Evolution of two-dimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations

The evolution of lump solutions for the Zakharov-Kuznetsov equation and the surface electromigration equation, which describes mass transport along the surface of nanoconductors, is studied. Approximate equations are developed for these equations, these approximate equations including the important effect of the dispersive radiation shed by the lumps as they evolve. The approximate equations show that lump-like initial conditions evolve into lump soliton solutions for both the Zakharov-Kuznetsov equation and the surface electromigration equations. Solutions of the approximate equations, within their range of applicability, are found to be in good agreement with full numerical solutions of the governing equations. The asymptotic and numerical results predict that localized disturbances will always evolve into nanosolitons. Finally, it is found that dispersive radiation plays a more dominant role in the evolution of lumps for the electromigration equations than for the Zakharov-Kuznetsov equation.     

An exact solution of the Einstein-Maxwell equations referring to a magnetic dipole

There is a formal similarity between stationary exterior solutions of the Einstein equations and static magnetic solutions of the Einstein-Maxwell theory. This is particularly evident for axially symmetric fields, and one finds that the sets of equations governing the two cases can be transformed one into the other by simple transformations of the dependent variables.     

Complex solitary waves and soliton trains in KdV and mKdV equations

We demonstrate the existence of complex solitary wave and periodic solutions of theKorteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. The solutions ofthe KdV (mKdV) equation appear in complex-conjugate pairs and are even (odd) under thesimultaneous actions of parity (??) and time-reversal (??) operations. The corresponding localized solitons arehydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishingintensity. The ????-odd complex soliton solution is shown to beiso-spectrally connected to the fundamental sech² solution through supersymmetry. Physically, thesecomplex solutions are analogous to the experimentally observed grey solitons of non-liner Schödinger equation, governing the dynamics of shallow waterwaves and hence may also find physical verification.     

Nonlinear coupled-mode equations

Nonlinear coupled-mode equations governing the modal coupling of a two-mode coupled system (such as twin core couplers) are integrable; power swapping in such a system follows a periodical manner and can be expressed analytically. When three or more modes (for systems such as multiple-core couplers) are involved, the nonlinear coupled-mode equations are no longer integrable and chaotic power swapping is expected. A numerical approach is required, in general, to solve such nonlinear coupled systems involving the coupling of three or more modes. We find, however, that for certain structural configurations, such as triple-core couplers with the cores arranged in the shape of an isosceles triangle, the nonlinear coupled-mode equations for multiple-core couplers can be solved analytically under a resonant condition. The analytical solution indicates that power swapping among, for example, the three cores placed in the shape of an isosceles triangle can be aperiodic at high power, although power may flow from core to core periodically at low power.     

Poloidal–toroidal decomposition in a finite cylinder. I: Influence matrices for the magnetohydrodynamic equations

The Navier–Stokes equations and magnetohydrodynamics equations are written in terms of poloidal and toroidal potentials in a finite cylinder. This formulation insures that the velocity and magnetic fields are divergence-free by construction, but leads to systems of partial differential equations of higher order, whose boundary conditions are coupled. The influence matrix technique is used to transform these systems into decoupled parabolic and elliptic problems. The magnetic field in the induction equation is matched to that in an exterior vacuum by means of the Dirichlet-to-Neumann mapping, thus eliminating the need to discretize the exterior. The influence matrix is scaled in order to attain an acceptable condition number.     

Coupled scalar field equations for nonlinear wave modulations in dispersive media

