Traveling Wave System Research Articles

Bifurcation and Exact Solutions of a Concatenation Physical Model

To find the exact explicit solution of the concatenation model, the corresponding differential system of the amplitude component is used, which is a planar dynamical system with a singular straight line. In this paper, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007], its corresponding traveling wave system is solved and analyzed, obtaining the corresponding phase portraits and showing the dynamical behavior of the amplitude component. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons, are found explicitly.

The soliton and periodic solutions for the perturbed sine–cosine–Gordon equation

In the paper, the existence of soliton and periodic solutions is studied for the perturbed sine–cosine–Gordon equation. The corresponding traveling wave system is converted to a regularly Hamilton system via using bifurcation theory of differential equations and the geometric singular perturbation theory. Then by using Melnikov's method and symbolic computation, periodic wave solutions, solitary solutions, and kink and antikink solutions are obtained. The wave speed selection principles are given, respectively. Numerical simulations are performed to illustrate our theoretical analysis.

New solitary waves in a convecting fluid

It is critical to examine the effects of small perturbations on solitary wave solutions to nonlinear wave equations because of the existence of the unavoidable perturbations in real applications. In this work, we aim to study the solitary wave solutions for the dissipative perturbed mKdV equation which is proposed to model the evolution of shallow wave in a convecting fluid. By using the geometric singular perturbation theorem, the three-dimensional traveling wave system is reduced to a planar dynamical system with regular perturbation. By examining the homoclinic bifurcations via Melnikov method, we show that not only some solitary waves with particularly chosen wave speed persist but also some new type of solitary waves appear under small perturbation. The theoretical analysis results are illustrated by numerical simulations.

On the data assimilation of initial distribution for 2-dimensionalshallow-water equation model

Abstract The utilization of data assimilation (DA) techniques is prevalent in marine meteorology for the purpose of estimating the complete state of the system. This is done to address the practical limitations associated with measurement data, which can only be specified at finite number of discrete points within a limited domain. We develop an efficient DA algorithm to reconstruct the initial state of the shallow-water equations (SWE) within a 2-dimensional rectangular domain using sparse spatial measurement data. Our algorithm takes into account both the complete Coriolis force and the ocean bottom topography in the SWE model, resulting in accurate recovery of the initial status. After establishing the uniqueness of the solution to the nonlinear SWE with appropriate boundary conditions, we proceed to establish the conservation laws for the suitably defined energy quantity for this traveling wave system. This generalization of the known conservation laws for the simplified SWE system, which ignores the Coriolis force and topography, allows us to reveal the influence of nonconstant sea floor topography on wave propagation. In order to restore the initial state through the minimization of a cost functional using DA techniques, we proceed by deriving the adjoint problem for our iteration process. Additionally, we establish a discrete scheme for the governing equations in the Arakawa C-grid framework, from which we rigorously derive the error associated with energy conservation in discrete form. The numerical implementations are also provided to validate our proposed scheme through the verification of energy conservation and the reconstruction effect of the initial state for various configurations.

Peakon, Periodic Peakons, Compactons and Bifurcations of nonlinear Schrödinger’s Equation with Kudryashov’s Law of Refractive Index

In this paper, we consider the nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the traveling wave system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including peakon, periodic peakon, solitary wave solutions and compactons) under different parameter conditions.

Bifurcations and Exact Solutions of Optical Soliton Models in Fifth-Order Weakly Nonlocal Nonlinear Media

For the optical soliton model in fifth-order weakly nonlocal nonlinear media, to find its exact explicit solutions, the corresponding traveling wave system is formulated as a planar dynamical system with a singular straight line. Then, by using techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze the planar system and find the corresponding phase portraits, the dynamical behavior of the amplitude component can be assessed. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink, and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons are found with precise formulations.

Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity

In this paper, the bifurcation theory of dynamical system is applied to investigate the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. We mainly consider the case of $\alpha\neq 2\beta$ which is not discussed in previous work. By overcoming some difficulties aroused by the singular traveling wave system, such as bifurcation analysis of nonanalytic vector field, tracking orbits near the full degenerate equilibrium and calculation of complicated elliptic integrals, we give a total of 20 explicit exact traveling wave solutions of the time-space fractional complex Ginzburg-Landau equation and classify them into 11 categories. Some new traveling wave solutions of this equation are obtained including the compactons and the bounded solutions corresponding to some bounded manifolds.

