiang University, Hangzhou 3127, China 2 School of Electronics Information, Zhejiang University of Media and Communications, Hangzhou 3118, China (Received November 22, 21; revised manuscript received January 6, 211 Abstract We obtain exact spatiotemporal similaritons to a (3+1-dimensional inhomogeneous nonlinear Schrödinger equation, which describes the propagation of optical pulses in a cubic-quintic nonlinearity medium with distributed dispersion and gain. A one-to-one correspondence between such self-similar waves and solutions of the elliptic equation is found when a certain compatibility condition is satisfied. Based on exact solutions, we discuss evolutional behaviors of self-similar cnoidal waves and chirped similaritons in two kind of typical soliton control systems. Moreover, the comparison between chirped similaritons and chirp-free solitons is given. PACS numbers: Dp, Tg, 5.45.Yv Key words: spatiotemporal similaritons, (3+1-dimensional cubic-quintic nonlinear Schrödinger equation, dynamical behaviors 1 Introduction It is well known that the existence of localized structures in optics requires a careful balance among dispersion, diffraction, and nonlinearity, and solutions are often sensitive to beam instabilities. In this regard, however, research in guided wave optics has recently identified nonlinear self-similar propagation as a robust means of avoiding beam or pulse breakup at high power. 1 3 The envelopes of such nonlinear waves (or similaritons maintain their overall shape, but the field amplitude, width, and a phase chirp evolve on propagation inside nonlinear media. Although temporal similaritons 1 4 have thus far attracted more attention, spatial similaritons have also been discovered in graded-index waveguide amplifiers. 5 6 Recently, spatiotemporal localizations 7 9 have flourished into a research area of great importance and interest in many different contexts of nonlinear optics. Mathematically, all these cases are described by the inhomogeneous nonlinear Schrödinger equation (NLSE. Recently, the (1+1-dimensional NLSEs with distributed coefficients have been extensively investigated in various cases. 1 2,4 6,1 15 For example, optical soliton dispersion, amplification, soliton pulse width managements, and energy control were studied in 1. Serkin et al. 11 discovered bright and dark solitary nonlinear Bloch waves in dispersion managed fiber systems and soliton lasers. Hao et al. 12 and Wang et al. 13 discussed the (1+1-dimensional NLSE with spatial inhomogeneity. Luo et al. 14 gave exactly controllable transmission of nonautonomous optical solitons. Nonautonomous solitons was also discussed in Bose Einstein condensation. 15 Based on elliptic functions like cn, exact cnoidal waves solutions in the form of periodic arrays of pulses was also investigated. 4 Furthermore, discrete (1+1-dimensional NLSEs 16 and higher-order NLSEs 17 were also studied. However, relatively less research work for high dimensional NLSE was carried out. Maybe part of the reason is that all localized solutions are usually unstable for (2+1- and (3+1-dimensional (3+1D constantcoefficient NLSE due to the weak and strong collapses. 18 Even the radially symmetric solution at the critical power (or the critical exponent, known as the Townes soliton, is unstable. However, different situations are found in nonlinear optics with temporally or spatially modulated parameters. In a variational and numerical treatment, Adhikari 19 has shown that the 3D spatiotemporal optical solitons can be stabilized by a rapidly oscillating scattering length or the dispersion coefficient in a Kerr medium with cubic nonlinearity. And he also studied the interaction between solitons in three dimensions. 2 Thus, the study of the (3+1D NLSE with distributed coefficients has been one of the central issues in the field of nonlinear optics. 