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The time(T)- and time-space(TS)-fractional higher-order Schrödinger(FHNLS) equations are studied, which can describe the dispersion and nonlinearity of optical pulse propagating in nonuniform fiber systems. The generalized fractional wave transformation is appropriately used to convert the T- and TS-FHNLS equations into some ordinary differential equations, and the real and imaginary parts are separated respectively. A few of new exact analytical solutions of the fractional parameters α and β are obtained with the homogeneous balance method, including time-fractional and time-space fractional bright and dark solutions, and some fractional order effects on soliton solutions are considered. Some dynamic behaviors of the hyperbolic soliton solutions of the fractional-typed sech and tanh are obtained, and some novel forms of the dark solitary wave and bright solitary wave are presented with the parameters α and β . And some novel toroidal, spherical bright and dark solitons are derived by the toroidal and spherical coordinate transformations. Further, four kinds of time-space fractional non-autonomous soliton solutions of the higher-order Gross–Pitaevskii(GP) model are derived, we find that the value of α decreases, the numbers of oscillations or singularities increase for small time values and decrease for large time values. These results are significantly different from the classical higher-order nonlinear Schrödinger and GP equations, which can help us to understand the solitary wave transmission pattern and apply to the physical manipulation. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related physical experiments in the fractional media. [ABSTRACT FROM AUTHOR]

Some space-shifted bright soliton solutions in terms of determinants for the space-shifted nonlocal coupled nonlinear Schrödinger (NCNLS) equations are constructed by using the improving Hirota's bilinear method. A few of 1-bright and 2-bright solitons of the NCNLS equations are derived with a function e 2 i γ t and the space-shifted term x 0 . The influence of the space-shifted parameters for the solution is significant, and some novel dynamic behaviors of the space-shifted solutions are presented. The two-bright solitons admit some novel patterns, whose amplitudes increase or decrease with time. The bright-breather soliton solutions are derived through a long wave limit of the obtained bright soliton solutions, and their collision dynamics are also investigated. And some bright solitons can occur elastic collisions, which shows that some expressions of the soliton amplitudes are independent on the phase shift factor x 0 . Some novel toroidal and spherical bright solitons, breather solitons are derived by the toroidal and spherical coordinate transformations. [ABSTRACT FROM AUTHOR]

The deep ocean, Earth's untouched expanse, presents immense challenges for exploration due to its extreme pressure, temperature, and darkness. Unlike traditional marine robots that require specialized metallic vessels for protection, deep-sea species thrive without such cumbersome pressure-resistant designs. Their pressure-adaptive forms, unique propulsion methods, and advanced senses have inspired innovation in designing lightweight, compact soft machines. This perspective addresses challenges, recent strides, and design strategies for bioinspired deep-sea soft robots. Drawing from abyssal life, it explores the actuation, sensing, power, and pressure resilience of multifunctional deep-sea soft robots, offering game-changing solutions for profound exploration and operation in harsh conditions. High pressure and low temperature are the greatest challenges faced by scientists to explore deep oceans, which remain largely unknow to us today. Li et al. review these challenges and give insight into designing soft robots, inspired by deep-sea creatures, that enable resilient operations in harsh conditions. [ABSTRACT FROM AUTHOR]

In this paper, the exact solutions of generalized nonlinear Schrödinger (GNLS) equation are obtained by using Darboux transformation(DT). We derive some expressions of the 1-solitons, 2-solitons and n-soliton solutions of the GNLS equation via constructing special Lax pairs. And we choose different seed solutions and solve the GNLS equation to obtain the soliton solutions, breather solutions and rational wave solutions. Based on these obtained solutions, we consider the elastic interactions and dynamics between two solitons. [ABSTRACT FROM AUTHOR]

The PT-symmetric Gross-Pitaevskii(GP) equation is presented, which is an important and useful model in Bose-Einstein condensation(BEC). The generalized Gross-Pitaevskii(GGP(n,n)) equation has several kinds of potentials including Gaussian, harmonic and radial potentials, and it is a generalization GP equation.Then, some physically relevant solutions are derived, a kind of flat-top soliton solution is considered for the nonautonomous GGP(n,n) equation with Gaussian-harmonic-radial PT-symmetric potential. Especially, some novel flat-top bright(FTB) solitons are found, these FTB solitons can exist stably with Gaussian-harmonic-radial PT-symmetric potentials in a broad range. We investigate the interaction dynamics of between the FTB soliton and FTB soliton, the FTB soliton and bright soliton through addressing numerically. Intriguingly, the FTB solitons can admit some novel features and are different from these usual features of solitons, which have not a effected by other external waves. These results are useful for the possibility of some relative experiments and potential applications. [ABSTRACT FROM AUTHOR]

