2001
DOI: 10.1103/physreve.64.016603
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Traveling solitons in the parametrically driven nonlinear Schrödinger equation

Abstract: We show that the parametrically driven nonlinear Schrödinger equation has wide classes of travelling soliton solutions, some of which are stable. For small driving strengths stable nonpropagating and moving solitons co-exist while strongly forced solitons can only be stable when moving sufficiently fast. PACS number(s): 05.45.Yv, 05.45.Xt

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Cited by 44 publications
(54 citation statements)
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“…Some parts of the stability analysis of the traveling dark solitons [28] can be carried over to the case of the traveling solitons of the NLS equation with a driving term. Namely, one can show [29] that a pair of linearized eigenvalues crosses from the imaginary to real axis at the valueṼ where dP /dṼ = 0. The sign of the derivative dP /dṼ required for stability depends on the type of the soliton; some classes of solitons require dP /dṼ < 0, whereas other classes are stable when dP /dṼ > 0.…”
Section: U + δU = R[u(xt); Xt]mentioning
confidence: 99%
“…Some parts of the stability analysis of the traveling dark solitons [28] can be carried over to the case of the traveling solitons of the NLS equation with a driving term. Namely, one can show [29] that a pair of linearized eigenvalues crosses from the imaginary to real axis at the valueṼ where dP /dṼ = 0. The sign of the derivative dP /dṼ required for stability depends on the type of the soliton; some classes of solitons require dP /dṼ < 0, whereas other classes are stable when dP /dṼ > 0.…”
Section: U + δU = R[u(xt); Xt]mentioning
confidence: 99%
“…If we fix h and continue in γ towards γ = 0, the separation distance between the solitons in the complex grows without bounds; hence we conjecture that symmetric multisoliton complexes do not exist for γ = 0. (There are nonsymmetric complexes with γ = 0 though; see [14]. )…”
Section: Stationary Two-soliton Complexesmentioning
confidence: 99%
“…The soliton ψ + is stable when the difference h − γ is small but loses its stability to a time-periodic soliton when h exceeds a certain limit h Hopf (γ). The solitons ψ + and ψ − can form a variety of bound states, or complexes [10,14,15]. (For example, in the previous paper [1] we mentioned a complex ψ (−+−) , that is, a symmetric stationary association of two solitons ψ − and one ψ + .)…”
Section: Stationary Two-soliton Complexesmentioning
confidence: 99%
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