7 9 Note that for the NLSE, Kerr nonlinearity is considered. However, when the incident fields become stronger, non-kerr nonlinearities (e.g. quintic nonlinearity come into play changing essentially the physical features and the stability of optical soliton propagation. Historically, the study of topological quasi-soliton solutions for the inhomogeneous cubic-quintic (CQ NLSE began with the pioneering work of Serkin et al. 21 In this paper, we will employ an analytical method to discuss similaritons in the Supported by the National Natural Science Foundation of China under Grant No , and by the Ministry of Science and Technology of China under Grant No. 21DFA469 Corresponding author, chenyix1979@126.com c 211 Chinese Physical Society and IOP Publishing Ltd

2 872 Communications in Theoretical Physics Vol. 55 context of a 3D cubic-quintic nonlinear media. 2 Model and Similarity Transformation Cubic-quintic nonlinearity arises from a nonlinear correction to the refractive index of a medium in the form δn = n 2 u 2 n 4 u 4, and the coefficients n 2, n 4 > determine the nonlinear response of the medium. CQ nonlinearity can be derived by doping a fiber with two appropriate semiconductor materials. In this paper, we extend the 3D NLS model discussed in 22 by considering the quintic nonlinearity in the form 23 iu z β(z( u + u tt + χ(z u 2 u + ν(z u 4 u = iγ(zu, (1 where u(z, x, y, t is the spatiotemporal field envelope propagating along z, t is the retarded time, and is the transverse Laplacian (diffraction operator acting on (x, y. The functions β, χ, ν, and γ denote the diffraction/dispersion, cubic and quintic nonlinearities and gain coefficients, respectively. All coordinates are made dimensionless by the choice of coefficients. We define the complex periodic wave u of Eq. (1 in terms of its amplitude and phase u(z, x, y, t = ρ(z, x, y, texpiϕ(z, x, y, t. (2 Substituting u into Eq. (1, we obtain the following coupled equations: ρ z β2ρ xϕ x + 2ρ y ϕ y + 2ρ t ϕ t + ρ( ϕ + ϕ tt γρ =, ρϕ z β ρ + ρ tt ρ(ϕ 2 x + ϕ2 y + ϕ2 t + χρ3 + νρ 5 =. (3 The standard leading order analysis requires the following transformation 24 Thus Eq. (3 turns into ρ = 1/2. (4 z + β xϕ x + yϕ y + tϕ t + ( ϕ + ϕ tt 2γ =, 2 2ϕ z + β 1 ( 2x 4 + 2y + 2z + 1 2(ϕ 2 tt 2 x + ϕ2 y + ϕ2 t + 2χ 3 + 2ν 4 =. (5 We seek travelling wave solutions to Eq. (5 and assume the functions to be of the form: = f 1 (z + f 2 (zuθ(z, x, y, t, ϕ = a(z(x 2 + y 2 + t 2 + b(z(x + y + t + e(z, θ = k 1 (zx + k 2 (zy + k 3 (zt + ω(z, (6 where U satisfies the first kind elliptic equation U 2 θ = c + c 2 U 2 + c 4 U 4, (7 where U = U(θ, θ = θ(z, x, y, t. This choice is interesting because it allows us to determine the width related to k 1 (z p(z, k 2 (z q(z, k 3 (z r(z, and its group velocity related to ω(z. e(z, b(z, and a(z are the phase offset function, frequency shift, and chirp factor, respectively. Substituting Eq. (3 into Eq. (5 and requiring that x l U n, y l U n, t l U n, and c + c 2 U 2 + c 4 U 4 (l =, 1, 2; n =, 1, 2, 3 of each term be separately equal to zero, we can obtain the following set of equations f j,z + 6aβf j 2γf j =, k iz + 2aβk i =, ω z + bβ(k 1 + k 2 + k 3 =, (9 a z + 2βa 2 =, b z + 2βab =, 8e z β (k k2 2 + k2 3 c 2 24 f 1 χ 48 f 1 2 ν + 12 β b 2 =, (11 8f 2 1 e z 8 ν f β f2 2 (k2 1 + k2 2 + k2 3 c 8 χ f β f2 1 b2 =, (12 8 ν f β (k k k 2 3c 4 =, 2 f 2 2χ + 8 f 1 f 2 2ν + β (k k k 2 3f 1 c 4 =, 3χ + 8f 1 ν =, (13 where j = 1, 2, i = 1, 2, 3, and the subscript means its derivative with respect to z. Solving this set of equation self-consistently, one can obtain that f j (z = f j α 3 exp 2 γ(sds, (14 a(z = 1 2 a α, b(z = αb, k i (z = αk i, ω(z = ω b k i α (8 (1 β(sds, (15

