Conservation Laws in Higher-Order Nonlinear Optical Effects

Conservation laws of the nonlinear Schrödinger equation are studied in the presence of higher-order nonlinear optical effects including the third-order dispersion and the selfsteepening. In a context of group theory, we derive a general expression for infinitely many conserved currents and charges of the coupled higher-order nonlinear Schrödinger equation. The first few currents and charges are also presented explicitly. Due to the higher-order effects, conservation laws of the nonlinear Schrödinger equation are violated in general. The differences between the types of the conserved currents for the Hirota and the Sasa-Satsuma equations imply that the higher-order terms determine the inherent types of conserved quantities for each integrable cases of the higher-order nonlinear Schrödinger equation. 1 Electronic address; [email protected] 2 Electronic address; [email protected] 3 Electronic address; [email protected] In the ultrafast optical signal system, the higher-order nonlinear effects such as the third-order dispersion, the self-steepening, and the self-frequency shift become important if the pulses are shorter than T0 ≤ 100fs [1]. The use of optical pulses with distinct polarizations and/or frequencies also require the consideration of nonlinear cross-couplings between different modes of pulses. Inclusion of both the higher-order and the cross-coupling effects lead to the study on the coupled higher-order nonlinear Schrödinger equations (CHONSE) which are not in general integrable except for special cases of coupling constants. Those integrable cases of coupling constants have been classified in association with Hermitian symmetric spaces [2]. It is also well known that soliton equations which can be integrated by the inverse scattering method possess infinite number of conserved quantities. For example, the nonlinear Schrödinger equation (NSE) has infinite number of conserved charges in addition to the ones corresponding to the energy and the intensityweighted mean frequency. However, the effect of the higher-order and the cross coupling terms on the conservation laws has not been considered up to now. In this paper, we make a systematic study on the conservation laws in the presence of the higher-order and the cross-coupling terms. We first indicate that except for the energy conservation, other conservation laws of the NSE such as the conservation of the intensity-weighted mean frequency do not hold due to the higher-order effects any more, unless the higher-order terms are of a unique type. In the case of integrable CHONSE, we derive general expressions of infinite number of conserved currents and charges from the Lax pair formulation utilizing the properties of the Hermitian symmetric space. From the general expressions, explicit forms of the first few conserved currents and the associated charges of the Hirota and the Sasa-Satsuma equations are calculated. We then explain the correlations of conservation laws between the two integrable cases of the higher-order extension of the NSE. In order to illustrate the issue, we first consider the NSE including the higher-order terms. In a mono-mode optical fiber, the propagation of a ultrashort pulse is governed by the higher-order nonlinear Schrödinger equation [3] ∂̄ψ = i(γ1∂ ψ + γ2|ψ| ψ) + γ3∂ ψ + γ4∂(|ψ| ψ) + γ5∂(|ψ| )ψ, (1) where ∂̄ ≡ ∂/∂z̄ and ∂ ≡ ∂/∂z are derivatives in retarded time coordinates (z̄ = x, z = t− x/v), and ψ is the slowly varying envelope function. The real coefficients γi (i = 1, 2, 3, 4) in the first four terms on the right hand side of Eq. (1) specify in sequence the effects 1 of the group velocity dispersion, the self-phase modulation, the third order dispersion, and the self-steepening. With appropriate scalings of space, time, and field variables, one can readily normalize Eq. (1) so that γ1 = 1, γ2 = 2, γ3 = 1 which we assume from now on. The remaining coefficient γ5 in the last term is complex in general. The real and the