Inverse scattering transform analysis of Stokes–anti-Stokes stimulated Raman scattering

Gerdjikov, Vladimir S.; Kostov, N. A.

doi:10.1103/physreva.54.4339

Cited by 10 publications

(9 citation statements)

References 40 publications

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“…The choice s 1 = −s 2 = −1 leads to the noncompact real form so(2, 3) and s 1 = s 2 = −1 gives another noncompact one-so (1,4). If in this last case we identify Q 01 = Q pol , Q 10 = −E s , Q 11 = E p and Q 12 = −E a then we obtain the system studied in [23] which describes Stockes-anti-Stockes wave generation. Here Q pol is the normalized effective polarization of the medium and E p , E s and E a are the normalized pump, Stockes and anti-Stockes wave amplitudes respectively.…”

Section: Real Forms Of Bmentioning

confidence: 99%

N -wave interactions related to simple Lie algebras. ℤ ₂ -reductions and soliton solutions

et al. 2001

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Abstract.The reductions of the integrable N -wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra g are analyzed. The Zakharov-Shabat dressing method is extended to the case when g is an orthogonal algebra. Several types of one soliton solutions of the corresponding N -wave equations and their reductions are studied. We show that to each soliton solution one can relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave equations related to so(5) which find applications in Stockes-anti-Stockes wave generation.

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Section: Real Forms Of Bmentioning

confidence: 99%

N -wave interactions related to simple Lie algebras. ℤ ₂ -reductions and soliton solutions

et al. 2001

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“…then we obtain the system similar to that studied in [1,11] which describes Stokes-anti-Stokes wave generation. Here Q pol (x, t) is the normalized effective polarization of the medium and E p (x, t), E s (x, t) and E a (x, t) are the normalized pump, Stokes and anti-Stokes wave amplitudes respectively.…”

Section: Physical Applications Of 4-wave Equation With a Z 2 Reductionmentioning

confidence: 99%

N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions

Gerdjikov

Kostov²,

Valchev³

2007

SIGMA

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Abstract. We consider N -wave type equations related to the orthogonal algebras obtained from the generic ones via additional reductions. The first Z 2 -reduction is the canonical one. We impose a second Z 2 -reduction and consider also the combined action of both reductions. For all three types of N -wave equations we construct the soliton solutions by appropriately modifying the Zakharov-Shabat dressing method. We also briefly discuss the different types of one-soliton solutions. Especially rich are the types of one-soliton solutions in the case when both reductions are applied. This is due to the fact that we have two different configurations of eigenvalues for the Lax operator L: doublets, which consist of pairs of purely imaginary eigenvalues, and quadruplets. Such situation is analogous to the one encountered in the sine-Gordon case, which allows two types of solitons: kinks and breathers. A new physical system, describing Stokes-anti Stokes Raman scattering is obtained. It is represented by a 4-wave equation related to the B 2 algebra with a canonical Z 2 reduction.

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“…It is described by the following Hamiltonian: In the case η = −1 if we identify q 10 = Q, q 11 = E p , q 21 = E a and q 1 = E s , where Q is the normalized effective polarization of the medium and E p , E s and E a are the normalized pump, Stokes and anti-Stokes wave amplitudes respectively, then we obtain the system of equations generalizing the one studied in [23] which describes Stokesanti-Stokes wave generation. This approach allowed us to derive a new Lax pair for (4.19).…”

Section: )mentioning

confidence: 99%

“…In the case when the reduction is of the type (2.1), i.e. p α = s α q * α then (6.21) is equivalent to 23) and can be interpreted as relations between the densities |q α | 2 of the 'particles' of type α. For the other types of reductions such interpretation is not so obvious.…”

Section: )mentioning

confidence: 99%

Reductions ofN-wave interactions related to low-rank simple Lie algebras: I. ℤ₂-reductions

Gerdjikov¹,

Grahovski²,

Kostov³

2001

J. Phys. A: Math. Gen.

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Abstract. The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem. We show how the second order reductions of the N -wave interactions related to low-rank simple Lie algebras g can be embedded also in the Weyl group of g. This allows us to display along with the well known ones a number of new types of integrable N -wave systems. Some of the reduced systems find applications to nonlinear optics.

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Inverse scattering transform analysis of Stokes–anti-Stokes stimulated Raman scattering

Abstract: Zakharov-Shabat-Ablowitz-Kaup-Newel-Segur (ZS-AKNS) representation for Stokes-anti-Stokes stimulated Raman scattering (SRS) is proposed. Periodical waves, solitons and self-similarity solutions are derived. Transient and bright threshold solitons are discussed.

Cited by 10 publications

References 40 publications

N -wave interactions related to simple Lie algebras. ℤ ₂ -reductions and soliton solutions

N -wave interactions related to simple Lie algebras. ℤ ₂ -reductions and soliton solutions

N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions

Reductions ofN-wave interactions related to low-rank simple Lie algebras: I. ℤ₂-reductions

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Inverse scattering transform analysis of Stokes–anti-Stokes stimulated Raman scattering

Abstract: Zakharov-Shabat-Ablowitz-Kaup-Newel-Segur (ZS-AKNS) representation for Stokes-anti-Stokes stimulated Raman scattering (SRS) is proposed. Periodical waves, solitons and self-similarity solutions are derived. Transient and bright threshold solitons are discussed.

Cited by 10 publications

References 40 publications

N -wave interactions related to simple Lie algebras. ℤ 2 -reductions and soliton solutions

N -wave interactions related to simple Lie algebras. ℤ 2 -reductions and soliton solutions

N-Wave Equations with Orthogonal Algebras: Z2 and Z2 × Z2 Reductions and Soliton Solutions

Reductions ofN-wave interactions related to low-rank simple Lie algebras: I. ℤ2-reductions

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N -wave interactions related to simple Lie algebras. ℤ ₂ -reductions and soliton solutions

N -wave interactions related to simple Lie algebras. ℤ ₂ -reductions and soliton solutions

N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions

Reductions ofN-wave interactions related to low-rank simple Lie algebras: I. ℤ₂-reductions