Dynamics of diverse data-driven solitons for the three-component coupled nonlinear Schrödinger model by the MPS-PINN method

Nonlinear Dynamics - We improve the physical information neural network by adding multiple parallel subnets to predict seven types of soliton dynamics, such as one soliton, two solitons and soliton...     

Reconstruction of stability for Gaussian spatial solitons in quintic–septimal nonlinear materials under $${{\varvec{\mathcal {P}}}}{\varvec{\mathcal {T}}}$$ P T -symmetric potentials

Nonlinear Dynamics - Gaussian spatial soliton solutions of both the constant-coefficient and variable-coefficient (2 + 1)-dimensional nonlinear Schrödinger equations in...     

Elastic interactions between multi-valued foldons and anti-foldons for the (2+1)-dimensional variable coefficient Broer–Kaup system in water waves

With the help of the improved tanh-function method, some exact variable separation solutions for a (2+1)-dimensional variable coefficient Broer–Kaup system in water waves are found. The detailed investigation indicates that these seemly independent variable separation solutions actually depend on each other. Based on the exact variable separation solution, completely and noncompletely elastic interactions between multi-valued foldons and anti-foldons are studied analytically and graphically.     

New Exact Solutions of （1 ＋ 1）-Dimensional Coupled Integrable Dispersionless System

This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more generalvariable separation solutions with two arbitrary functions of (1+1)-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p(x) and q(t) inour solutions, the annihilation phenomena of the flat-basin soliton, arch-basin soliton, and flat-top soliton are discussed.     

Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrödinger equation

Modern methods of nonlinear dynamics including time histories, phase portraits, power spectra, and Poincaré sections are used to characterize the stability and bifurcation regions of a cantilevered pipe conveying fluid with symmetric constraints at the point of contact. In this study, efforts are made to demonstrate the importance of characterizing the system at the arbitrarily positioned symmetric constraints rather than at the tip of the cantilevered pipe. Using the full nonlinear equations of motion and the Galerkin discretization, a nonlinear analysis is performed. After validating the model with previous results using the bifurcation diagrams and achieving full agreement, the bifurcation diagram at the point of contact is further investigated to select key flow velocities of interest. In addition to demonstrating the progression of the selected regions using primarily phase portraits, a detailed comparison is made between the tip and the point of contact at the key flow velocities. In doing so, period doubling, pitchfork bifurcations, grazing bifurcations, sticking, and chaos that occur at the point of contact are found to not always occur at the tip for the same flow speed. Thus, it is shown that in the case of cantilevered pipes with constraints, more accurate characterization of the system is obtained in a specified range of flow velocities by characterizing the system at the point of contact rather than at the tip.     

Nonautonomous solitons in the continuous wave background of the variable-coefficient higher-order nonlinear Schro¨dinger equation

We reduce the variable-coefficient higher-order nonlinear Schrdinger equation (VCHNLSE) into the constantcoefficient (CC) one. Based on the reduction transformation and solutions of CCHNLSE, we obtain analytical soliton solutions embedded in the continuous wave background for the VCHNLSE. Then the excitation in advancement and sustainment of soliton arrays, and postponed disappearance and sustainment of the bright soliton embedded in the background are discussed in an exponential system.     

Spatiotemporal Rogue Waves for the Variable-Coefficient (3+1)-Dimensional Nonlinear Schrödinger Equation

With the help of the similarity transformation connected the variable-coefficient (3+1)-dimensional nonlinear Schrödinger equation with the standard nonlinear Schrödinger equation, we firstly obtain first-order and second-order rogue wave solutions. Then, we investigate the controllable behaviors of these rogue waves in the hyperbolic dispersion decreasing profile. Our results indicate that the integral relation between the accumulated time T and the real time t is the basis to realize the control and manipulation of propagation behaviors of rogue waves, such as sustainment and restraint. We can modulate the value T₀ to achieve the sustained and restrained spatiotemporal rogue waves. Moreover, the controllability for position of sustainment and restraint for spatiotemporal rogue waves can also be realized by setting different values of X₀.     

Exact travelling wave solutions of the discrete sine–Gordon equation obtained via the exp-function method

In this paper, we generalize the exp-function method, which was used to find new exact travelling wave solutions of nonlinear partial differential equations (NPDEs) or coupled nonlinear partial differential equations, to nonlinear differential–difference equations (NDDEs). As an illustration, two series of exact travelling wave solutions of the discrete sine–Gordon equation are obtained by means of the exp-function method. As some special examples, these new exact travelling wave solutions can degenerate into the kink-type solitary wave solutions reported in the open literature.