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The evolution of spatial solitons in the photovoltaic photorefractive crystal can be governed by the specific coupled nonlinear Schrödinger equations. Under the photovoltaic field with the external bias field, the coupled cn–sn-type periodic wave solution and the corresponding photorefractive bright–dark soliton pair were constructed to describe the evolution of beam. The influence of the external bias field on solitonic dynamics is analyzed. In the photovoltaic crystal, coupled sn–cn-type, sn–dn-type periodic wave solutions, solution constructed by products of elliptic functions and the corresponding dark–bright soliton pair and coupled double-peaked soliton solutions are found to describe the evolution of a spatial-phase-modulated photovoltaic soliton and a non-phase-modulated beam.

Because of the complexity and difficulty of realizing a multi-wavelength soliton state, reports on its internal dynamic characteristics are scarce. In this study, the switching and periodic soliton explosion processes of the multi-wavelength soliton state in a negative dispersion passively mode-locked fiber laser are realized. The generation of the multi-wavelength soliton state undergoes the process of noise, oscillation, and stable mode-locking, and the splitting and annihilation of solitons with different group velocities directly impact the generation and disappearance of three wavelengths. Positive and negative dispersion lead to different group velocities of solitons. The presence and displacement of solitons with different group velocities cause soliton collisions, which lead to soliton explosions. A soliton experiences relative phase oscillation, chaos, and oscillation, as well as convergence and separation before and after an explosion. With an increase in parameters related to pump power, single-soliton oscillation, multi-wavelength solitons, and chaos are found in experiments and simulations, proving the relevance and reliability between simulation and experimental results. This work promotes the dynamical study of multi-soliton collisions in nonlinear science and the development of chaos theory in multi-comb lasers.

We follow our interest in a nonautonomous (2+1)-dimensional coupled nonlinear Schrödinger equation with partially nonlocal nonlinear effect and a linear potential, and get a relational expression mapping nonautonomous equation into autonomous one. Further applying the Darboux method, we find affluent vector and scalar solutions, including the crossed double-Ma breather solution. Regulating values of initial width, initial chirp and diffraction parameters so that the maximal value of accumulated time changes to compare with values of peak positions, we actualize the controlling effect of vector and scalar crossed double-Ma breathers including the complete shape, crest shape and nascent shape excitations in different linear potentials.

We use the physics-informed neural network to solve a variety of femtosecond optical soliton solutions of the high-order nonlinear Schrödinger equation, including one-soliton solution, two-soliton solution, rogue wave solution, W-soliton solution and M-soliton solution. The prediction error for one-soliton, W-soliton and M-soliton is smaller. As the prediction distance increases, the prediction error will gradually increase. The unknown physical parameters of the high-order nonlinear Schrödinger equation are studied by using rogue wave solutions as data sets. The neural network is optimized from three aspects including the number of layers of the neural network, the number of neurons, and the sampling points. Compared with previous research, our error is greatly reduced. This is not a replacement for the traditional numerical method, but hopefully to open up new ideas.

A (3 + 1)-dimensional partially nonlocal nonlinear Schrödinger equation is considered, and approximate spatiotemporal Hermite–Gaussian soliton solutions are obtained using the Hirota method. Based on these results, some basic characteristics of spatiotemporal Hermite–Gaussian solitons are studied.

We consider a ( 2 + 1 )-dimensional nonautonomous-coupled nonlinear Schrödinger equation, which includes the partially nonlocal nonlinearity under linear and harmonic potentials. Via a projecting expression between nonautonomous and autonomous equations, and utilizing the bilinear method and Darboux transformation method, we find diversified exact solutions. These solutions contain the nonlocal rogue wave and Akhmediev or Ma breather solutions, and the combined solution which describes a rogue wave superposed on an Akhmediev or Ma breather. By adjusting values of diffraction, width and phase chirp parameters of wave, the maximum value of the accumulated time can be modulated. When we compare the maximum value of the accumulated time with that of the excitation position parameters, we study the management of scalar and vector rogue waves, such as the excitations of full shape, early shape and climax shape for rogue waves.

Under parity-time symmetric potentials, different-order nonlinearities such as cubic, quintic and septimal nonlinearities, altogether with their combinations and second-order and fourth-order dispersions/diffractions are simultaneously considered to form three-dimensional optical solitons. Based on some high-order nonlinear Schrödinger equations, three-dimensional analytical optical soliton solutions are found. In the defocusing cubic nonlinear case, three-dimensional optical soliton without fourth-order diffraction/dispersion is stable than that with fourth-order diffraction/dispersion. However, in the defocusing cubic and focusing quintic nonlinear case, the stability situation of soliton is just on the contrary. Among all combinations of nonlinearity, the stability of three-dimensional optical soliton in the cubic-quintic nonlinear case is better than that in the cubic nonlinear case, but worse than that in the cubic-quintic-septimal nonlinear case. In the quintic-septimal nonlinear case, three-dimensional optical soliton is unstable and will collapse ultimately.

Hierarchies of Peregrine solution and breather solution are derived in a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation with partial nonlocality. Based on these solutions, we study the control of the excitation of Peregrine solution and breather solution in different planes. In particular, the localized Peregrine solution and breather solution are firstly reported in two-dimensional space. It is expected that our analysis and results may give new insight into higher-dimensional localized rogue waves in nonlocal media.

A \\[(3+1)\\]-dimensional coupled nonlocal nonlinear Schrödinger equation in the inhomogeneous \\[{\\mathcal {PT}}\\]-symmetric and strongly nonlocal nonlinear media is studied, and analytical vector spatiotemporal localized solutions are obtained. From these solutions, Gaussian soliton clusters, multipole soliton clusters, and nested soliton clusters can be constructed. The expansion behavior and periodic expansion and compression of spatiotemporal localizations are also investigated in the diffraction decreasing system and periodic modulation system, respectively.

A (2+1)-dimensional N -coupled nonlinear Schrödinger equation with spatially modulated cubic–quintic nonlinearity and transverse modulation is studied, and vector multipole and vortex soliton solutions are analytically obtained. When the modulation depth q is chosen as 0 and 1, vector multipole and vortex solitons are constructed, respectively. The number of “petals” for the multipole solitons and vortex solitons is related to the value of the topological charge m , and the number of layers in the multipole solitons and vortex solitons is determined by the value of the soliton order number n .