Solutions to a Novel Casimir Equation for the Ito System

We discuss two classes of solutions to a novel Casimir equation associated with the Ito system, a coupled nonlinear wave equation. Both travelling wave
solutions and separable self-similar solutions are discussed. In a number of cases, explicit exact solutions are found. Such results, particularly the exact solutions, are useful in that they provide us a baseline of comparison to any numerical simulations.Besides, such solutions provide us a glimpse of the behavior of the Ito system,and hence the behavior of a type of nonlinear wave equation, for certain parameter regimes.     

Nonlinear tunneling effect in the (2+1)-dimensional cubic-quintic nonlinear Schrödinger equation with variable coefficients

By means of the similarity transformation, we obtain exact solutions of the(2+1)-dimensional generalized nonlinear Schrödinger equation, which describes thepropagation of optical beams in a cubic-quintic nonlinear medium with inhomogeneousdispersion and gain. A one-to-one correspondence between such exact solutions andsolutions of the constant-coefficient cubic-quintic nonlinear Schrödinger equation existswhen two certain compatibility conditions are satisfied. Under these conditions, wediscuss nonlinear tunneling effect of self-similar solutions. Considering the fluctuationof the fiber parameter in real application, the exact balance conditions do not satisfy,and then we perform direct numerical analysis with initial 5% white noise for the brightsimilariton passing through the diffraction barrier and well. Numerical calculationsindicate stable propagation of the bright similariton over tens of diffraction lengths.     

The constructive technique and its application in solving a nonlinear reaction diffusion equation

A mathematical technique based on the consideration of a nonlinear partial differential equation together with an additional condition in the form of an ordinary differential equation is employed to study a nonlinear reaction diffusion equation which describes a real process in physics and in chemistry. Several exact solutions for the equation are acquired under certain circumstances.     

Spatiotemporal self-similar solutions for the nonautonomous (3+1)-dimensional cubicben quintic Grossben Pitaevskii equation

With the help of similarity transformation, we obtain analytical spatiotemporal self-similar solutions of the nonautonomous (3+1)-dimensional cubic-quintic Gross-Pitaevskii equation with time-dependent diffraction, nonlinearity, harmonic potential and gain or loss when two constraints are satisfied. These constraints between the system parameters hint that self-similar solutions form and transmit stably without the distortion of shape based on the exact balance between the diffraction, nonlinearity and the gain/loss. Based on these analytical results, we investigate the dynamic behaviours in a periodic distributed amplification system.     

Spatiotemporal self-similar solutions for the nonautonomous （3＋1）-dimensional cubic-quintic Gross-Pitaevskii equation

With the help of similarity transformation,we obtain analytical spatiotemporal self-similar solutions of the nonautonomous(3+1)-dimensional cubic-quintic Gross-Pitaevskii equation with time-dependent diffraction,nonlinearity,harmonic potential and gain or loss when two constraints are satisfied.These constraints between the system parameters hint that self-similar solutions form and transmit stably without the distortion of shape based on the exact balance between the diffraction,nonlinearity and the gain/loss.Based on these analytical results,we investigate the dynamic behaviours in a periodic distributed amplification system.     

New applications of the homogeneous balance principle

The homogeneous balance principle has been widely applied to the exploration of nonlinear transformation, exact solutions (especially solitary wave solution), dromion and similarity reduction to the nonlinear partial differential equations in mathematical physics. In this paper, we use the homogeneous balance principle to derive B?cklund transformations for nonlinear partial differential equations that have more nonlinear terms and more highest-order partial derivative terms. With the aid of the B?cklund transformations derived here, we could obtain exact solutions to the nonlinear partial differential equations. The Davey－Stewartson equation and the Nizhnik－Novikov－Veselov equation are considered as the examples.     

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential

An improved homogeneous balance principle and an F-expansiontechnique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schrödinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented.     

Similarity transformations and exact solutions for a family of higher-dimensional generalized Boussinesq equations

In this Letter, we investigate explicitly exact solutions of the higher-dimensional generalized Boussinesq equation. We firstly reduce this equation to one nonlinear ODE and a set of two nonlinear homogeneous PDEs via semi-traveling wave similarity transformation. And then we study solutions of the obtained nonlinear ODE and the set of two nonlinear homogeneous PDEs, respectively. Finally, we can obtain many types of exact solutions of higher-dimensional generalized Boussinesq equation via the semi-traveling wave similarity transformations. These solutions contain an arbitrary function which leads to abundant structures.     

Heat transfer for boundary layers with cross flow

An analysis is presented to study the dual nature of solutions for the forced convective boundary layer flow and heat transfer in a cross flow with viscous dissipation terms in the energy equation. The governing equations are transformed into a set of three self-similar ordinary differential equations by similarity transformations. These equations are solved numerically using the very efficient shooting method. This study reveals that the dual solutions of the transformed similarity equations for velocity and temperature distributions exist for certain values of the moving parameter, Prandtl number, and Eckert numbers. The reverse heat flux is observed for larger Eckert numbers; that is, heat absorption at the wall occurs.     

