A Vector Fokas-Lenells System from the Coupled Nonlinear Schrödinger Equations

With the aid of the spectral gradient method of Fuchssteiner, the compatible pair of Hamiltonian operators for the coupled NLS hierarchy is rediscovered. This result enables us to construct a hierarchy, which contains a vector generalization of Fokas-Lenells system. The vector Fokas-Lenells system is shown to be bi-Hamiltonian and to possess a Lax pair.     

Heavy ion–acoustic rogue waves in magnetized electron–positron multi-ion plasmas

Non-linear heavy ion-acoustic waves (HIAWs) are studied in a homogeneous magnetized four-component multi-ion plasma composed of inertial heavy negative ions, light positive ions, and inertia-less non-extensive electrons and positrons. The non-linear Schrödinger equation is derived in this model using the perturbation method. The criteria for modulational instability of HIAWs and the basic features of finite-amplitude heavy ion acoustic rogue waves (HIARWs) are investigated. The presence of the magnetic field was found to reduce the amplitude of HIARWs and enhances the stability. It is interesting to note that increasing positive ion mass causes decreases in the amplitude and width of rogue waves, which is opposite behaviour to that demonstrated in the previous study of these waves in an unmagnetized plasma. Furthermore, it is also shown that striking parameters, such as the non-extensive parameter, the positron number density, the electron number density, the electron temperature, and the magnetic field parameter, play an undeniable role on the stability of waves packets. The findings of the present investigation may be of wide relevance to some plasma environments, such as active galactic nuclei, pulsar magnetospheres, and other magnetic confinement systems.     

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

Rogue waves: New forms enabled by GPU computing

A method to determine parameters governing periodic Riemann theta function rogue-wave solutions to the nonlinear Schrödinger equation is presented. A map of parameter values leading to candidate solutions is developed. In addition to candidate solutions, an overview of qualitative aspects of the solution space can be gained from this map. Based on these findings, several new extreme wave solutions are presented. Although the computations required to determine the map are quite demanding, it is shown that these computations can be efficiently accelerated with a parallel computing architecture. A general purpose computing on a graphics processor unit (GPGPU) implementation yielded a 400× acceleration over a single threaded high level implementation. This acceleration enabled exploration and examination of the solution space, which otherwise would not have been possible. In addition, the solution methodology presented here can be extended to explore other classes of solutions.     

Analytical solutions and rogue waves in (3+1)-dimensional nonlinear Schrödinger equation

Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrödinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

环形振子势的精确解

将环形振子势的Schrödinger方程在球坐标系中进行变量分离.然后求解了角向方程和径向方程,给出了精确的能谱方程.获得了用普遍的缔合Legendre多项式表示的归一化的角向波函数和用合流超几何函数表示的归一化的径向波函数.     

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrödinger equation. Firstly, the generalized nonlinear Schrödinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.     

Extreme wave formation in unidirectional sea due to stochastic wave phase dynamics

The authors consider a stochastic model based on the interaction and phase coupling amongst wave components that are modified envelope soliton solutions to the nonlinear Schrödinger equation. A probabilistic study is carried out and the resulting findings are compared with ocean wave field observations and laboratory experimental results. The wave height probability distribution obtained from the model is found to match well with prior data in the large wave height region. From the eigenvalue spectrum obtained through the Inverse Scattering Transform, it is revealed that the deep-water wave groups move at a speed different from the linear group speed, which justifies the inclusion of phase correction to the envelope solitary wave components. It is determined that phase synchronization amongst elementary solitary wave components can be critical for the formation of extreme waves in unidirectional sea states.     

On Darboux transformations for the derivative nonlinear Schrödinger equation

We consider Darboux transformations for the derivative nonlinear Schrödinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the derivative nonlinear Schrödinger equation are given as explicit examples.     

Modulation of ion sound excitations in electron–ion plasmas with double spectral index distribution function

Theoretical investigation of amplitude modulation of ion sound waves is presented here for an electron–ion plasma where the electrons are dictated by the double spectral index (r, q) distribution function. Using the standard reductive perturbative technique, a non-linear Schrödinger (NLS) equation is derived that describes the evolution of modulated ion sound envelope excitations. Stability analysis of the NLS equation shows that the ion sound waves remain stable for the flat-topped and kappa-like distributions, but they can become unstable for the spiky electron velocity distribution. It is shown that changing the electron population in regions of low and high phase space density regions results in remarkable features that have no equivalent in ion sound waves with Boltzmannian electrons. Different types of localized ion sound excitations are plotted for the different shapes of the distribution functions controlled by the double spectral indices, and the underlying physics is discussed in detail. The present investigation may be beneficial to understand ion sound excitations in space plasmas where the distribution functions of the shapes presented here are frequently encountered by the satellite missions.