3 No. 5 Communications in Theoretical Physics 873 e(z = e + α 8 k 2 i (6f 1 2 c 4 + f 2 2c 2 4b 2 β(sds, (16 where a, b, e, ω, k i, f j, j = 1, 2, i = 1, 2, 3 are all arbitrary constants and α = (1 s β(sds 1 is the normalized chirp function, which is related to the wave front curvature and presents a measure of the phase chirp imposed on the wave. The subscript denotes the initial values of the corresponding parameters at distance z =. One should note the universal influence of the chirp function α on the solutions. The chirp function is related only to the diffraction or dispersion coefficient; however, it affects all of the parameters. In the case when there is no chirp, s =, and α = 1, the parameters k i and b are all constants. In the presence of chirp, they all acquire the prescribed z dependence. The chirp also influences the form of the amplitude through the dependence of f j and θ on α. It should also be emphasized that the constants c, c 2, and c 4 have the following relation χ = c 4f 1 ( 3 k2 i β f2 2 α exp 2 f 4 1 c 4 + f 2 1 f2 2 c 2 + f 4 2 c =, (17 and the nonlinearity coefficients χ(z and ν(z is not arbitrary but depend on α, β, and γ: β ( 3 3 c 4 k2 i γ(sds, ν = 8f 2 2 α 4 exp 4 γ(sds, (18 which imply that exact solutions can exist in a lossy medium only when the nonlinearity coefficients χ and ν grow exponentially. These equations can be conveniently understood as two integrability conditions on Eq. (1: 1 dχ χ dz + 1 dα α dz 1 dβ β dz + 2γ(z =, 1 dν ν dz + 4 dα α dz 1 dβ + 4γ(z =. (19 β dz Hence, the solutions found can exist only under certain conditions and the system parameter functions β, γ, χ, and ν cannot be all chosen independently. For example, if β(z and γ(z are chosen to be the free parameters, then χ(z and ν(z will be determined from Eq. (18 or (19. Thus, we have proven the following result: the substitution exp u(z, t = γ(sds z 1 s f β(sds3/2 1 + f 2 U(θ 1/2 exp{ia(x 2 + y 2 + t 2 + b(x + y + t + e}, (2 where θ = ω + αk 1 x + k 2 y + k 3 t b (k 1 + k 2 + k 3 β(sds, leads to Eq. (7, with the conditions (17 and (19. The solutions of Eq. (1 can be obtained from those of Eq. (7 via the transformation (2. 3 Exact Spatiotemporal Similaritons for 3D CQNLSE The reduction of the 3D CQNLSE (1 to Eq. (7 helps one to find exact solutions, as the latter equation is solvable in terms of elliptic functions. Then, if a solution to Eq. (7 is known, one can construct exact solutions to the underlying 3D CQNLSE (1. Equation (7 possesses a series of Jacobian elliptic function solutions and the corresponding solitary wave solutions, which are shown in Table 1 with certain c, c 2, c Table 1 Jacobian elliptic functions. Solution c c 2 c 4 U m = m = (1 + m 2 m 2 sn sin tanh 2 1 m 2 2m 2 1 m 2 cn cos sech 3 m m 2 1 dn 1 sech 4 m 2 (1 + m 2 1 ns cosec coth 5 m 2 2m m 2 nc sec cosh m 2 m 2 1 nd 1 cosh m 2 1 m 2 sc tan sinh 8 1 m 2 2 m 2 1 cs cot csch 9 1 (1 + m 2 m 2 cd cos 1 1 m 2 (1 + m 2 1 dc sec 1 Thus the solution of Eq. (1 can be obtained from the solution of elliptic equation (7 by exploiting a one-to-one correspondence. Employing the transformations (2 with the conditions (17 and (19, one can obtain the following periodic wave solutions and the corresponding bright and dark similaritons for Eq. (1 as follows Solution 1 U(θ = sn(θ u(z, t = exp γ(sds 1 s β(sds3/2 µ 1 ± µ 1 sn(θ, m 1/2 expiϕ(z, x, y, t, (21

