3D bright-bright Peregrine triple-one structures in a nonautonomous partially nonlocal vector nonlinear Schrödinger model under a harmonic potential

Similar dynamical properties of coupled solitons have been heavily reported in previous literatures. This paper focuses on the different excitation management of two components for 3D bright-bright Peregrine triple-one structures in a nonautonomous partially nonlocal vector nonlinear Schrödinger model under a harmonic potential by contrasting the maximum accumulated time value with the excited value for each Peregrine peak. Furthermore, the different diffraction values in two transverse directions affect the bright-bright Peregrine triple-one structure configurations and amplitude and width of the Peregrine structure. This study extends the management of rogue wave to the partially nonlocal phenomena of optical wave, matter wave and other nonlinear waves.

Abundant nonlinear evolutional systems, including nonlinear Schrödinger system(NLSS) [27], Hirota system [28,29] and their coupled versions [30], have analytical PS solutions, including PS, PS triplet, PS quartet and corresponding vector solutions [31]. These structures theoretically found also were verified by the experiments in water tank [32,33] and optical fiber [34,35]. These experiments in turn require theoretical research to control the excitation of PS. Under this requirement, the control for PS have been theoretically investigated on the basis of nonautonomous NLSS [36,37].
In recent year, nonlocal soliton dynamics was paid attention with the help of the nonlocal NLSS [38]. More recently, the study focused on nonautonomous partially nonlocal(NPN) soliton dynamics [39][40][41] by utilizing the partially nonlocal NLSS [42,43]. Rogue waves and their coupled structures were also discussed by using the NPN-NLSS [42,43]. However, in these literatures [41,44,45], dynamical properties of coupled solitons show similar behaviors by using similar forms of coupled component solutions. Therefore, some questions are focused on: how different behaviors of dynamical properties of coupled NPN soliton component are studied by using coupled NPN-NLSS? How different influence for different two transverse diffractions on the NPN soliton dynamics is found?
This manuscript aims to consider two questions above by using the coupled NPN-NLSS with two different transverse diffractions as where complex functions U k ðt; x; y; zÞ; k ¼ 1; 2; with spatiotemporal variables t, x, y, z represent two components of optical envelopes or order parameters in BECs. Terms including b 1 ðtÞ and b 2 ðtÞ respectively depict two transverse x and y-directional diffractions, and terms including q 1 ðtÞ and q 2 ðtÞ respectively portray two transverse x and y-directional harmonic modulation of the refractive index or potential. Term including vðtÞ depicts the z-directional nonlocal and two transverse xÀ and y-directional localized nonlinearity. Coupled nonlinearity coefficients c k;k ; c k;3Àk describe polarized eigenmodes. When c k;k ¼ c k;3Àk ¼ 1, the self-phase and cross-phase modulation are same. Moreover, c k;k ¼ 1:5c k;3Àk ; 0:5c k;3Àk \c k;k \1:5c k;3Àk ; c k;k ¼ 0:5c k;3Àk depict respectively polarized eigenmodes with linearity, ellipticity and circularity in optics [46,47] or nonlinear coupling intra-and interspecies atomic interactions in BEC [48,49]. Based on Eq. (1) with U 1 ¼ U 2 and b 1 ðtÞ ¼ b 2 ðtÞ; q 1 ðtÞ ¼ q 2 ðtÞ ¼ 0; c k;k ¼ c k;3Àk ¼ 1, soliton and vortex structures were discussed in Ref. [50]. Based on Eq. (1) with b 1 ðtÞ ¼ b 2 ðtÞ; q 1 ðtÞ ¼ q 2 ðtÞ ¼ 0; c k;k ¼ c k;3Àk ¼ 1, spatiotemporal solitons were studied [51], where only similar dynamical behaviors of coupled solitons were given. In this manuscript, we will study different dynamical behaviors of coupled NPN soliton components.

Solution via reduction processing with Darboux method
The reduction processing begins with the substitution of the transformation U k ðt; x; y; zÞ ¼ aðzÞcðtÞN k ½gðx; y; tÞ; dðtÞ exp ½ijðt; x; y; zÞ; ð2Þ with the spatiotemporal amplitude modulations aðzÞand cðtÞ, reduction parameter gðx; y; tÞ, accumulated time dðtÞ and phase jðt; x; y; zÞ, into Eq.(1) with the complex function N m ðg; dÞ meeting the coupled NLSS ð3Þ with real constants B and C. Thus partial differential equations are produced as which result in physical quantities in Eq.(2) as (i) Spatial amplitude modulation with the Hermite polynomial H q ðbzÞ for the nonnegative integer q, and the temporal amplitude modulation as (ii) Reduction parameter and (iv) Phase Moreover, this reduction (2) with (7)-(11) is valid when and with W 1 ðtÞ ¼ 1=lðtÞ and W 2 ðtÞ ¼ 1=mðtÞ.
and R ¼ ða; d; f Þ T with the matrix transpose sign ''T 00 and the complex conjugation sign ''*'' for the complex spectral parameter k.
With the help of nonrecursive Darboux transformation (DT) method [52], the pth-order vector Peregrine solutions read with the seed solutions N 10 and N 20 in the form of the plane wave, ''y 00 denoting the complex-conjugate transpose and 1 Â p row vectors L k ðk ¼ 1; 2; 3Þ as where K ðlÞ represent the Taylor expansion coefficients of the factorized column vector KðkÞ ¼ H À1 RðkÞ with H ¼ diagð1; N Ã 10 =jN 10 j; N Ã 20 =jN 20 jÞ when the spectral parameter is k 0 . The elements K jl of p Â p matrix K can be expressed from the Taylor expansion by , and applying the relation (2) with the nonrecursive DT method above, vector secondorder (p ¼ 2) Peregrine solution of Eq. (1) has M 0 ¼l 1 þ 2l 2 N þ 4il 3 ð-À 2iNÞ; N 0 ¼e 1 l 1 þ e 2 l 2 þ e 3 l 3 þ l 4 þ 2l 5 N þ 4il 6 ð-À 2iNÞ; where

