Higher-dimensional Chen-Lee-Liu equation and asymmetric peakon soliton

Integrable systems play a crucial role in physics and mathematics.In particular,the traditional(1+1)-dimensional and(2+1)-dimensional integrable systems have received significant attention due to the rarity of integrable systems in higher dimensions.Recent studies have shown that abundant higher-dimensional integrable systems can be constructed from(1+1)-dimensional integrable systems by using a deformation algorithm.Here we establish a new(2+1)-dimensional Chen-Lee-Liu(C-L-L)equation using the deformation algorithm from the(1+1)-dimensional C-L-L equation.The new system is integrable with its Lax pair obtained by applying the deformation algorithm to that of the(1+1)-dimension.It is challenging to obtain the exact solutions for the new integrable system because the new system combines both the original C-L-L equation and its reciprocal transformation.The traveling wave solutions are derived in implicit function expression,and some asymmetry peakon solutions are found.

CALCULATION OF TWO-DIMENSIONAL PARABOLIC STABILITY EQUATIONS

Two dimensional parabolic stability equations (PSE) are numerically solved using expansions in orthogonal functions in the normal direction.The Chebyshev polynomials approximation,which is a very useful form of orthogonal expansions, is applied to solving parabolic stability equations. It is shown that results of great accuracy are effectively obtained.The availability of using Chebyshev approximations in parabolic stability equations is confirmed.

Rogue Waves in the(2+1)-Dimensional Nonlinear Schrodinger Equation with a Parity-Time-Symmetric Potential

The (2+1)-dimension nonlocal nonlinear Schrödinger (NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al. [Phys. Rev. Lett. 110 (2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the plane.

Painlevé Property and Exact Solutions to a (2+1) Dimensional KdV-mKdV Equation

A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.

On a Class of Two Dimensional Singular Integral Equations

We mainly consider a class of two dimensional normal type singular integral equations, the solutions as well as the conditions of solvability of which are obtained .The methods used consist of transferring them to several one dimensional Riemann boundary value problems, and then solving the latter.

Symmetries of(2+1)-Dimensional KdV-Ito Equation in the Bilinear Form^1

A set of symmetries of a generalized (2+l)-dimensional bilinear equation is given by a formal serins formula. There exist four truncated symmetries for the KdV-Ito model. These trun-cated symmetries with four arourary functions of time t constitute an infinnite-dirmensional Lin algebra which contains two types of the Virasoro subalgebra.

Painleve Analysis for(2+1)Dimensional Non-Linear Schrodinger Equation

This paper investigates a real version of a (2 + 1) dimensional nonlinear Schr?dinger equation through adoption of Painlevé test by means of which the (2 + 1) dimensional nonlinear Schr?dinger equation is studied according to the Weiss et al. method and Kruskal’s simplification algorithms. According to Painlevé test, it is found that the number of arbitrary functions required for explaining the Cauchy-Kovalevskaya theorem exist. Finally, the associated B?cklund transformation and bilinear form is directly obtained from the Painlevé test.

Adaptive Sparse Grid Discontinuous Galerkin Method:Review and Software Implementation

This paper reviews the adaptive sparse grid discontinuous Galerkin(aSG-DG)method for computing high dimensional partial differential equations(PDEs)and its software implementation.The C++software package called AdaM-DG,implementing the aSG-DG method,is available on GitHub at https://github.com/JuntaoHuang/adaptive-multiresolution-DG.The package is capable of treating a large class of high dimensional linear and nonlinear PDEs.We review the essential components of the algorithm and the functionality of the software,including the multiwavelets used,assembling of bilinear operators,fast matrix-vector product for data with hierarchical structures.We further demonstrate the performance of the package by reporting the numerical error and the CPU cost for several benchmark tests,including linear transport equations,wave equations,and Hamilton-Jacobi(HJ)equations.

REMARKS ON NONLINEAR GALERKIN METHOD FOR KURAMOTO-SIVASHINSKY EQUATION

This paper is concentrated on a nonlinear Galerkin method with sm small- scale components for Kuramoto-Sivashmsky equation, in which convergence results and the analysis of error estimates are given, The conclusion shows that this choce of modes is efficient .for The method modifred.

Attractors of Dissipative Soliton Equation

In the paper ,we study longtime dynamic behavior of dissipative soliton equation existence of attractor ,geometrical structure of attractor dynamic behavior under the parametric perturbation of dissipative soliton equation.estimate of fractal dimension of attractor.

THE INERTIAL FRACTAL SET FOR WEAKLY DAMPED FORCED KORTEWEG-DE-VRIES EQUATION

In this paper we consider weakly damped forced Korteweg-de-Vries equation withnon-self-adjoint operator. The existence of inertial fractal set M of this equation is proved, the estimates of the upper bounds of fractal dimension for M are alsoobtained.

