Exact Solutions to (3+1) Conformable Time Fractional Jimbo–Miwa,Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations

Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.     

Generalized Orthogonal Polynomials, Discrete KP and Riemann–Hilbert Problems

Classically, a single weight on an interval of the real line leads to moments, orthogonal polynomials and tridiagonal matrices. Appropriately deforming this weight with times t= (t ₁, t ₂, …), leads to the standard Toda lattice and τ-functions, expressed as hermitian matrix integrals. This paper is concerned with a sequence of t-perturbed weights, rather than one single weight. This sequence leads to moments, polynomials and a (fuller) matrix evolving according to the discrete KP-hierarchy. The associated τ-functions have integral, as well as vertex operator representations. Among the examples considered, we mention: nested Calogero–Moser systems, concatenated solitons and m-periodic sequences of weights. The latter lead to 2m+ 1-band matrices and generalized orthogonal polynomials, also arising in the context of a Riemann–Hilbert problem. We show the Riemann–Hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower–times upper–triangular matrix. Received: 8 September 1998 / Accepted: 27 April 1999     

Novel Wronskian Condition and New Exact Solutions to a （3＋1）-Dimensional Generalized KP Equation

Utilizing the Wronskian technique, a combined Wronskian condition is established for a （3＋1）-dimensional generalized KP equation. The generating functions for matrix entries satisfy a linear system of new partial differential equations. Moreover, as applications, examples of Wronskian determinant solutions, including N-soliton solutions, periodic solutions and rational solutions, are computed.     

Traveling waves in rational expressions of exponential functions to the conformable time fractional Jimbo–Miwa and Zakharov–Kuznetsov equations

The conformable time fractional Jimbo–Miwa and Zakharov–Kuznetsov equations are solved by the generalized form of the Kudryashov method. A simple compatible wave transformation is employed to reduce the dimension of the equations to one. The predicted solution is of the form of a rational expression of two finite series at both the numerator and the denominator. The terms of both series are of the powers of some functions having exponential expressions satisfying a particular ODE. The exact solutions are expressed explicitly in terms of powers of some exponential functions in form of rational expressions.     

Variable Separation and Exact Solutions to Generalized Nonlinear Diffusion Equations

We study in detail a method to find the generalized nonlinear diffusion equations,which can be solved by means of the variable separation approach.A complete list of canonical forms for such equations,which admit the functional separable solutions,is botained and some exact solutions to the resulting equations are described. A number of methods have been proven to be effective for finding symmetry reductions and constructing exact solutions to nonlinear diffusion equations.     

Variable Separation and Derivative-Dependent Functional Separable Solutions to Generalized KdV Equations

Using the generalized conditional symmetry approach, a complete list of canonical forms for the Korteweg de-Vries type equations with which possessing derivative-dependent functional separable solutions (DDFSSs) is obtained. The exact DDFSSs of the resulting equations are explicitly exhibited.     

Generalized Euler–Poincaré Equations on Lie Groups and Homogeneous Spaces,Orbit Invariants and Applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler–Poincaré equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler–Poincaré equations that have not yet been considered in the literature as well as integrable equations like Camassa–Holm, Degasperis–Procesi, μCH and μDP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.     

New travelling wave and anti-kink wave solutions of space-time fractional (3+1)-Dimensional Jimbo–Miwa equation

In this work, the new analytical exact solution of a highly nonlinear fractional partial differential equation (FPDE) has been presented; specifically space-time fractional (3+1)-dimensional Jimbo–Miwa (JM) equation. As a consequence of the applied extended (G'/G)-expansion method, the new analytical exact solution of the governing equation has been acquired successfully. Moreover, the physical natures of the solutions have been analyzed here by means of numerical simulations.     

On the Weak Solutions to the Maxwell–Landau–Lifshitz Equations and to the Hall–Magneto–Hydrodynamic Equations

In this paper we deal with weak solutions to the Maxwell–Landau–Lifshitz equations and to the Hall–Magneto–Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall–Magneto–Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally we examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes.     

A Note on Symmetries and Generalized W∞ Algebra of the Modified KP Equation

Some remarks on the paper Symmetries and generalized W algebra of the modified KP equation (S. Y. Lou and G. J. Ni, Lett. Math. Phys. 34 (1995), 327–331) are given. It is pointed out that if we consider the inverse operator of a differential operator to be a linear operator, the vector fields nv(f) defined in the above Letter are not certain to be symmetries of the modified KP equation under consideration.     

Lax Equations, Singularities and Riemann–Hilbert Problems

The existence of singularities of the solution for a class of Lax equations is investigated using a development of the factorization method first proposed by Semenov-Tyan-Shanski? (Funct Anal Appl 17(4):259?C272, 1983) and Reymann and Semenov-Tian-Shansky (1994). It is shown that the existence of a singularity at a point t?=?t _i is directly related to the property that the kernel of a certain Toeplitz operator (whose symbol depends on t) be non-trivial. The investigation of this question involves the factorization on a Riemann surface of a scalar function closely related to the above-mentioned operator. Two examples are presented which show different aspects of the problem of computing the set of singularities of the solution to the system considered. The relation between the Riemann surfaces of the classical and Lax formulation is also considered.     

