The two-parameter soliton family for the interaction of a fundamental and its second harmonic

Abstract

Additional Information:• This is a pre-print. The definitive version: GRIMSHAW, KUZNETSOV and SHAPIRO, 2001. The two-parameter soliton family for the interaction of a fundamental and its second harmonic. Physica D, 152, Metadata Record: https://dspace.lboro.ac.uk/2134/766Please cite the published version.The two-parameter soliton family for the interaction of a fundamental and its second harmonic For a system of interacting fundamental and second harmonics the soliton family is characterized by t wo independent parameters, a soliton potential and a soliton velocity. It is shown that this system, in the general situation, is not Galilean invariant. As a result, the family of movable solitons cannot be obtained from the rest soliton solution by applying the corresponding Galilean transformation. The region of soliton parameters is found analytically and con rmed by n umerical integration of the steady equations. On the boundary of the region the solitons bifurcate. For this system there exist two kinds of bifurcation: supercritical and subcritical. In the rst case the soliton amplitudes vanish smoothly as the boundary is approached. Near the bifurcation point the soliton form is universal, determined from the nonlinear Schrodinger equation. For the second type of bifurcations the wave amplitudes remain nite at the boundary. In this case the Manley-Rowe i n tegral increases inde nitely as the boundary is approached, and therefore according to the VK-type stability criterion, the solitons are unstable.

Solitons and collapses: two evolution scenarios of nonlinear wave systems

“…To conclude this section, we discuss one example in which, based on the combined method, it is possible to draw a more or less definitive conclusion about the soliton stability by numerically integrating system (73), (74). The dependences of H and N (for one-dimensional soliton solutions) on l was numerically found in [56] for a nonzero frequency detuning O T 0. Numerical integration has shown that for O`0, both dependences are monotonic: N increases with l and H decreases.…”

Solitons in quadratic media

“…More recently, the same questions concerning solitonic states have been scrutinized for ( ) χ 2 media (see [34]), where the nonlinear susceptibility tensor only involves quadratic terms, thus leading to quadratic NLS equations (see [23], [27], [35], [38], [40], [41], [42], [45]). This case is particularly mathematically interesting since the corresponding Cauchy problems are globally well-posed for any data in the relevant Sobolev classes and so the possible instability of solitary waves cannot be due, as in the cubic case, to finite time blow-up.…”

Numerical Method for 2d Soliton Solution at SHG in Media With Time‐dependent Combined Nonlinearity

“…As it is well known, soliton is a name for a solitary wave, which does not change in the direction of propagation of optical pulse in medium. Modern laser equipments make it possible to realize the so-called colored solitons, when optical waves at several frequencies exist and propagate simultaneously along the nonlinear medium [2,3,4,5,6,9,10,11,12,13,14,15,16,17].…”