THE EXISTENCE OF GLOBAL SOLUTION AND "BLOW UP" PHENOMENON FOR A SYSTEM OF MULTI DIMENSIONAL SYMMETRIC REGULARIZED WAVE EQUATIONS

The existence and uniqueness of the global smooth solution to the initial-boundary valueproblem of a system of multi-dimensions SRWE are proved. The sufficient conditions of 'blowingup' of the solution are given.

Existence of Solutions for Three Dimensional Stationary Incompressible Euler Equations with Nonvanishing Vorticity

In this paper,solutions with nonvanishing vorticity are established for the three dimensional stationary incompressible Euler equations on simply connected bounded three dimensional domains with smooth boundary.A class of additional boundary conditions for the vorticities are identified so that the solution is unique and stable.

Multiple Positive Solutions for One Dimensional Third Order p-Laplacian Equations with Integral Boundary Conditions

In this paper,we consider the one dimensional third order p-Laplacian equation(Фp(u″))′+h(t)f(t,u(t))=0 with integral boundary conditions u(0)-αu′(0)=∫_(0)^(1) g1(s)u(s)ds,u(1)+βu′(1)=∫_(0)^(1)g2(s)u(s)ds,u″(0)=0.By using kernel functions and the Avery-Peterson fixed point theorem,we establish the existence of at least three positive solutions.

Darboux Transformation and Exact Solutions for Two Dimensional A_(2n-1)^((2)) Toda Equations

The Darboux transformation for the two dimensional A_(2n-1)^((2))Toda equations is constructed so that it preserves all the symmetries of the corresponding Lax pair.The expression of exact solutions of the equation is obtained by using Darboux transformation.

Dynamics for Three Dimensional Generalized Navier-Stokes Equations with Delay

This paper is concerned with the existence of pullback attractors for three dimensional generalized Navier-Stokes equations with delay.According to compact argument,the existence and uniqueness of weak solutions are proved by using Galerkin method,and the continuous dependence of solutions on initial values is also shown.Based on the asymptotic compactness via weak convergence method and pullback absorbing set on appropriate functional phase spaces,we get the existence of pullback attractors.

A Novel Numerical Method of O(h^(4))for Three-Dimensional Non-Linear Triharmonic Equations

In this article,we present two new novel finite difference approximations of order two and four,respectively,for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u,δ^(2)u/δn^(2)andδ^(4)u/δn^(4)are prescribed on the boundary.We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions.We require only 7-and 19-grid points on the compact cell for the second and fourth order approximation,respectively.The Laplacian and the biharmonic of the solution are obtained as by-product of the methods.We require only system of three equations to obtain the solution.Numerical results are provided to illustrate the usefulness of the proposed methods.

IMPLICITY LINEAR COLLOCATION METHOD AND ITERATED IMPLICITY LINEAR COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF HAMMERSTEIN FREDHOLM INTEGRAL EQUATIONS ON 2D IRREGULAR DOMAINS

In this work,we adapt and compare implicity linear collocation method and iterated implicity linear collocation method for solving nonlinear two dimensional Fredholm integral equations of Hammerstein type using IMQ-RBFs on a non-rectangular domain.IMQs show to be the most promising RBFs for this kind of equations.The proposed methods are mesh-free and they are independent of the geometry of domain.Convergence analysis of the proposed methods together with some benchmark examples are provided which support their reliability and numerical stability.

Local Well-posedness of the Derivative Schrödinger Equation in Higher Dimension for Any Large Data

In this paper,the authors consider the local well-posedness for the derivative Schrödinger equation in higher dimension ut-iΔu+|u|^(2)(→γ·▽u)+u^(2)(→λ·▽-u)=0,(x,t)∈R^(n)×R,→γ,→λ∈R^(n);n≥2 It is shown that the Cauchy problem of the derivative Schrödinger equation in higher dimension is locally well-posed in H^(s)(R^(n))(s>n/2)for any large initial data.Thus this result can compare with that in one dimension except for the endpoint space H^(n/2).

Estimation of the Postmortem Interval by Measuring Blood Oxidation‑reduction Potential Values

Accurate estimation of the postmortem interval(PMI)is an important task in forensic practice.In the last half-century,the use of postmortem biochemistry has become an important ancillary method in determining the time of death.The present study was carried out to determine the correlation between blood oxidation-reduction potential(ORP)values and PMIs,and to develop a three-dimensional surface equation to estimate the PMI under various temperature conditions.A total of 48 rabbits were placed into six groups and sacrificed by air embolism.Blood was obtained from the right ventricle of each rabbit,and specimens were stored at 10℃,15℃,20℃,25℃,30℃,and 35℃.At different PMIs(once every 4 h),the blood ORP values were measured using a PB-21 electrochemical analyzer.Statistical analysis and curve fitting of the data yielded cubic polynomial regression equations and a surface equation at different temperatures.Result:The results showed that there was a strong positive correlation between the blood ORP values at different temperatures and the PMI.This study provides another example of using a three-dimensional surface equation as a tool to estimate the PMI at various temperature conditions.