Generalized WDVV Equations for Br and Cr Pure N = 2 Super-Yang–Mills Theory

A proof that the prepotential for pure N = 2 Super-Yang–Mills theory associated with Lie algebras B _r and C _r satisfies the generalized WDVV (Witten–Dijkgraaf–Verlinde–Verlinde) system was given by Marshakov, Mironov, and Morozov. Among other things, they use an associative algebra of holomorphic differentials. Later Itô and Yang used a different approach to try to accomplish the same result, but they encountered objects of which it is unclear whether they form structure constants of an associative algebra. We show by explicit calculation that these objects are none other than the structure constants of the algebra of holomorphic differentials.     

Bäcklund transformation,infinitely-many conservation laws,solitary and periodic waves of an extended (3 + 1)-dimensional Jimbo–Miwa equation with time-dependent coefficients

In this paper, an extended (3+1)-dimensional Jimbo–Miwa equation with time-dependent coefficients is investigated, which comes from the second member of the Kadomtsev–Petviashvili hierarchy and is shown to be conditionally integrable. Bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived via the binary Bell polynomials and symbolic computation. With the help of the bilinear form, one-, two- and three-soliton solutions are obtained via the Hirota method, one-periodic wave solutions are constructed via the Riemann theta function. Additionally, propagation and interaction of the solitons are investigated analytically and graphically, from which we find that the interaction between the solitons is elastic and the time-dependent coefficients can affect the soliton velocities, but the soliton amplitudes remain unchanged. One-periodic waves approach the one-solitary waves with the amplitudes vanishing and can be viewed as a superposition of the overlapping solitary waves, placed one period apart.     

Generalized Euler–Lagrange Equations for Variational Problems with Scale Derivatives

We obtain several Euler–Lagrange equations for variational functionals defined on a set of Hölder curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives are considered.     

Generalized Beth–Uhlenbeck approach to mesons and diquarks in hot,dense quark matter

An important first step in the program of hadronization of chiral quark models is the bosonization in meson and diquark channels. This procedure is presented at finite temperatures and chemical potentials for the SU(2) flavor case of the NJL model with special emphasis on the mixing between scalar meson and scalar diquark modes which occurs in the 2SC color superconducting phase. The thermodynamic potential is obtained in the Gaussian approximation for the meson and diquark fields and it is given in the Beth–Uhlenbeck form. This allows a detailed discussion of bound state dissociation in hot, dense matter (Mott effect) in terms of the in-medium scattering phase shift of two-particle correlations. It is shown for the case without meson–diquark mixing that the phase shift can be separated into a continuum and a resonance part. In the latter, the Mott transition manifests itself by a change of the phase shift at threshold by π$\pi$ in accordance with Levinson’s theorem, when a bound state transforms to a resonance in the scattering continuum. The consequences for the contribution of pionic correlations to the pressure are discussed by evaluating the Beth–Uhlenbeck equation of state in different approximations. A similar discussion is performed for the scalar diquark channel in the normal phase. Further developments and applications of the developed approach are outlined.     

A Semi-Analytical Solution to Classic Yang–Mills Equations with Both Asymptotical Freedom and Confining Features

It is well known that confinings and asymptotic freedom are properties of quantum chromo-dynamics(QCD).But hints of these features can also be observed at purely classic levels.For this purpose we need to find solutions to the colorly-sourceful Yang–Mills equations with both confining and asymptotic freedom features.We provide such a solution in this paper which at the near-source region is of serial form,while at the far-away region is approximately expressed through simple elementary functions.From the solution,we derive out a classically non-perturbative beta function describing the running of efective coupling constant,which is linear in the couplings both in the infrared and ultraviolet region.     

Explicit Solutions of （2A-1）-Dimensional Canonical Generalized KP,KdV, and （2A-1）-Dimensional Burgers Equations with Variable Coefficients

In this paper, using the generalized （G1/G）-expansion method and the auxiliary differential equation method, we discuss the （2＋1）-dimensional canonical generalized KP （CGKP）, KdV, and （2＋1）-dimensional Burgers equations with variable coetticients. Many exact solutions of the equations are obtained in terms of elliptic functions, hyperbolic functions, trigonometric functions, and rational functions.     

A Statistical Approach to the Asymptotic Behavior of a Class of Generalized Nonlinear Schrödinger Equations

A statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear Schrödinger-type which govern non-integrable, non-singular dynamics. In a bounded domain the solutions of these equations have been shown numerically to tend in the long-time limit toward a Gibbsian statistical equilibrium state consisting of a ground-state solitary wave on the large scales and Gaussian fluctuations on the small scales. The main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit. The large deviation principle pertains to a process governed by a Gibbs ensemble that is canonical in energy and microcanonical in particle number. Some supporting Monte-Carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation, especially the high frequency growth of the dispersion relation. The large deviation principle for the process governed by the Gibbs ensemble is based on a large deviation principle for Gaussian processes, for which two independent proofs are given.This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0202309)This research was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation.This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0207064).