imaginary parts of γ5 are due to the effect of the frequency-dependent radius of fiber mode and the effect of the self-frequency shift by stimulated Raman scattering, respectively. It is well known that the above equation becomes integrable if γ4 = −γ5 = 6 (Hirota case) [4] or γ4 = −2γ5 = 6 (Sasa-Satsuma case) [5]. In the absense of higher-order terms (γ3 = γ4 = γ5 = 0), Eq. (1) possesses infinite number of conserved charges among which the first three charges [6] are Q1 = ∫ ∞ −∞ |ψ|dt , Q2 = i ∫ ∞ −∞ (ψ∗∂ψ − ∂ψ∗ψ)dt , Q3 = ∫ ∞ −∞ (∂ψ∗∂ψ − |ψ|)dt, (2) where Q1 represents conserved energy, and Q2 the mean frequency weighted by the intensity of optical pulses. In the conventional NSE where the time and the space coordinates are interchanged, Q1, Q2 and Q3 respectively correspond to conserved mass, momentum and energy. If we include higher-order terms, Qi are not necessarily conserved but subject to the relations; ∂̄Q1 = 0 , ∂̄Q2 = 2i(γ4 + γ5) ∫ ∞ −∞ ∂|ψ| , (ψ∗∂ψ − ∂ψ∗ψ)dt ∂̄Q3 = (3γ4 + 2γ5 − 6) ∫ ∞ −∞ ∂|ψ|∂ψ∂ψdt . (3) The calculations indicate that the charge Q1 which corresponds to energy is conserved for all values of γ4, γ5 while Q2 and Q3 are conserved provided γ4+γ5 = 0 and 3γ4+2γ5 = 6, respectively. Note that Q2 and Q3 are conserved simultaneously only for the specific value γ4 = −γ5 = 6 that is precisely the Hirota case. It is interesting to observe that integrability does not always imply the same types of conserved currents in the presence of higher-order terms. Another integrable case of the Sasa-Satsuma equation, where γ4 = −2γ5 = 6, in fact does not have Q2 and Q3 in Eq. (3) as the conserved charges. This consequence is rather remarkable in view of the fact that integrable equations possess infinite number of 2 conserved quantities. We will show, however, the Sasa-Satsuma equation also possesses infinitely many conserved charges of different types other than the ones of the Hirota equation. In case we include both the higher-order and the cross-coupling nonlinear effects, the propagating system is governed by a CHONSE. Without understanding physical settings, it would be meaningless to write down any general expression of the CHONSE. However, as explicitly derived in [2], there exists a group theoretic specification which admits a systematic classification of integrable cases of the CHONSE. In the following, we consider a group theoretic generalization of the NSE and define the CHONSE in association with a Hermitian symmetric space. By solving the linear Lax equations iteratively, we derive infinite number of conserved currents and charges for the CHONSE. For the later use, now we briefly review the definition of Hermitian symmetric spaces [7, 8] and the generalization of the NSE [2, 9] according to the Hermitian symmetric spaces. A symmetric space is a coset space G/K for Lie groups G ⊃ K whose associated Lie algebras g and k, with the decomposition: g = k⊕m, satisfy the commutation relations; [k, k] ⊂ k, [m, m] ⊂ k, [k, m] ⊂ m (4) A Hermitian symmetric space is the symmetric space G/K equipped with a complex structure. One can always find an element T in the Cartan subalgebra of g whose adjoint action defines a complex structure and also the subalgebra k as a kernel, i.e., k = {V ∈ g : [T, V ] = 0}. That is, the adjoint action J ≡ adT = [T, ∗] is a linear map J : m → m that satisfies the complex structure condition, J = −I, or [T, [T, M ]] = −M forM ∈ m. Then, we define a CHONSE as ∂̄E = ∂Ẽ − 2EẼ + α(∂E + β1E ∂E + β2∂EE ) (5) where E and Ẽ ≡ [T, E] are extended field variables belonging to m. The arbitrary constant α may be normalized to 1 by an appropriate scaling but we keep it in order to exemplify the higher-order effects. Also the cross-coupling effects between different modes of polarizations or frequencies are accommodated in the matrix form of E which is determined by each Hermitian symmetric space. For example, in the case where G/K = We restrict to symmetric spaces AIII, CI and DIII only so that the expression of CHONSE becomes simplified [2].