Bernstein-Greene-Kruskal modes in a three-dimensional plasma

Bernstein-Greene-Kruskal modes in a three-dimensional (3D) unmagnetized plasma are constructed. It is shown that 3D solutions that depend only on energy do not exist. However, 3D solutions that depend on energy and additional constants of motion (such as angular momentum) do exist. Exact analytical as well as numerical solutions are constructed assuming spherical symmetry, and their properties are contrasted with those of 1D solutions. Possible extensions to solutions with cylindrical symmetry with or without a finite magnetic guide field are discussed.     

Two-Dimensional Rossby Waves: Exact Solutions to Petviashvili Equation

The two-dimensional (2D) nonlinear Rossby waves described by the Petviashvili equation, which has been invoked as an ageostrophic extension of the barotropic quasi-geostrophic potential vorticity equation, can be investigated through the exact periodic-wave solutions for the Petviashvili equation, while the exact analytical periodic-wave solutions to the Petviashvili equation are obtained by using the Jacobi elliptic function expansion method. It is shown that periodic-wave 2D Rossby solutions can be obtained by this method, and in the limit cases, the 2D Rossby soliton solutions are also obtained.     

A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆2 ± λ2

In this paper, we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆² ± λ². Similar to the derivation of fundamental solutions, it is non-trivial to derive particular solutions for higher order differential operators. In this paper, we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D. The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration. Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.     

Abundant solutions of Wick-type stochastic fractional 2D KdV equations

A modified fractional sub-equation method is applied to Wick-type stochastic fractional two-dimensional （2D） KdV equations. With the help of a Hermit transform, we obtain a new set of exact stochastic solutions to Wick-type stochastic fractional 2D KdV equations in the white noise space. These solutions include exponential decay wave solutions, soliton wave solutions, and periodic wave solutions. Two examples are explicitly given to illustrate our approach.     

Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect

In this paper, a set of 3D general solutions to static problems of 1D hexagonal piezoelectric quasicrystals is obtained by introducing two displacement functions and utilizing the rigorous operator theory. All the physical quantities are expressed by five quasi-harmonic functions. Based on the general solutions and with the help of the superposition principle, fundamental solutions for infinite/half-infinite spaces are presented by trial-and-error technique. The general solutions can be conveniently used to solve the boundary value problems regarding dislocations, cracks and inhomogeneities. The fundamental solutions are of primary significance to development of numerical codes such as boundary element method.     

Direct quadrature method of moments solution of the Fokker–Planck equation

The direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the Fokker–Planck equation corresponding to stochastic nonlinear dynamical systems. The theoretical details of the solution procedure are first presented. The method is then used to solve Fokker–Planck equations for both 1D and 2D (noisy van der Pol oscillator) processes which possess nonlinear stochastic differential equations. Higher-order moments of the stationary solutions are computed and prove to be very accurate when compared to analytic (1D process) and Monte Carlo (2D process) solutions.     

New Types of Solutions for the Classic 3D SU (2) Yang–Mills Equations

Doklady Physics - New types of 3D solutions for the classic Yang–Mills equations in the Faddeev–Niemi reformulation are found. In a particular case, these solutions describe 3D vortices.     

Global regularity of the 4D restricted Euler equations

We are concerned with the critical threshold phenomena in the restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for the 3D and 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that the 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic.     

Single-scale two-dimensional-three-component generalized-Beltrami-flow solutions of incompressible Navier-Stokes equations

Generalized-Beltrami-flow (GBF) solutions, which are exact solutions of incompressible Navier-Stokes equations (NSE), are still rare. Most existing GBF solutions are either planar or axisymmetric cases. We derive analytically a series of single-scale two-dimensional-three-component (2D3C) GBF solutions under the framework of helical decomposition. These solutions yield a manifold of fixed points with infinite degrees of freedom in the solution space. The key of the derivation is to arbitrarily put different wave vectors at the same wave length, and to apply a novel parallel relation to any pair of these wave vectors. Although these solutions belong to a general class of 2D3C Euler solutions, to our knowledge there has been no publication focusing on these particular GBF forms. The significance of these GBF solutions is that the novel parallel relation implies new statistical relations on turbulence energy transfer and velocity phases.     

Modeling of Dark Solitons for Nonlinear Longitudinal Wave Equation in a Magneto-Electro-Elastic Circular Rod

In this paper, sub equation and expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation (LWE) in a magneto-electro-elastic circular rod. The proposed methods have been used to construct hyperbolic, rational, dark soliton and trigonometric solutions of the LWE in the magneto-electro-elastic circular rod. Arbitrary values are given to the parameters in the solutions obtained. 3D, 2D and contour graphs are presented with the help of a computer package program. Solutions attained by symbolic calculations revealed that these methods are effective, reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics.     

