NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s（R） for s 〉 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.

Analytical solutions of simplified modified Camassa-Holm equation with conformable and M-truncated derivatives:A comparative study

This paper extracts some analytical solutions of simplified modified Camassa-Holm(SMCH)equations with various derivative operators,namely conformable and M-truncated derivatives that have been recently introduced.The SMCH equation is used to model the unidirectional propagation of shallowwater waves.The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived.We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system.Furthermore,some 2D and 3D graphical illustrations of the obtained solutions have been presented.The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted.The derived solutions may contribute to comprehending wave propagation in shallow water.So,the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management,which are critical topics in the ocean and coastal engineering.

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.

A Modified Transitional Korteweg-De Vries Equation: Posed in the Quarter Plane

This paper is concerned with a modified transitional Korteweg-de Vries equation ut+f(t)u2ux+uxxx=0, (x,t)∈R+×R+with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either 1) f(t)≤0, f′(t)≥0or 2) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.

Gibbs Measure for the Higher Order Modified Camassa-Holm Equation

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann and Tzvetkov(Nonlinearity,23(2010),2771–2791),Tzvetkov(Probab.Theory Relat.Fields,146(2010),4679–4714),Burq,Thomann and Tzvetkov(Ann.Fac.Sci.Toulouse Math.,27(2018),527–597),the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.

Exact solutions of a time-fractional modified KdV equation via bifurcation analysis

The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.

Breather and its interaction with rogue wave of the coupled modified nonlinear Schrodinger equation

We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.

Dispersive propagation of optical solitions and solitary wave solutions of Kundu-Eckhaus dynamical equation via modified mathematical method

In this research work,we constructed the optical soliton solutions of nonlinear complex Kundu-Eckhaus(KE)equation with the help of modified mathematical method.We obtained the solutions in the form of dark solitons,bright solitons and combined dark-bright solitons,travelling wave and periodic wave solutions with general coefficients.In our knowledge earlier reported results of the KE equation with specific coefficients.These obtained solutions are more useful in the development of optical fibers,dynamics of solitons,dynamics of adiabatic parameters,dynamics of fluid,problems of biomedical,industrial phenomena and many other branches.All calculations show that this technique is more powerful,effective,straightforward,and fruitfulness to study analytically other higher-order nonlinear complex PDEs involves in mathematical physics,quantum physics,Geo physics,fluid mechanics,hydrodynamics,mathematical biology,field of engineering and many other physical sciences.

Solving Different Types of Differential Equations Using Modified and New Modified Adomian Decomposition Methods

The Modified Adomian Decomposition Method (MADM) is presented. A number of problems are solved to show the efficiency of the method. Further, a new solution scheme for solving boundary value problems with Neumann conditions is proposed. The scheme is based on the modified Adomian decomposition method and the inverse linear operator theorem. Several differential equations with Neumann boundary conditions are solved to demonstrate the high accuracy and efficiency of the proposed scheme.

Exact Traveling Wave Solutions to Phi-4 Equation and Joseph-Egri (TRLW) Equation and Calogro-Degasperis (CD) Equation by Modified (G'/G2)-Expansion Method

In this study, we will introduce the modified (G'/G2)-expansion method to explore some of the exact traveling wave solutions of some nonlinear partial differential equations namely, Phi-4 equation, Joseph-Egri (TRLW) equation, and Calogro-Degasperis (CD) equation. As a result, we have obtained solutions for the equations expressed in terms of trigonometric, hyperbolic and rational functions. Moreover, some selected solutions are plotted using some specific values for the parameters.

Blow-up criteria and periodic peakons for a two-component extension of the μ-version modified Camassa-Holm equation

Some two-component extensions of the modifiedμ-Camassa-Holm equation are proposed.We show that these systems admit Lax pairs and bi-Hamiltonian structures.Furthermore,we consider the blow-up phenomena for one of these extensions(2μmCH),and the periodic peakons of this system are derived.

A New Analytical Study of Modified Camassa-Holm and Degasperis-Procesi Equations

In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.

Quintessence anisotropic stellar models in quadratic and Born-Infeld modified teleparallel Rastall gravity

This study aims to discuss anisotropic solutions that are spherically symmetric in the quintessence field,which describe compact stellar objects in the modified Rastall teleparallel theory of gravity.To achieve this goal,the Krori and Barua arrangement for spherically symmetric components of the line element is incorporated.We explore the field equations by selecting appropriate off-diagonal tetrad fields.Born-Infeld function of torsion f(T)=β√λT+1-1 and power law form h(T)=δTn are used.The Born-Infeld gravity was the first modified teleparallel gravity to discuss inflation.We use the linear equation of state pr=ξρto separate the quintessence density.After obtaining the field equations,we investigate different physical parameters that demonstrate the stability and physical acceptability of the stellar models.We use observational data,such as the mass and radius of the compact star candidates PSRJ 1416-2230,Cen X-3,&4U 1820-30,to ensure the physical plausibility of our findings.

Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.

A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.

Constitutive equations of 1060 pure aluminum based on modified double multiple nonlinear regression model

In order to study the work-ability and establish the optimum hot formation processing parameters for industrial 1060 pure aluminum, the compressive deformation behavior of pure aluminum was investigated at temperatures of 523?823 K and strain rates of 0.005?10 s?1 on a Gleeble?1500 thermo-simulation machine. The influence rule of processing parameters (strain, strain rate and temperature) on flow stress of pure aluminum was investigated. Nine analysis factors consisting of material parameters and according weights were optimized. Then, the constitutive equations of multilevel series rules, multilevel parallel rules and multilevel series &parallel rules were established. The correlation coefficients (R) are 0.992, 0.988 and 0.990, respectively, and the average absolute relative errors (AAREs) are 6.77%, 8.70% and 7.63%, respectively, which proves that the constitutive equations of multilevel series rules can predict the flow stress of pure aluminum with good correlation and precision.

Approximate any 1-state solutions of the Dirac equation for modified deformed Hylleraas potential by using the Nikiforov—Uvarov method

We investigate the spin and pseudospin symmetries of the Dirac equation under modified deformed Hylleraas potential via a Pekeris approximation and the Nikiforov-Uvarov technique. A tensor interaction of Coulomb form is considered and its degeneracy-removing role is discussed in detail. The solutions are reported for an arbitrary quantum number in a compact form and useful numerical data are included.

New Jacobian Elliptic Function Solutions to Modified KdV Equation: Ⅰ

An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdVequation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu's allresults are obtained. When the modulus m → 1 or 0, we can find the corresponding six solitary wave solutions and sixtrigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian ellipticfunction solutions and can be applied to other nonlinear differential equations.

A Modified Form of Mild-Slope Equation with Weakly Nonlinear Effect

Nonlinear effect is of importance to waves propagating from deep water to shallow water. The non-linearity of waves is widely discussed due to its high precision in application. But there are still some problems in dealing with the nonlinear waves in practice. In this paper, a modified form of mild-slope equation with weakly nonlinear effect is derived by use of the nonlinear dispersion relation and the steady mild-slope equation containing energy dissipation. The modified form of mild-slope equation is convenient to solve nonlinear effect of waves. The model is tested against the laboratory measurement for the case of a submerged elliptical shoal on a slope beach given by Berkhoff et al. The present numerical results are also compared with those obtained through linear wave theory. Better agreement is obtained as the modified mild-slope equation is employed. And the modified mild-slope equation can reasonably simulate the weakly nonlinear effect of wave propagation from deep water to coast.