4 874 Communications in Theoretical Physics Vol. 55 where θ = ω + αk 1 x+k 2 y + k 3 t b (k 1 + k 2 + k 3 βdz, ϕ(z, x, y, t = (1/2a α(x 2 + y 2 + t 2 +b α(x+y + t + (α/8( 3 k2 i (5m2 1µ 2 4b 2 β(sds + e and a(z, b(z, e(z satisfy Eqs. (15 and (16. µ 1, a, b, e, ω, and k i (i = 1, 2, 3 are arbitrary form-factors, sn is the standard Jacobian elliptic sine functions with the modulus m ( < m < 1 and modulus parameter m describes the degree of the energy localization of the cnoidal waves. Solution 2 U(θ = cn(θ where u(z, t = exp γ(sds 1 s β(sds3/2 µ 2 ± µ 2 cn(θ, m 1/2 expiϕ(z, x, y, t, (22 θ = ω + α k 1 x + k 2 y + k 3 t b (k 1 + k 2 + k 3 ϕ(z, x, y, t = 1 2 a α(x 2 + y 2 + t 2 + b α(x + y + t α 8 βdz, k 2 i (4m 2 + 1µ b 2 β(sds + e and a(z, b(z, e(z satisfy Eqs. (15 and (16. µ 2, a, b, e, ω, and k i (i = 1, 2, 3 are arbitrary form-factors, cn is the standard Jacobian elliptic cosine functions with the modulus m ( < m < 1. The sn(θ, m and cn(θ, m functions have unity amplitude, oscillating character and can take both positive and negative signs. Period of the elliptical functions is given by 4K(m for cn(θ, m wave, and 2K(m for sn(θ, m wave, where K(m is the elliptical integral of the first kind. When m the integral K(m π/2, and when m 1 the integral K(m. When the localization parameter m is close to zero, sn(θ, m and cn(θ, m functions are well approximated by sin(θ and cos(θ with period 2π, that corresponds to the weak localization. With increase of the localization parameter m to 1 the contribution from the nonlinear terms increases, wave period goes to infinity and wave profile are well described by the hyperbolic type solution. Thus from solution (21, the kink similariton is with u(z, t = exp γ(sds 1 s β(sds3/2 µ 1 ± µ 1 tanh(θ 1/2 expiϕ(z, x, y, t, (23 θ = ω + α k 1 x + k 2 y + k 3 t b (k 1 + k 2 + k 3 ϕ(z, x, y, t = 1 2 a α(x 2 + y 2 + t 2 + b α(x + y + t + α 2 From solution (22, the corresponding similariton is with u(z, t = β(sds, k 2 i (µ 2 1 b2 β(sds + e. exp γ(sds 1 s β(sds3/2 µ 2 ± µ 2 sech(θ 1/2 expiϕ(z, x, y, t, (24 θ = ω + α k 1 x + k 2 y + k 3 t b (k 1 + k 2 + k 3 ϕ(z, x, y, t = 1 2 a α(x 2 + y 2 + t 2 + b α(x + y + t α 8 Note that the bright or dark similariton is decided by + branch and branch in solution (24, respectively. From solutions (23 and (24, we see the velocities of kink and bright or dark similariton are both determined by b (k 1 + k 2 + k 3 β(z, the chirps are both related to a α/2, the time shifts are both described by b (k 1 + k 2 + k 3 β(sds, the amplitudes are both determined by exp γ(sds/1 s β(sds3/2. The phase shifts are related to α 2 k 2 i (µ 2 1 b2 β(sds, β(sds, k 2 i α 8 (5µ b2 β(sds + e. k 2 i (5µ b 2 β(sds, respectively. Therefore, we can select the parameters β(z to control the formation of dark and bright similariton. Similarly, choosing c, c 2, c 4 from Table 1, we can get other eight families Jacobian elliptic function solutions and can obtain corresponding solitonic solutions or trigonometric function solutions. For the limit of length, we do not list them here. 4 Dynamical Behaviors of Self-Similar Waves The difference between soliton and similariton is that

5 No. 5 Communications in Theoretical Physics 875 pulses are chirped with s in the similariton case but chirp-free with s = in the soliton case. The soliton dynamics can be effectively controlled by specific soliton management conditions, e.g., by dispersion management of solitons, and soliton intensity management, etc. 1,1 12 Similarly to the soliton management, the similariton dynamics can also be controlled by the dispersion management function β(z. 4,6,9 Fig. 1 Propagation dynamics of (a chirped self-similar sn-wave intensity I = u 2, (b chirped kink similariton with s =.1 and (c chirp-free kink soliton with s =. The parameters are chosen ω =, σ = γ =.1, β =.1, p = r = b = η = µ 1 = µ 2 = 1, q = a = 2. Fig. 2 Propagation dynamics of (a chirped self-similar cn-wave intensity I = u 2, (b chirped bright similariton with s =.1 and (c chirp-free bright soliton with s =. The parameters are the same as that of Fig. 1. Fig. 3 Propagation dynamics of (a chirped self-similar cn-wave intensity I = u 2, (b chirped dark similariton with s =.1 and (c chirp-free dark soliton with s =. The parameters are the same