A review of the generic features as well as the exact analytical solutions of a class of coupled scalar field equations governing nonlinear wave modulations in dispersive media like plasmas is presented. The equations are derivable from a Hamiltonian function which, in most cases, has the unusual property that the associated kinetic energy is not positive definite. To start with, a simplified derivation of the nonlinear Schrödinger equation for the coupling of an amplitude modulated high-frequency wave to a suitable low-frequency wave is discussed. Coupled sets of time-evolution equations like the Zakharov system, the Schrödinger-Boussinesq system and the Schrödinger-Korteweg-de Vries system are then introduced. For stationary propagation of the coupled waves, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system of equations are then reviewed. A comparison between the various sets of governing equations as well as between their exact analytical solutions is presented. Parameter regimes for the existence of different types of localized solutions are also discussed. The generic system of equations has a Hamiltonian structure, and is closely related to the well-known Hénon-Heiles system which has been extensively studied in the field of nonlinear dynamics. In fact, the associated generic Hamiltonian is identically the same as the generalized Hénon-Heiles Hamiltonian for the case of coupled waves in a magnetized plasma with negative group dispersion. When the group dispersion is positive, there exists a novel Hamiltonian which is structurally same as the generalized Hénon-Heiles Hamiltonian but with indefinite kinetic energy. The above correspondence between the two systems has been exploited to obtain the parameter regimes for the complete integrability of the coupled waves. There exists a direct one-to-one correspondence between the known integrable cases of the generic Hamiltonian and the stationary Hamiltonian flows associated with the only integrable nonlinear evolution equations (of polynomial and autonomous type) with a scale-weight of seven. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex Korteweg-de Vries equation and the complexified classical dynamical equations has also been discussed.     

Collisionless Boltzmann equations and integrable moment equations

A collisionless Boltzmann equation, describing long waves in a dense gas of particles interacting via short-range forces, is shown to be equivalent to the Benney equations, which describe long waves in a perfect two-dimensional fluid with a free surface. These equations also describe, in a random phase approximation, the evolution, on long space and time scales, of multiply periodic solutions of the nonlinear Schrödinger equation. The derivative nonlinear Schrödinger equation is likewise shown to be related to an integrable system of moment equations.     

A deep learning method for solving third-order nonlinear evolution equations

It has still been difficult to solve nonlinear evolution equations analytically. In this paper, we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly. Specifically, the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters. In particular, numerical experiments on several third-order nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation, modified KdV equation, KdV–Burgers equation and Sharma–Tasso–Olver equation, demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.     

Analysis of the rogue waves in the blood based on the high-order NLS equations with variable coefficients

The research of rogue waves is an advanced field which has important practical and theoretical significances in mathematics, physics, biological fluid mechanics, oceanography, etc. Using the reductive perturbation theory and long wave approximation, the equations governing the movement of blood vessel walls and the flow of blood are transformed into high-order nonlinear Schrödinger (NLS) equations with variable coefficients. The third-order nonlinear Schrödinger equation is degenerated into a completely integrable Sasa-Satsuma equation (SSE) whose solutions can be used to approximately simulate the real rogue waves in the vessels. For the first time, we discuss the conditions for generating rogue waves in the blood vessels and effects of some physiological parameters on the rogue waves. Based on the traveling wave solutions of the fourth-order nonlinear Schrödinger equation, we analyze the effects of the higher order terms and the initial deformations of the blood vessel on the wave propagation and the displacement of the tube wall. Our results reveal that the amplitude of the rogue waves are proportional to the initial stretching ratio of the tube. The high-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave steepness.     

On the variational form of evolution equations

In this paper a new definition is given for representability in the form of a variational derivative; a particular case of this definition yields the classical expression for the variational derivative. A close connection will be demonstrated between the variational derivative and the exterior differentiation operator d. For the case of functions of one variable one obtains effective computational criteria which solve the “hidden variability” problem. The rigorous mathematical theory is illustrated as applied to the study of a few nonlinear equations of mathematical physics.     

An approximate global solution of Einstein’s equations for a rotating finite body

We obtain an approximate global stationary and axisymmetric solution of Einstein’s equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find second-order approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to obtain a global solution. The resulting metric depends on three arbitrary constants: mass density, rotational velocity and the star radius at the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined by these three parameters. It is easy to check that Kerr’s metric cannot be the exterior part of that metric.     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

Trilocal structures. V. Wave equations

In addition to the usual centroid-time wave equation, a trilocal structure will need to satisfy two relative-time wave equations. When the trilocal wave function is expanded in tree functions, each of the three wave equations becomes an infinite matrix equation, but when the four auxiliary conditions (defined in earlier articles in this series) are introduced, each wave equation reduces to a set of 16 linear homogeneous equations in 16 unknown expansion coefficients (the first 16 coefficients in the tree expansion). The 48 linear equations, in the 16 unknownC _j , are given explicitly. Every 16-by-16 determinant, formed from any 16 of these 48 linear homogeneous equations, must vanish if the trilocal structure is to be an acceptable solution; this requirement will be used in later calculations.     