Closed-form analytical solutions of the time difference of arrival source location problem for minimal element monitoring arrays

Closed-form analytical solutions are found for the time difference of arrival (TDOA) source location problem. Solutions are found for both two-dimensional (2D) and three-dimensional (3D) source location by formulating the TDOA equations in, respectively, polar and spherical coordinate systems, with the radial direction coincident with the assumed geodesic path of signal propagation to a reference sensor. Quadratic equations for TDOA 2D and 3D source location based on the spherical intersection (SX) scheme, in some cases permitting dual physical solutions, are found for three and four sensor element monitoring arrays, respectively. A method of spherical intersection subarrays (SXSAs) is developed to derive from these quadratic equations globally unique closed-form analytical solutions for TDOA 2D and 3D source location, for four and five sensor element monitoring arrays, respectively. Errors in 2D source location for introduced bias in time differences of arrival are shown to have a strong geometrical dependence. The SXSA and SX methods perform well in terms of accuracy and precision at high levels of arrival time bias for both 2D and 3D source location and are much more efficient than nonlinear least-squares schemes. The SXSA scheme may have particular applicability to accurately solving source location problems in demanding real-time situations.     

New Symmetry Reductions,Dromions—Like and Compacton Solutions for a 2D BS（m,n） Equations Hierarchy with Fully Nonlinear Dispersion

We have found two types of important exact solutions,compacton solutions,which are solitary waves with the property that after colliding with their own kind,they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction,in the (1 1)D,(1 2)D and even (1 3)D models,and dromion solutions (exponentially decaying solutions in all direction) in many (1 2)D and (1 3)D models.In this paper,symmetry reductions in (1 2)D are considered for the break soliton-type equation with fully nonlinear dispersion (called BS(m,n) equation)ut b(u^m)xxy 4b(u^n δx^-1uy)x=0,which is a generalized model of (1 2)D break soliton equation ut buxxy 4buuy 4buxδx^-1uy=0,by using the extended direct reduction method.As a result,six types of symmetry reductions are obtained.Starting from the reduction equations and some simple transformations,we obtain the solitary wavke solutions of BS(1,n) equations,compacton solutions of BS(m,m-1) equations and the compacton-like solution of the potential form (called PBS(3,2)) ωxt b(ux^m)xxy 4b(ωx^nωy)x=0.In addition,we show that the variable ∫^x uy dx admits dromion solutions rather than the field u itself in BS(1,n) equation.     

Generalized Wronskian Solutions to Differential-Difference KP Equation

A new method for constructing the Wronskian entries is proposed and applied to the differential-difference Kadomtsev-Petviashvilli (DΔKP) equation. The generalized Wronskian solutions to it are obtained, including rational solutions and Matveev solutions.     

Rotating dyonic dipole black rings: exact solutions and thermodynamics

New rotating dyonic dipole black ring solutions are derived in 5D Einstein-dilaton gravity with antisymmetric forms. The black rings are analyzed and their thermodynamics is discussed. New dyonic black string solutions are also presented.     

Soliton Solutions of Integrable Hierarchies and Coulomb Plasmas

Some direct relations are given between soliton solutions of integrable hierarchies and thermodynamic quantities of the Coulomb plasmas on the plane. We find that certain soliton solutions of the Kadomtsev–Petviashvili (KP) and B-type KP (BKP) hierarchies describe 2D one- or two-component lattice plasmas at special boundary conditions and fixed temperatures. It is shown that different reductions of integrable hierarchies describe pure or dipole Coulomb gases on 1D submanifolds embedded in the 2D space.     

A new class of nodal stationary states in a 2D Heisenberg Ferromagnet

A new class of nodal topological excitations in a 2D Heisenberg model is studied. The solutions correspond to a nodal singular point of the gradient field of the azimuthal angle. An analytical solution is found for the isotropic case. The effect of in-plane exchange anisotropy is studied numerically. This results in solutions which are analogues of the conventional out-of-plane solitons in 2D magnets.     

Bright,dark optical and other solitons to the generalized higher-order NLSE in optical fibers

In this paper, we study the generalized higher-order nonlinear Schrödinger equation analytically. We use two integral schemes for conducting this study. Dark, bright, combined dark–bright optical, singular soliton, soliton-like and trigonometric function solutions are successfully constructed. We give the constraint conditions for the existence of valid solutions. The 2D, 3D and the contour graphs for the dark and bright solitons are plotted.     

Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.     

Exact solutions for the shape of a 2D conducting drop moving through a dielectric medium at an angle to the external electric field

Possible equilibrium configurations of the surface of a drop of a conducting liquid, moving relative to a dielectric medium at a certain angle to the external electric field, are considered in the 2D formulation. A two-parametric family of exact particular solutions to this problem is obtained using the conformal mapping method. These solutions are characterized by a considerable strain of the 2D drop surface (the highest possible ratio of its longitudinal and transverse sizes is 11/2).     

Accessible solitons in three-dimensional parabolic cylindrical coordinates

A class of self-similar beams, named three-dimensional (3D) spatiotemporal parabolic accessible solitons, are introduced in the 3D highly nonlocal nonlinear media. We obtain exact solutions of the 3D spatiotemporal linear Schrödinger equation in parabolic cylindrical coordinates by using the method of separation of variables. The 3D localized structures are constructed with the help of the confluent hypergeometric Tricomi functions and the Hermite polynomials. Based on such an exact solution, we graphically display three different types of 3D beams: the Gaussian solitons, the ring necklace solitons, and the parabolic solitons, by choosing different mode parameters. We also perform direct numerical simulation to discuss the stability of local solutions. The procedure we follow provides a new method for the manipulation of spatiotemporal solitons.