Optical temporal rogue waves in the generalized inhomogeneous nonlinear Schrödinger equation with varying higher-order even and odd terms

Abstract

We investigate optical temporal rogue waves of the generalized nonlinear Schrödinger (NLS) equation with higher-order odd and even terms and space-and time-modulated coefficients, which includes the NLS equation, Lakshmanan-Porsezian-Daniel (LPD) equation, Hirota equation, Chen-Lee-Liu equation, and Kaup-Newell derivative NLS equation. Based on the similarity reduction method, the generalized NLS equation can be reduced to the integrable LPDHirota equation under a set of constraint conditions, from which the solutions of generalized NLS equation can be obtained in the basis of solutions of the LPDHirota equation and the similarity transformation. In particular, the first-and second-order self-similar rogue wave solutions of the generalized NLS equation are derived under different parameters, and the contour profiles and density evolutions of self-similar rogue wave solutions are given to study their wave structures and dynamic properties. At the same time, the motions of the hump and valleys related to the self-similar rogue waves are also given, from which we can control and manage the self-similar rogue waves by adjusting the third-order dispersion term. These results may be useful in nonlinear optics and related fields. KeywordsThe generalized nonlinear Schrödinger (NLS) equation · The LPD-Hirota equation · Similarity reduction method · Rogue wave solutions · Motions of the hump and valleys

Soliton solutions and traveling wave solutions of the two-dimensional generalized nonlinear Schrödinger equations

“…, (14) f (x, y, t) = 1 + 2 f 2 (x, y, t) + 4 f 4 (x, y, t) + ..., (15) h(x, y, t) = 1 + 2 h 2 (x, y, t) + 4 h 4 (x, y, t) + ..., (16) where g j , ( j = 1, 3, 5, ...) are the complex functions of x, y and t, and f n , h n , (n = 2, 4, 6, ...) are the real ones. Substituting expression ( 14)-( 16) into ( 9)- (11) and collecting the coefficients of the same power of , we have from Eq. ( 9)…”

“…The explicit values for the functions g 3 , g 5 , g 7 , f 2 , f 4 , f 6 , f 8 , h 2 , h 4 , h 6 , h 8 are determined from Eqs. ( 9)- (11). Note that g j = 0 for j = 9, 11, 13... and f n = 0, h n = 0 for n = 10, 12, 14....…”

“…One of the nonlinear equations is the nonlinear Schrödinger (NLS) equation which arises from a wide variety of fields, such as weakly nonlinear dispersive water waves, quantum field theory and nonlinear optics [1][2][3][4]. Different modifications and generalizations of the NLS equations were proposed and studied [5][6][7][8][9][10][11]. There are various methods to study nonlinear equations, such as the Darboux transformation [12][13][14][15][16], the Hirota method [17][18][19][20][21], the sine-cosine [22,23], the extended tanh method [24][25][26], and so on.…”

Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrodinger equation

“…In recent years, many authors [18][19][20] have reported the higher-order rogue wave solutions, some important physical properties and applications for the NLS equation. In addition, various extensions of NLS equation have also been studied, such as pair-transition-coupled NLS equation [21], variable coefficient NLS equation [22][23][24], three-component NLS equations [25], three-component coupled derivative NLS equations [26], and n-component NLS equations [27]. General high-order solitons of three different types of nonlocal NLS equations in the reverse-time, PT-symmetric and reverse-space-time were derived by using a Riemann-Hilbert treatment [28].…”

Lump solitons in a higher-order nonlinear equation in2+1dimensions

“…[6] and [7] as well as in Refs. [9] and [10]. The relevance of third-order dispersion terms in the context of the self-induced Raman effect have been pointed out by * Corresponding author: pilar@usal.es…”