The coupled nonlinear Schrödinger (CNLS) equations are the important models for the study of the multicomponent Bose-Einstein condensates (BECs). In this paper, we study the four-components CNLS equations via Darboux transformation and obtain the N-soliton solutions with zero seed and non-zero seed solutions( q i = 0 or q i = e - 2 i t ) . The 1-soliton solution and 2-soliton solution are calculated on complex wave backgrounds, the dark-bright-bright-bright soliton solutions and dark-dark-bright-bright soliton solutions are constructed. We can obtain a new class of dark-bright-bright-bright soliton solutions, which admit one-valley dark soliton in component q 1 and triple-hump bright solitons in the other three components. The collision properties between dark-dark-bright-bright solitons are considered, and the vector solitons are expected to be much more abundant than those of previously reported vector soliton collisions. [ABSTRACT FROM AUTHOR]

The non-autonomous discrete bright–dark soliton solutions(NDBDSSs) of the 2 + 1-dimensional Ablowitz–Ladik (AL) equation are derived. We analyze the dynamic behaviors and interactions of the obtained 2 + 1-dimensional NDBDSSs. In this paper, we present two kinds of different methods to control the 2 + 1-dimensional NDBDSSs. In first method, we can only control the wave propagations through the spatial part, since the time function has not effect in the phase part. In second method, we can control the wave propagations through both the spatial and temporal parts. The different propagation phenomena can also be produced through two kinds of managements. We obtain the novel " π "-shape non-autonomous discrete bright soliton solution(NDBSS), the novel " ⋏ "-shape non-autonomous discrete dark soliton solution(NDDSS) and their interaction behaviors. The novel behaviors are considered analytically, which can be applied to the electrical and optical fields. [ABSTRACT FROM AUTHOR]

A generalized nonlocal nonlinear Hirota (GNNH) equation with variable coefficients is presented, which can be reduced into the nonlocal Hirota equation with the self-induced PT-symmetric potential. Especially, the nonlocal Gross–Pitaevskii (NGP) equation with the self-induced PT-symmetric potential is derived from the GNNH equation. Then, we obtain some novel non-autonomous breather solutions and rogue waves of GNNH equation via similarity and Hirota methods, and consider some controllable behaviors of these non-autonomous wave solutions. Furthermore, some properties of the non-autonomous rational (NR) waves are investigated analytically for the NGP equation. The trajectories of peaks and depressions of the non-autonomous rogue waves are produced by means of analytical method, and the dynamical stabilities of the NR solution are derived through the numerical method. The obtained results are different from the solutions of the local nonlinear equations. Some different propagation phenomena can also be produced through manipulating non-autonomous rogue waves, which can present the potential applications for the rogue wave phenomena in nonlocal wave models. [ABSTRACT FROM AUTHOR]

We study the dynamics of discrete rogue waves in an array of nonlinear waveguides. Some non-autonomous discrete rogue wave solutions and their interactions in the coupled Ablowitz–Ladik (CAL) equation with variable coefficients are considered. And these problems possess complicated wave propagations in time and differ from the single discrete rogue waves. Furthermore, we predict the long-living rogue wave solutions of the discrete CAL equation. In addition, the controllable behaviors of this non-autonomous rogue wave system with the nonlinearity management parameters are discussed. At last, the numerical simulations on the evolution and collision of two rogue waves are performed to verify the prediction of the analytical formulations. These results may be useful to explain some nonlinear wave phenomena in certain electrical and optical systems. [ABSTRACT FROM AUTHOR]

We consider the nonautonomous discrete vector bright-dark solutions and their controllable behaviors in the coupled Ablowitz-Ladik equation with variable coefficients, which possesses complicated wave propagation in time. Based on the differential-difference symmetry transformation and the Lamé polynomial solutions, we use the Jacobi elliptic functions $$\hbox {sn}(n, m), \hbox {cn}(n, m)$$ , $$\hbox {dn}(n, m)$$ and present the nonautonomous discrete vector bright-dark solutions, which are localized in space and keep the localization longer in time. Moreover, we also exhibit the wave propagation of nonautonomous Lamé polynomial solutions of higher order and their dynamics for some chosen parameters and functions. And the managements and dynamic behaviors of these solutions are investigated analytically. [ABSTRACT FROM AUTHOR]

