Bright and Dark Solitons in Generalized Kerr-Like Media

Abstract

w U ni versity of T echnolo gy K oszy kowa 75, 00-662 W arszawaIn the pap er the propagation of short pulses in nonli near K err-like me dium is considered . Pulses are describ ed by the nonlinear Schr odinger equation in w hich the generalizati on of nonlinear disp ersion term and the self-frequency shif t are taken into account. T he solution of the equation in the form of quadrature is derived. Both the phase and pro Ùle of the pulse intensity are obtained. T he cases of propagating bright and dark solitons in saturable K err-like medium are analyzed. T w o mo dels of saturation are considered. T he explic it analytical expressions describing the K err limit are rep orted. T he relations b etw een the medium and propagatio n parameters are thoroughly discussed .PACS numb ers: 42. 65. T g, 42. 82. Et 1. I n t r o d u ct io n I n no nl i near Kerr di electri c stro ng l i ght p ul ses of specia l shap e (sol i to ns) pro pagate wi tho ut changi ng i ts for m due to the com p ensati on of two e˜ects | the l i near gro up vel ocit y di spersion ( G V D ) and nonl i near p ol ari zati on of the medi um (self-pha se m odul atio n, SP M) [1{ 3]. Pro pagati on of these pul ses i s descri b ed by no nl i near Schr odi nger equa ti on (N LSE).The stro ng electri c Ùeld of sol i to n can cause ma ny other e˜ects in the Kerr di electri c. One of these e˜ects | the thi rd or hi gher order di spersion | i s l i near [4{ 6]; the others are no nl i near. In the pap er two of such e˜ects are considered | nonl i near di spersio n (ND ) [6{ 9] and self-frequency shif t (SFS) [10,11]. The NLSE equati on general i zed by these (G NLSE) term s al so p ossesse s sol uti ons i n the form of sol i to ns, but special requi rem ents b etween para m eters of the medi um and sol i to n are needed i n order to preserve the ori gina l pro Ùle fro m the pure Kerr di electri c [6, 12{ 16].In the presence of stro ng soli to n Ùeld the nonl i near perm i tti vi ty of the m edi um can devi ate fro m the sim pl e Kerr dep endence reveal ing satura ti o n. The wa y we shoul d m odi fy i t depends on m odel of satura ti on. By no w the tw o-level m odel [17], the exp onenti al mo del [18 ], \ square two-level" [19], and pol yno m ial (7 7)

Fast Magic‐Angle‐Spinning NMR Reveals the Evasive Hepatitis B Virus Capsid C‐Terminal Domain**

“…Still, line broadening can be induced by internal motions, in particular on intermediate time scales, [4] at a correlation time comparable with the inverse magic‐angle‐spinning (MAS) rotation frequency ( τ c about 1 μs), which leads to incoherent, i.e. relaxation‐induced, line broadening [7–11] . This is shown in Figure 1a, where incoherent line broadening is calculated for an isolated 1 H‐ 15 N two‐spin system.…”

Fast Magic‐Angle‐Spinning NMR Reveals the Evasive Hepatitis B Virus Capsid C‐Terminal Domain**

“…relaxation-induced, line broadening. [7][8][9][10][11] This is shown in Figure 1a, where incoherent line broadening is calculated for an isolated 1 H- 15 N two-spin system. It can be seen that for motions at a time scale similar to the MAS frequency, substantial line broadening can occur, notably at slower MAS frequencies.…”

“…Still, line broadening can be induced by internal motions, in particular on intermediate time scales, [4] at a correlation time comparable with the inverse magic‐angle‐spinning (MAS) rotation frequency ( τ c about 1 μs), which leads to incoherent, i.e. relaxation‐induced, line broadening [7–11] . This is shown in Figure 1a, where incoherent line broadening is calculated for an isolated 1 H‐ 15 N two‐spin system.…”

Making the invisible visible: fast magic-angle-spinning NMR reveals the evasive hepatitis B virus capsid functional C-terminal domain

“…Still, line broadening can be induced by internal motions, in particular on intermediate time scales [4] , at a correlation time comparable with the inverse magic-angle-spinning (MAS) rotation frequency (𝝉c about 1 µs), which leads to incoherent, i.e. relaxation-induced, line broadening [7][8][9][10][11] . This is shown in Figure 1b, where incoherent line broadening is calculated for an isolated 1 H-15 N two-spin system.…”