as that of Fig. 1. As the first example, major attention will be paid on a periodic distributed amplification system 1,4,6,9 1 with the periodic varying diffraction/dispersion parameter β(z = β cos(ηzexp(σz, (25 but constant values of gain γ = γ. The parameters η and σ control the rate of dispersion change inside the fiber. In particular, the constant dispersion can be treated by α = σ =. Figures 1 3 show propagation dynamics of cnoidal waves and their corresponding similaritons. The parameters are chosen ω =, σ = γ =.1, s = β =.1, p = r = b = η = µ 1 = 1, q = a = 2, µ 2 = 1 for Figs. 1 and 2 and µ 2 = 1 for Fig. 3. The propagation properties of the snakelike kink and bright or dark similaritons are exhibited as follows. The velocities are all determined by.4 exp(.1z cos(z, the chirps are both related to 1 exp(.1zsin(z 1 4 exp(.1zcos(z, the time shifts are both described by.4 exp(.1z1 sin(z + cos(z.4, the amplitudes are both determined by exp(.1z1

6 876 Communications in Theoretical Physics Vol. 55 exp(.1zsin(z 1 4 exp(.1zcos(z 3/2. Compared the chirped similaritons with chirp-free solitons, one can see that the linear chirp leads to the periodical change of the pulse amplitude. Next, let us consider the compression problem of the laser pulse in a dispersion/diffraction decreasing fiber with the dispersion/diffraction and nonlinearity parameter 2 according to β(z = β exp( σz, χ(z = χ exp(δz, (26 where β >, χ >, and σ. In optical system (26, Eq. (19 yields the distributed gain γ(z = δ 2 + σ(λ 1 λ 1 + exp( σz, (27 with the parameter λ = σ/s β. It is clear to see that for chirped self-similar waves (s if the nonlinearity parameter χ(z is constant and λ = 1, i.e. δ = and s = σ/β, the gain can be completely balanced out. For chirp-free solitons (s =, if σ = δ/2, then the gain can be also completely balanced out. Fig. 4 (a Parameters of optical system (26; (b Amplitude and width of similaritons with δ =.2 and δ =.2. The other parameters are taken as ω =, β = χ =.6, σ = s =.1, p = r = b = 1, q = 2. Fig. 5 (a Kink similariton, (b bright similariton, and (c dark similariton. The other parameters are taken as ω =, β = χ =.6, σ = s =.1, δ =.2, p = r = b = µ 1 = µ 2 = 1, q = 2. When the parameters are taken as ω =, β = χ =.6, σ = s =.1, δ =.2, p = r = b = µ 1 = µ 2 = 1, q = 2, the parameters of optical system are shown in Fig. 4(a. In this dispersion/diffraction decreasing fiber system, the cubic and quintic nonlinearities and gain gradually decrease with the increasing distance z. When σ = δ/2, if σ >, δ >, then the amplitude of self-similar waves decreases for increasing z, whereas if σ >, δ <, then the amplitude of self-similar waves grows for increasing z. The width of chirped similaritons in Fig. 5 decreases for increasing z See Fig. 4(b. Figure 5 displays the evolutional scenario of the kink and bright or dark similaritons, where the amplitude raises and the width decreases dramatically for increasing z. The interaction of nonlinearities, diffraction/dispersion and gain causes the shape of input pulse to converge asymptotically to a pulse whose shape is self-similar. 5 Conclusions Via the general similarity transformation connecting with the solvable elliptic equation, we have obtained explicit self-similar cnoidal wave and similariton solutions of the (3+1-dimensional cubic-quintic nonlinear Schrödinger equation with spatially distributed coefficients. An exact balance condition between the dispersion/diffraction, cubic and quintic nonlinearities and the gain/loss has been obtained. Under this condition, we discuss dynamical behaviors of self-similar cnoidal waves and similaritons in two kind of typical soliton control systems. Compared the chirped similaritons with chirp-free solitons

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