Radiationless Higher-Order Embedded Solitons

Abstract

The soliton solutions of a higher-order nonlinear Schrödinger equation including nonlinearities of orders 2b þ 1 and 4b þ 1, as well as second-and fouth-order dispersive terms, are obtained. It is shown that these solitons may be (or may not be) embedded, depending on the coefficients of the equation. The radiationless character of the embedded solitons is explained by analyzing the Fourier transform of the equation. The stability of these solitons is studied analytically and numerically. In the analytical approach, we combined the Vakhitov-Kolokolov stability criterion with the standard variational method. This combined technique predicts that the standard solitons are unstable and the embedded ones might be stable. The numerical results confirm these predictions, and show that the embedded and standard solitons respond very differently to perturbations.

Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion

“…In the case when the inequality (17) holds, it is obvious that the solitons defined by Equations (81)-(84) are embedded, since in this case the wavenumbers permitted by the linear dispersion relation (15) cover the entire real axis (as explained in Section 2), and therefore the solitons' wavenumbers H 1,2 will necessarily be contained in the range of this dispersion relation. On the other hand, when the inequality (18) holds, the range of the dispersion relation will contain a band of forbidden wavenumbers, and the boundaries of this band are the values k 1 and k 2 given by Equation (20). In this case it is not evident if the solitons defined by Equations (81)-(84) are embedded or not.…”

“…However, if desired, it is also possible to use the procedure shown in Refs. [18] and [20] to construct a rigorous proof (not just a qualitative one) which shows that the absence of resonances between the solitons and the radiation modes is the consequence of a delicate balance between the linear and the nonlinear terms of Equation (6). This procedure consists in taking the FT of Equation (6), using a function of the form (52) to calculate the FT of the nonlinear terms.…”

“…(a) −iu ttt and/or u 4t : these higher-order derivatives are necessary to describe sub-picosecond pulses [1][2][3][4][5][6][7][8][9][10][11][12]; in particular, conditions for including u 4t and discarding −iu ttt are discussed in [7,9], (b) |u| 4 u: this higher-order nonlinearity is used when we want to describe the propagation of pulses when the light intensity approaches the values which produce the "saturation" of the refractive index [13][14][15][16][17][18][19][20], (c) i(|u| 2 u) t : this term is necessary to describe the self-steepening of the optical pulses [21][22][23], (d) u(|u| 2 ) t : this term is associated with the effect of Raman scattering [24][25][26][27].…”

Soliton dynamics of a high-density Bose-Einstein condensate subject to a time varying anharmonic trap

“…In the present article we will use a variational method, as these methods have been very successful to obtain approximate solutions of nonlinear equations similar to Eq. (3), which describe the propagation of light pulses in optical fibers and liquid crystals [29][30][31][32][33].…”

Stability and Metastability of Trapless Bose-Einstein Condensates and Quantum Liquids

“…As a starting point, we emphasize that our stability studies should not be confused with a conventional stability analysis of optical solitons and other objects whose evolution is also governed by nonlinear wave equations [57][58][59][60][61][62]. The underlying physics of quantum molecular gases and liquids is very different from that of optical fibers and other electromagnetic (EM) materials.…”