3D bright-bright Peregrine triple-one structure
In the frame of the constant-coefficient coupled NLSS (3), bright-bright Peregrine triple-one structure can be constructed. In Fig. 1, components N 1 and N 2 both show bright Peregrine triple-one structures with different excitation directions. Here we use respectively the sign I,II,III to label the Peregrine triple structures, and i to label the Peregrine one structure. They are excited separately at d III ¼ 17 The analysis of excitations for Peregrine triple-one structure can be conducted by utilizing the exponential diffraction control condition [53,54] Fig. 2. From Fig. 2a, b, two components U 1 and U 2 both show bright Peregrine triple-one structures, but have different configurations. For the fixed value of b 20 , with the enlarging value of b 10 , besides the configuration appears the difference (c.f. Fig. 2a, b, c and d), the amplitude adds and the width along the y-direction declines. For the fixed value of b 10 , with the increasing value of b 20 , besides the configuration appears the difference (c.f. Fig. 2c, d, e and f), the amplitude declines and the width along the y-direction increases.
The isosurface figures of entirely excited brightbright Peregrine triple-one structures for two components U 1 and U 2 and total intensity are shown in Fig. 3. When q ¼ 0 in Fig. 3a, b, the single layer structure is produced along the z-direction. With the enlarging value of k, the Peregrine triple-one structure along the z-direction turns into the k þ 1-layer configuration. For q ¼ 1; 2, two and three layer structures for two components U 1 and U 2 and total intensity are respectively exhibited along the z-direction in Fig. 3d, f, g and i.
When d M is slightly larger than d III , the dragged-tail Peregrine structures labelled as ''III'' in bright-bright Peregrine triple-one structures ( Fig. 1a and b) for two components U 1 and U 2 are engendered in Fig. 4a and   Fig. 1a and b) for two components U 1 and U 2 both display the peak maintenance, namely, they both are excited to their maximum amplitudes in Fig. 4c and d, where other Peregrine structures [labelled as ''II,i,I''] are also excited to entire formations. For d M \d III , the Peregrine structures labelled as ''III'' in bright-bright Peregrine tripleone structures ( Fig. 1a and b) for two components U 1 and U 2 both show the beginning excitation in Fig. 4e and f, where other Peregrine structures [labelled as ''II,i,I''] are also excited to entire formations.
Similarly, when d M is slightly larger than d II ' d i , the dragged-tail Peregrine structures labelled as ''II'' and ''i'' in bright-bright Peregrine triple-one structures ( Fig. 1a and b) for two components U 1 and U 2 are engendered in Fig. 5a  If d M is larger and smaller than d I , the Peregrine structures labelled as ''I'' for two components U 1 and U 2 are both excited to entire formations and beginning shapes, respectively. If d M equals to d I , the Peregrine structures labelled as ''I'' for two components U 1 and U 2 both display the peak maintenance. Due to the length restrictions, we do not show the related plots, which similar to the excitation control of the single rogue waves in Ref. [55].

Conclusions
In summary, via the reduction processing with the nonrecursive Darboux method, analytical solutions of the coupled NPN-NLSS under a harmonic potential are derived. Different excitation management of two components for 3D bright-bright Peregrine triple-one structures is reported by contrasting the maximum accumulated time value with the excited value for each Peregrine peak. Furthermore, the different diffraction values in two transverse directions affect the bright-bright Peregrine triple-one structure configurations and amplitude and width of the Peregrine structure. For the fixed value of b 20 , with the enlarging value of b 10 , besides the configuration appears the difference (c.f. Fig. 2a, b, c and d), the amplitude adds and the width along the y-direction declines. For the fixed value of b 10 , with the increasing value of b 20 , besides the configuration appears the difference (c.f. Fig. 2c, d, e and f), the amplitude declines and the width along the y-direction increases. This study extends the management of rogue wave to the partially nonlocal phenomena of optical wave, matter wave and other nonlinear waves.
The neural network model can be regarded as a test function to get exact analytical solutions of the nonlinear model [8,[56][57][58][59] and many localized structures are studied by this method. Therefore, in the future work, we can also try to investigate the bright-bright Peregrine triple-one structures obtained in this paper via the neural network method such as the bilinear neural network method.