Exact Traveling Wave Solutions for Generalized Camassa-Holm Equation by Polynomial Expansion Methods

We formulate efficient polynomial expansion methods and obtain the exact traveling wave solutions for the generalized Camassa-Holm Equation. By the methods, we obtain three types traveling wave solutions for the generalized Camassa-Holm Equation: hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. At the same time, we have shown graphical behavior of the traveling wave solutions.

Orbital Stability of Peakons for a Generalized Camassa-Holm Equation

We investigate the orbital stability of the peakons for a generalized Camassa-Holm equation (gCH). Using variable transformation, a planar dynamical system is obtained from the gCH equation. It is shown that the planar system has two heteroclinic cycles which correspond two peakon solutions. We then prove that the peakons for the gCH equation are orbitally stable by using the method of Constantin and Strauss.

Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes

In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.

A GENERALIZED SCALAR AUXILIARY VARIABLE METHOD FOR THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS

This paper develops a generalized scalar auxiliary variable(SAV)method for the time-dependent Ginzburg-Landau equations.The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations.In this method,the system is decoupled and linearized to avoid solving the non-linear equation at each step.The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability,and this is confirmed by the numerical result,and also shows that the numerical algorithm is stable.

Dynamics and Exact Solutions of (1 + 1)-Dimensional Generalized Boussinesq Equation with Time-Space Dispersion Term

We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.

The Global Attractor and Its Dimension Estimation of Generalized Kolmogorov-Petrovlkii-Piskunov Equation

In this paper, the initial boundary value problem of a class of nonlinear generalized Kolmogorov-Petrovlkii-Piskunov equations is studied. The existence and uniqueness of the solution and the bounded absorption set are proved by the prior estimation and the Galerkin finite element method, thus the existence of the global attractor is proved and the upper bound estimate of the global attractor is obtained.

Adequate Closed Form Wave Solutions to the Generalized KdV Equation in Mathematical Physics

In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, the electro-hydro-dynamical model for local electric field, signal processing waves through optical fibers, etc. We determine the useful and further general exact traveling wave solutions of the above mentioned NLDEs by applying the exp(−τ(ξ))-expansion method by aid of traveling wave transformations. Furthermore, we explain the physical significance of the obtained solutions of its definite values of the involved parameters with graphic representations in order to know the physical phenomena. Finally, we show that the exp(−τ(ξ))-expansion method is convenient, powerful, straightforward and provide more general solutions and can be helping to examine vast amount of travelling wave solutions to the other different kinds of NLDEs.

An integrable generalization of the Fokas–Lenells equation:Darboux transformation, reduction and explicit soliton solutions

Under investigation is an integrable generalization of the Fokas–Lenells equation, which can be derived from the negative power flow of a 2 × 2 matrix spectral problem with three potentials. Based on the gauge transformation of the matrix spectral problem, one kind of Darboux transformation with multi-parameters for the three-component coupled Fokas–Lenells system is constructed. As a reduction, the N-fold Darboux transformation for the generalized Fokas–Lenells equation is obtained, from which the N-soliton solution in a compact Vandermonde-like determinant form is given. Particularly,the explicit one-and two-soliton solutions are presented and their dynamical behaviors are shown graphically.

Explicit Peaked Wave Solutions to the Generalized Camassa-Holm Equation

By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.

Comparative Analysis of the Generalized Omega Equation and Generalized Vertical Motion Equation

Research on vertical motion in mesoscale systems is an extraordinarily challenging effort.Allowing for fewer assumptions,a new form of generalized vertical motion equation and a generalized Omega equation are derived in the Cartesian coordinate system(nonhydrostatic equilibrium)and the isobaric coordinate system(hydrostatic equilibrium),respectively.The terms on the right-hand side of the equations,which comprise the Q vector,are composed of three factors:dynamic,thermodynamic,and mass.A heavy rain event that occurred from 18 to 19 July 2021 in southern Xinjiang was selected to analyze the characteristics of the diagnostic variable in the generalized vertical motion equation(Qz)and the diagnostic variable in the generalized Omega equation(Qp)using high-resolution model data.The results show that the horizontal distribution of the Qz-vector divergence at 5.5 km is roughly similar to the distribution of the Qp-vector divergence at 500 hPa,and that both relate well to the composite radar reflectivity,vertical motion,and hourly accumulated precipitation.The Qz-vector divergence is more effective in indicating weak precipitation.In vertical cross sections,regions with alternating positive and negative large values that match the precipitation are mainly concentrated in the middle levels for both forms of Q vectors.The temporal evolutions of vertically integrated Qz-vector divergence and Qp-vector divergence are generally similar.Both perform better than the classical quasigeostrophic Q vector and nongeostrophic Q vector in indicating the development of the precipitation system.