Self-Similar Solutions of the Boltzmann Equation and Their Applications

We consider a class of solutions of the Boltzmann equation with infinite energy. Using the Fourier-transformed Boltzmann equation, we prove the existence of a wide class of solutions of this kind. They fall into subclasses, labelled by a parameter a, and are shown to be asymptotic (in a very precise sense) to the self-similar one with the same value of a (and the same mass). Specializing to the case of a Maxwell-isotropic cross section, we give evidence to the effect that the only self-similar closed form solutions are the BKW mode and the two solutions recently found by the authors. All the self-similar solutions discussed in this paper are eternal, i.e., they exist for –<t<, which shows that a recent conjecture cannot be extended to solutions with infinite energy. Eternal solutions with finite moments of all orders, and different from a Maxwellian, are also studied. It is shown that these solutions cannot be positive. Moreover all such solutions (partly negative) must be asymptotically (for large negative times) close to the exact eternal solution of BKW type.     

Multi-bump, self-similar, blow-up solutions of the Ginzburg-Landau equation

For the Ginzburg-Landau equation (GL), we establish the existence and local uniqueness of two classes of multi-bump, self-similar, blow-up solutions for all dimensions 2<d<4 (under certain conditions on the coefficients in the equation). In numerical simulation and via asymptotic analysis, one class of solutions was already found; the second class of multi-bump solutions is new.In the analysis, we treat the GL as a small perturbation of the cubic nonlinear Schrödinger equation (NLS). The existence result given here is a major extension of results established previously for the NLS, since for the NLS the construction only holds for d close to the critical dimension d=2.The behaviour of the self-similar solutions is described by a nonlinear, non-autonomous ordinary differential equation (ODE). After linearisation, this ODE exhibits hyperbolic behaviour near the origin and elliptic behaviour asymptotically. We call the region where the type of behaviour changes the mid-range. All of the bumps of the solutions that we construct lie in the mid-range.For the construction, we track a manifold of solutions of the ODE that satisfy the condition at the origin forward, and a manifold of solutions that satisfy the asymptotic conditions backward, to a common point in the mid-range. Then, we show that these manifolds intersect transversely. We study the dynamics in the mid-range by using geometric singular perturbation theory, adiabatic Melnikov theory, and the Exchange Lemma.     

Spatiotemporal Self-Similar Solutions of the Generalized (3+1)-dimensional Nonlinear Schrödinger Equation with Polynomial Nonlinearity of Arbitrary Order

We construct analytical self-similar solutions for the generalized (3+1)-dimensional nonlinear Schrödinger equation with polynomial nonlinearity of arbitrary order. As an example, we list self-similar solutions of quintic nonlinear Schrödinger equation with distributed dispersion and distributed linear gain, including bright similariton solution, fractional and combined Jacobian elliptic function solutions. Moreover, we discuss self-similar evolutional dynamic behaviors of these solutions in the dispersion decreasing fiber and the periodic distributed amplification system.     

Spatial similaritons in the (2+1)-dimensional inhomogeneous cubic-quintic nonlinear Schrödinger equation

Along the idea of the similarity transformation, analytical spatial similaritons to a (2+1)-dimensional inhomogeneous cubic-quintic nonlinear Schrödinger equation with distributed diffraction and gain are derived when some certain compatibility conditions are satisfied. Based on these exact solutions, we investigate dynamic behaviors of self-similar cnoidal waves and chirped similaritons in the hyperbolically and Gaussian decreasing diffraction waveguides.     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

Investigation of soliton solutions with different wave structures to the(2+1)-dimensional Heisenberg ferromagnetic spin chain equation

The principal objective of this article is to construct new and further exact soliton solutions of the(2+1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets and explains their ordering in ferromagnetic materials.These solutions are exerted via the new extended FAN sub-equation method.We successfully obtain dark,bright,combined bright-dark,combined dark-singular,periodic,periodic singular,and elliptic wave solutions to this equation which are interesting classes of nonlinear excitation presenting spin dynamics in classical and semi-classical continuum Heisenberg systems.3D figures are illustrated under an appropriate selection of parameters.The applied technique is suitable to be used in gaining the exact solutions of most nonlinear partial/fractional differential equations which appear in complex phenomena.     

An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations

In this paper, we present a method to solve difference differential equation(s). As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).     

Self-similar cnoidal and solitary wave solutions of the (1+1)-dimensional generalized nonlinear Schrödinger equation

With the help of the similarity transformation and the solvable stationary nonlinear Schrödinger equation (NLSE),we obtain exact chirped and chirp-free self-similar cnoidal wave and solitary wave solutions of the generalized NLSE exhibitingspatial inhomogeneity, inhomogeneous nonlinearity and gain or loss at the same time. As an example, we investigate their propagationdynamics in a nonlinear optical system, and present a series of interesting properties of optical waves.     

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving...     

A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model

Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony （mBBM） model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations.     

Some new solutions derived from the nonlinear (2+1)-dimensional Toda equation---an efficient method of creating solutions

This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential--difference equations (NLDDEs) and lattice equation. By using this method via symbolic computation system MAPLE, we obtained abundant soliton-like and/or period-form solutions to the (2+1)-dimensional Toda equation. It seems that solitary wave solutions are merely special cases in one family. Furthermore, the method can also be applied to other nonlinear differential--difference equations.