High-Order Soliton Solutions and Hybrid Behavior for the (2 + 1)-Dimensional Konopelchenko-Dubrovsky Equations

In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton solution, we first study the evolution from N-soliton to T-order (T=1,2) breather wave solutions via the paired-complexification of parameters, and then we get the N-order rational solutions, M-order (M=1,2) lump solutions, and the hybrid behavior between a variety of different types of solitons combined with the parameter limit technique and the paired-complexification of parameters. Meanwhile, we also provide a large number of three-dimensional figures in order to better show the degeneration of the N-soliton and the interaction behavior between different N-solitons.

Positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg-de Vries equations

Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations are decomposed into multi-solitons as well as the(1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.

High-Order Solitons and Hybrid Behavior of (3 + 1)-Dimensional Potential Yu-Toda-Sasa-Fukuyama Equation with Variable Coefficients

In this paper, some exact solutions of the (3 + 1)-dimensional variable-coefficient Yu-Toda-Sasa-Fukuyama equation are investigated. By using Hirota’s direct method and symbolic computation, we obtained N-soliton solution. By using the long wave limit method, the N-order rational solution can be obtained from N-order soliton solution. Then, through the paired complexification of parameters, the lump solution is obtained from N-order rational solution. Meanwhile, we obtained a hybrid solution between 1-lump solution and N-soliton (N=1,2) by using the long wave limit method and parameter complex. Furthermore, four different sets of three-dimensional graphs of solitons, lump solutions and hybrid solutions are drawn by selecting four different sets of coefficient functions which include one set of constant coefficient function and three sets of variable coefficient functions.

Dynamic response of axially moving Timoshenko beams: integral transform solution

The generalized integral transform technique （GITT） is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations （ODEs） in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.

General M-lumps, T-breathers, and hybrid solutions to (2+1)-dimensional generalized KDKK equation

A special transformation is introduced and thereby leads to the N-soliton solution of the(2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt(KDKK) equation.Then,by employing the long wave limit and imposing complex conjugate constraints to the related solitons,various localized interaction solutions are constructed,including the general M-lumps,T-breathers,and hybrid wave solutions.Dynamical behaviors of these solutions are investigated analytically and graphically.The solutions obtained are very helpful in studying the interaction phenomena of nonlinear localized waves.Therefore,we hope these results can provide some theoretical guidance to the experts in oceanography,atmospheric science,and weather forecasting.

Trajectory equation of a lump before and after collision with other waves for generalized Hirota–Satsuma–Ito equation

Based on the Hirota bilinear and long wave limit methods,the hybrid solutions of m-lump with n-soliton and nbreather wave for generalized Hirota–Satsuma–Ito(GHSI)equation are constructed.Then,by approximating solutions of the GHSI equation along some parallel orbits at infinity,the trajectory equation of a lump wave before and after collisions with n-soliton and n-breather wave are studied,and the expressions of phase shift for lump wave before and after collisions are given.Furthermore,it is revealed that collisions between the lump wave and other waves are elastic,the corresponding collision diagrams are used to further explain.

Separation of contact stress components by moire interferometry

A hybrid technique of combining moire measurement and analytical solution is developed to separate the normal and the tangential components of distributed contact stresses between two co-plane bodies. The moire interfe-rometry offers the displacement fields near the deformed contact zone, from which the tangential strains and boundary slopes of the deformed configurations can be evaluated. Those experimental results provide boundary conditions for the discrete integration of Flamant＇s solutions, to inversely compute the separated components of the contact stresses.

Resonance Y-type soliton solutions and some new types of hybrid solutions in the(2+1)-dimensional Sawada–Kotera equation

In this paper,based on N-soliton solutions,we introduce a new constraint among parameters to find the resonance Y-type soliton solutions in(2+1)-dimensional integrable systems.Then,we take the(2+1)-dimensional Sawada–Kotera equation as an example to illustrate how to generate these resonance Y-type soliton solutions with this new constraint.Next,by the long wave limit method,velocity resonance and module resonance,we can obtain some new types of hybrid solutions of resonance Y-type solitons with line waves,breather waves,high-order lump waves respectively.Finally,we also study the dynamics of these interaction solutions and indicate mathematically that these interactions are elastic.

Multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas–Lenells equation

In this letter,we investigate multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas–Lenells equation over a nonzero background.First,we obtain 2 n-soliton solutions with a nonzero background via n-fold Darboux transformation,and find that these soliton solutions will appear in pairs.Particularly,2 n-soliton solutions consist of n‘bright’solitons and n‘dark’solitons.This phenomenon implies a new form of integrability:even integrability.Then interactions between solitons with even numbers and breathers are studied in detail.To our best knowledge,a novel nonlinear superposition between a kink and 2 n-soliton is also generated for the first time.Finally,interactions between some different smooth positons with a nonzero background are derived.

General M-lump,high-order breather,and localized interaction solutions to(2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation

The(2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation is a significant physical model.By using the long wave limit method and confining the conjugation conditions on the interrelated solitons,the general M-lump,high-order breather,and localized interaction hybrid solutions are investigated,respectively.Then we implement the numerical simulations to research their dynamical behaviors,which indicate that different parameters have very different dynamic properties and propagation modes of the waves.The method involved can be validly employed to get high-order waves and study their propagation phenomena of many nonlinear equations.

Soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation

Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(gKDKK) equation by using the velocity resonance, module resonance, and long wave limits methods. By selecting some specific parameters, we can obtain soliton molecules and asymmetric soliton molecules of the gKDKK equation. And the interactions among N-soliton molecules are elastic. Furthermore, some novel hybrid solutions of the gKDKK equation can be obtained, which are composed of lumps,breathers, soliton molecules and asymmetric soliton molecules. Finally, the images of soliton molecules and some novel hybrid solutions are given, and their dynamic behavior is analyzed.

A review of the technologies for wave energy extraction

The main objective of this article is to provide a comprehensive picture of existing wave technologies being used for wave energy extraction.The overview will explain their potential and also the challenges wave technologies face.The article will also briefly discuss the benefits of combined offshore wind-wave projects,also known as hybrids.Key factors and impacts on relevant existing wave technologies will be outlined,including capacity factor and capture width.Finally the levelized cost of energy(LCOE)targets for the most promising technologies will be discussed.

Plug-in and plug-out dispatch optimization in microgrid clusters based on flexible communication

With large-scale development of distributed generation(DG) and its potential role in microgrids, the microgrid cluster(MGC) becomes a useful control model to assist the integration of DG. Considering that microgrids in a MGC, power dispatch optimization in a MGC is dif-ficult to achieve. In this paper, a hybrid interactive communication optimization solution(HICOS) is suggested based on flexible communication, which could be used to solve plug-in or plug-out operation states of microgrids in MGC power dispatch optimization. HICOS consists of a hierarchical architecture: the upper layer uses distributed control among multiple microgrids, with no central controller for the MGC, and the lower layer uses a central controller for each microgrid. Based on flexible communication links among microgrids, the optimal iterative information are exchanged among microgrids, thus HICOS would gradually converge to the global optimal solution.While some microgrids plug-in or plug-out, communication links will be changed, so as to unsuccessfully reach optimal solution. Differing from changeless communication links in traditional communication networks, HICOS redefines the topology of flexible communication links to meet the requirement to reach the global optimal solutions.Simulation studies show that HICOS could effectively reach the global optimal dispatch solution with non-MGC center. Especially, facing to microgrids plug-in or plug-out states, HICOS would also reach the global optimal solution based on refined communication link topology.