Bifurcations and Exact Solutions of the Generalized Radhakrishnan–Kundu–Lakshmanan Equation with the Polynomial Law

In this paper, we investigate the generalized Radhakrishnan–Kundu–Lakshmanan equation with polynomial law using the method of dynamical systems. By using traveling-wave transformation, the model can be converted into a singular integrable traveling-wave system. Then, we discuss the dynamical behavior of the associated regular system and we obtain bifurcations of the phase portraits of the traveling-wave system under different parameter conditions. Finally, under different parameter conditions, we obtain the exact periodic solutions, and the peakon, homoclinic and heteroclinic solutions.

Spatiotemporal propagation of a time‐periodic reaction–diffusion SI epidemic model with treatment

AbstractThis work is concerned with the spatiotemporal propagation phenomena for a time‐periodic reaction‐diffusion susceptible‐infectious (SI) epidemic model with treatment in terms of the asymptotic speed of spread and periodic traveling waves. First, the asymptotic speed of spread is characterized and the spreading properties of the model are analyzed by combining the periodic principal eigenvalue problem, comparison method, and the uniform persistence idea for a dynamical system. Second, by constructing suitable super‐ and subsolutions for truncation problems corresponding to the traveling wave system, the existence of periodic traveling waves is established via the fixed point theorem twice. It turned out that the asymptotic speed of spread coincides with the minimum wave speed of periodic traveling waves. Finally, via numerical simulation, the effects of some important parameters (such as diffusion rate, treatment rate, etc.) on the spreading speed are discussed, and the asymptotic properties of the periodic traveling waves are explored.

Dynamical Discussion and Diverse Soliton Solutions via Complete Discrimination System Approach Along with Bifurcation Analysis for the Third Order NLSE

The purpose of this study is to introduce the wave structures and dynamical features of the third-order nonlinear Schr\"{o}dinger equations (TONLSE). We take the original equation and, using the traveling wave transformation, convert it into the appropriate traveling wave system, from which we create a conserved quantity known as the Hamiltonian. The Jacobian elliptic function solution (JEF), the hyperbolic function solution, and the trigonometric function solution are just a few of the optical soliton solutions to the equation that may be found using the complete discrimination system (CDS) of polynomial method (CDSPM) and also transfer the JEF into solitary wave (SW) soltions. It also includes certain dynamic results, such as bifurcation points and critical conditions for solutions, that might be utilized to explore the dynamic features of the equation employing the CDSPM. This method could also be used for qualitative analysis. The qualitative analysis is used to illustrate the equilibrium points and phase potraits of the equation. Phase portraits are visual representations used in dynamical systems to illustrate a system's behaviour through time. They can provide crucial information about a system's stability, periodic behaviour, and the presence of attractors or repellents.

Bifurcations, Exact Peakon, Periodic Peakons and Solitary Wave Solutions of Generalized Camassa–Holm–Degasperis–Procosi Type Equation

For the generalized Camassa–Holm–Degasperis–Procosi (CH–DP) type equation, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to analyze its corresponding traveling wave systems, which depend on four parameters, it is found that under different parameter conditions its bifurcation portraits exhibit all possible exact explicit bounded solutions, such as solitary wave solutions, periodic wave solutions, peakon as well as periodic peakons. A total of 30 explicit exact parametric representations of the traveling wave system of the CH–DP type equation are presented.

Bifurcations and Exact Traveling Wave Solutions of the Generalized Serre–Green–Naghdi System with Weak Coriolis Effect and Surface Tension

For the generalized Serre–Green–Naghdi system with weak Coriolis effect and surface tension, by using the dynamical system methods and singular traveling wave theory developed by Li and Chen [2007] to its associate traveling wave system, under different parameter conditions, all possible bounded solutions (solitary wave solutions, periodic wave solutions, peakons, periodic peakons as well as compacton solution families) are obtained. Exact explicit parametric representations are given.

Numerical Solution of the Electric Field and Dielectrophoresis Force of Electrostatic Traveling Wave System.

Electrostatic traveling wave (ETW) methods have shown promising performance in dust mitigation of solar panels, particle transport and separation in in situ space resource utilization, cell manipulation, and separation in biology. The ETW field distribution is required to analyze the forces applied to particles and to evaluate ETW design parameters. This study presents the numerical results of the ETW field distribution generated by a parallel electrode array using both the charge simulation method (CSM) and the boundary element method (BEM). A low accumulated error of the CSM is achieved by properly arranging the positions and numbers of contour points and fictitious charges. The BEM can avoid the inconvenience of the charge position required in the CSM. The numerical results show extremely close agreement between the CSM and BEM. For simplification, the method of images is introduced in the implementation of the CSM and BEM. Moreover, analytical formulas are obtained for the integral of Green's function along boundary elements. For further validation, the results are cross-checked using the finite element method (FEM). It is found that discrepancies occur at the ends of the electrode array. Finally, analyses are provided of the electric field and dielectrophoretic (DEP) components. Emphasis is given to the regions close to the electrode surfaces. These results provide guidance for the fabrication of ETW systems for various applications.