The Darboux transformation (DT) for the super-integrable hierarchy has an essential difference from the general system. As we know, the super-integrable soliton equation hierarchies with four potentials are discussed. Starting from the spectral problems of super-AKNS hierarchy and super-Dirac hierarchy, a DT method for two super-integrable hierarchies is constructed, which is more complex than the general integrable system. Soliton solutions of super-Schrödinger equation and super-Dirac equation are presented by using DT, which contain some bright, dark and breather wave soliton solutions. Then, the properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. [ABSTRACT FROM AUTHOR]

We consider nonautonomous rogue wave solutions of a $$(3+1)$$ -dimensional (3D) nonlinear Schrödinger equation (NLSE) with time-space modulation terms, the dispersion and the nonlinear coefficients engendering temporal dependency. Similarity transformation is used to convert the nonautonomous equation into autonomous NLSE; we obtain the multi-rogue wave solutions employing the generalized Darboux transformation. Particularly, the rogue wave solutions possess several free parameters. Then, the first-order and second-order nonautonomous rogue wave solutions are considered for the 3D NLSE with variable coefficients. At last, the numerical simulations on the evolution and collision of rogue wave solutions are performed to verify the prediction of the analytical formulations. The obtained nonautonomous rogue wave solutions can be used to describe the dynamics waves in the nonlinear optical fibers and Bose-Einstein condensates, respectively. [ABSTRACT FROM AUTHOR]

The study of soliton interactions is a significance for improving pulse qualities in nonlinear optics. In this paper, a generalized coupled cubic-quintic nonlinear Schrödinger (GCCQNLS) equation with the group-velocity dispersion, fiber gain-or-loss and nonlinearity coefficient functions is studied, which describes the evolution of a slowly varying wave packet envelope in the inhomogeneous optical fiber. In particular, based on the similarity transformation, we report several families of nonautonomous wave solutions of the GCCQNLS equation. It is reported that there are possibilities to manipulate the interactions of nonautonomous wave solution through manipulating nonlinear and gain/loss functions. Interactions between the different-type bright two solitons have been asymptotically analyzed and presented. And, the two parabolic-type bright solitons propagating with the opposite directions both change their directions after the interaction. Interactions between the linear-, parabolic- and periodic-type bright two solitons are elastic. At last, the numerical simulations on the evolution and collision of two soliton solutions are performed to verify the prediction of the analytical formulations. We present the general approach can provide many possibilities to manipulate soliton waves experimentally and consider the potential applications for the optical self-routing, non-Kerr media and Bose-Einstein condensates (BEC). [ABSTRACT FROM AUTHOR]

We present a brand new method to construct the Schrödinger-like equations from a solution in this paper. Some Schrödinger-like equations with sources are derived by using a generalized solution, which have the form $$i\frac{\partial \Psi }{\partial x}+\frac{1}{2}\frac{\partial ^2\Psi }{\partial t^2}+|\Psi |^2\Psi +\alpha (x,t)\Psi +\beta (x,t)e^{ipx}=0$$ . The abundant analytical solutions of the Schrödinger-like equations are considered, including the bright rogue wave solution, dark rogue wave solution, Bell-shaped soliton solution, the interactions of two solitons, and other special soliton solutions. And we prove that the equation with source term has a weak solution. At last, the numerical simulations on the evolution and solitons collision of rogue wave solutions are performed to verify the prediction of the analytical formulations. The results could be of interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids. [ABSTRACT FROM AUTHOR]

The analytical matter rogue wave solutions are reported for the coupled Gross-Pitaevskii equation by using the similarity transformation and Darboux transformation. We study the effect of the time-dependent linear and quadratic potentials (flying-bird potential) on the profiles and dynamics of non-autonomous rogue wave solution. A non-autonomous rogue wave and bright-dark rogue wave solutions are constructed and exhibited. The managements of external potential and the dynamic behaviors of the rogue wave solutions are investigated analytically. We present the general approach and use it to calculate non-autonomous rogue wave solutions and consider the potential applications for the rogue wave phenomena. [ABSTRACT FROM AUTHOR]

A few of discrete integrable coupling systems(DICSs) of previous papers are linear discrete integrable couplings(LDICS). We take a special matrix Lie algebra system(non-semisimple) to construct the Lax pairs, and establish a method for deriving the nonlinear discrete integrable coupling systems(NDICS). From the Lax pairs of the generalized Toda(G-Toda) spectral problem, we can derive a novel NDICS, which is a real NDICS. For the obtained lattice integrable coupling equation, we establish a Darboux transformation (DT) with 4 × 4 Lax pairs, and apply the gauge transformation to a specific equation, then the explicit solutions of the lattice integrable coupling equation are given, which contains discrete soliton solution, breather solution and rogue wave solution. Furthermore, we can derive the discrete explicit solutions with free parameters to depict their dynamic behaviors. [ABSTRACT FROM AUTHOR]