构造非线性发展方程无穷序列类孤子精确解的一种方法

辅助方程法已构造了非线性发展方程的有限多个新精确解. 本文为了构造非线性发展方程的无穷序列类孤子精确解, 分析总结了辅助方程法的构造性和机械化性特点. 在此基础上,给出了一种辅助方程的新解与Riccati方程之间的拟Bäcklund变换. 选择了非线性发展方程的两种形式解,借助符号计算系统 Mathematica,用改进的(2+1) 维色散水波系统为应用实例,构造了该方程的无穷序列类孤子新精确解. 这些解包括无穷序列光滑类孤子解, 紧孤立子解和尖峰类孤立子解.     

On geometric approach to Lie symmetries of differential-difference equations

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach.     

A new family of evolution water-wave equations possessing two-soliton solutions

We find a new family of fifth-order water-wave equations having common invariant manifold of the fourth order. These evolution equations are nonintegrable except for two cases corresponding to the Sawada–Kotera and Kaup–Kupershmidt equations. The invariant manifold of the family is an autonomous equation F-VI from the Cosgrove's classification of fourth-order ODEs having the Painlevé property. Two-parameter solutions of the equation F-VI allow to find two-soliton solutions for this family of evolution equations.     

Laplace and the era of differential equations

Between about 1790 and 1850 French mathematicians dominated not only mathematics, but also all other sciences. The belief that a particular physical phenomenon has to correspond to a single differential equation originates from the enormous influence Laplace and his contemporary compatriots had in all European learned circles. It will be shown that at the beginning of the nineteenth century Newton's “fluxionary calculus” finally gave way to a French-type notation of handling differential equations. A heated dispute in the Philosophical Magazine between Challis, Airy and Stokes, all three of them famous Cambridge professors of mathematics, then serves to illustrate the era of differential equations. A remark about Schrödinger and his equation for the hydrogen atom finally will lead back to present times.     

A Self-Similar Dynamics in Viscous Spheres

We study the evolution of radiating and viscous fluid spheres assuming an additional homothetic symmetry on the spherically symmetric space-time. We match a very simple solution to the symmetry equations with the exterior one (Vaidya). We then obtain a system of two ordinary differential equations which rule the dynamics, and find a self-similar collapse which is shear-free and with a barotropic equation of state. Considering a huge set of initial self-similar dynamics states, we work out a model with an acceptable physical behavior.     

On noncommutative minisuperspace and the Friedmann equations

In this Letter we present a noncommutative version of scalar field cosmology. We find the noncommutative Friedmann equations as well as the noncommutative Klein–Gordon equation, interestingly the noncommutative contributions are only present up to second order in the noncommutative parameter. Finally we conclude that if we want a noncommutative minisuperspace with a constant noncommutative parameter as viable phenomenological model, the noncommutative parameter has to be very small.     

耗散Jaynes-Cummings模型中微观与唯象学主方程方法的比较

在有耗散的Jaynes-Cummings模型中,对原子与腔复合系统初始处于贝尔态或是原子激发态而腔场为真空态时分别运用微观与唯象学主方程进行求解,并分析这两类主方程在不同耦合区域的区别．结果显示,在强耦合和大失谐情况下,微观主方程求解得到的原子基态的布居数呈线性增长,但唯象学主方程获得的结果却呈非线性增加．在弱耦合区域,这两类主方程求解得到的原子基态布居数都出现快速增加,但是用唯象学主方程要比用微观主方程得到的布居数增加得更快．通过分析与讨论,该结果可能对耗散的腔QED中更好的实现逻辑门提供理论参考．