The motion of small particles in electrostatic travelling waves for transport and separation

Electrostatic travelling waves can be used to transport small particles across a surface. This technology has received particular attention for dust mitigation on solar panels and for manipulating small particles on the Moon as part of in-situ resource utilization processes. Electrostatic travelling wave systems have no moving parts and are particularly well suited to low humidity environments. Here, we analyse the motion of small particles with the aim of exploiting differences in motion and velocity to separate particles by size. We investigate the effects of voltage, frequency, particle size and charge, wavelength and initial conditions on the properties of particle motion, such as particle velocity, levitation height and motion mode. We calculate the electrostatic fields using accurate boundary conditions based on the Fourier expansion method, which shows more detail near the surface of electrodes. We solve the equations of motion using the implicit Runge-Kutta method, and measure the particle charge with a free fall system. We show that the numerical results have a good agreement with the analytical results of a particle moving in a certain mode. We have observed three modes of motion with a high-speed camera, and these results provide guidance for the development of electrostatic travelling wave systems for various applications.

Bifurcations and Exact Bounded Solutions of Some Traveling Wave Systems Determined by Integrable Nonlinear Oscillators withq-Degree Damping

For a class of nonlinear diffusion–convection–reaction equations, the corresponding traveling wave systems are well-known nonlinear oscillation type of systems. Under some parameter conditions, the first integrals of these nonlinear oscillators can be obtained. In this paper, the bifurcations, exact solutions and dynamical behavior of these nonlinear oscillators are studied by using methods of dynamical systems. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink and anti-kink wave solutions, as well as limit cycles, are obtained. Most important and interestingly, a new global bifurcation phenomenon of limit bifurcation is found: as a key parameter is varied, so that singular points (except the origin) disappear, a planar dynamical system can create a stable limit cycle.

Bifurcation of Some Novel Wave Solutions for Modified Nonlinear Schrödinger Equation with Time M-Fractional Derivative

In this paper, we investigate the time M-fractional modified nonlinear Schrödinger equation that describes the propagation of rogue waves in deep water. Periodic, solitary, and kink (or anti-kink) wave solutions are discussed using the bifurcation theory for planar integrable systems. Some new wave solutions are constructed using the first integral for the traveling wave system. The degeneracy of the obtained solutions is investigated by using the transition between orbits. We visually explore some of the solutions using graphical representations for different values of the fractional order.

Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, which is different from the previous works, we obtain the phase portraits of the the corresponding traveling wave systems. In addition, we derive the exact parametric representations of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, peakon solutions, periodic peakon solutions and compacton solutions under different parameter conditions.

Bifurcations, Exact Peakon, Periodic Peakons and Solitary Wave Solutions of the Cubic Camassa–Holm Type Equation

For the cubic Camassa–Holm type equation, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to analyze its corresponding traveling wave system, it was found that under different parameter conditions, its bifurcation portraits exhibit all possible exact explicit bounded solutions (solitary wave solutions, periodic wave solutions, peakon as well as periodic peakons). A total of 19 explicit exact parametric representations of the traveling wave system of the Camassa–Holm type equation are presented.

Existence of traveling waves in a delayed convecting shallow water fluid model

<abstract><p>This paper investigates a delayed shallow water fluid model that has not been studied in previous literature. Applying geometric singular perturbation theory, we prove the existence of traveling wave solutions for the model with a nonlocal weak delay kernel and local strong delay convolution kernel, respectively. When the convection term contains a nonlocal weak generic delay kernel, the desired heteroclinic orbit is obtained by using Fredholm theory and linear chain trick to prove the existence of two kink wave solutions under certain parametric conditions. When the model contains local strong delay convolution kernel and weak backward diffusion, under the same parametric conditions to the previous case, the corresponding traveling wave system can be reduced to a near-Hamiltonian system. The existence of a unique periodic wave solution is established by proving the uniqueness of zero of the Melnikov function. Uniqueness is proved by utilizing the monotonicity of the ratio of two Abelian integrals.</p></abstract>

Wave solutions to an integrable negative order KdV equation

In this paper, we study the traveling wave solutions to the second representative equation of the negative-order Korteweg–de Vries hierarchy. The corresponding traveling wave system is a singular planar dynamical system with a singular straight line. By using the method of dynamical systems, explicit solutions are given. But this integrable negative order Korteweg–de Vries flow has no traveling solitary wave solution.