MOLECULES AND NEW INTERACTIONAL STRUCTURES FOR A(2+1)-DIMENSIONAL GENERALIZED KONOPELCHENKO-DUBROVSKY-KAUP-KUPERSHMIDT EQUATION

Soliton molecules(SMs)of the(2+1)-dimensional generalized KonopelchenkoDubrovsky-Kaup-Kupershmidt(gKDKK)equation are found by utilizing a velocity resonance ansatz to N-soliton solutions,which can transform to asymmetric solitons upon assigning appropriate values to some parameters.Furthermore,a double-peaked lump solution can be constructed with breather degeneration approach.By applying a mixed technique of a resonance ansatz and conjugate complexes of partial parameters to multisoliton solutions,various kinds of interactional structures are constructed;There include the soliton molecule(SM),the breather molecule(BM)and the soliton-breather molecule(SBM).Graphical investigation and theoretical analysis show that the interactions composed of SM,BM and SBM are inelastic.

A LARGE DEVIATION PRINCIPLE FOR THE STOCHASTIC GENERALIZED GINZBURG-LANDAU EQUATION DRIVEN BY JUMP NOISE

In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise.Here,we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle,which was initially proposed by Matoussi,Sabbagh and Zhang(2021).

Influence of dissipation on solitary wave solution to generalized Boussinesq equation

This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.

From Generalized Hamilton Principle to Generalized Schrodinger Equation

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.

Solitons and Bifurcations for the Generalized Tzitzéica Type Equation in Nonlinear Fiber Optics

Solitons and bifurcations for the generalized Tzitzéica type equation are studied by using the theory of dynamical systems and Hamilton function. With the help of Maple and bifurcation theory of differential equations, the bifurcation parameter conditions and all the bifurcation phase portraits are obtained. Because the same energy value of the Hamiltonian function is corresponding to the same orbit, thus the periodic wave solutions, bright soliton and dark soliton solutions are defined.

Exact Traveling Wave Solutions of the Generalized Fractional Differential mBBM Equation

By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its equilibrium points and first integral. Based on this, the phase portraits of the corresponding plane dynamic system are given. According to the phase diagram characteristics of the dynamic system, the periodic solution corresponds to the limit cycle or periodic closed orbit. Therefore, according to the phase portraits and the properties of elliptic functions, we obtain exact explicit parametric expressions of smooth periodic wave solutions. This method can also be applied to other fractional equations.

Semigroup of Weakly Continuous Operators Associated to a Generalized Schrödinger Equation

In this work, we prove the existence and uniqueness of the solution of the generalized Schrödinger equation in the periodic distributional space P’. Furthermore, we prove that the solution depends continuously respect to the initial data in P’. Introducing a family of weakly continuous operators, we prove that this family is a semigroup of operators in P’. Then, with this family of operators, we get a fine version of the existence and dependency continuous theorem obtained. Finally, we provide some consequences of this study.

The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation

A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the 3 pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs（R） with s 〉 is established via a limiting procedure. Provided that the initial momentum （1 - 02）u0 satisfies the sign condition,the existence and uniqueness of global solutionsfor the equation are shown to be true in the space C.

Blow-Up for a Periodic Two-Component Camassa-Holm Equation with Generalized Weakly Dissipation

In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.

BIFURCATIONS OF THE TRAVELLING WAVE SOLUTIONS TO A GENERALIZED CAMASSA-HOLM EQUATION

In this paper, we study some generalized Camassa-Holm equation. Through the analysis of the phase-portraits, the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave and compactons were discussed. In some certain parametric conditions, many exact solutions to the above travelling waves were given. Further-more, the 3D and 2D pictures of the above travelling wave solutions are drawn using Maple software.