On the standing wave for a class of nonlinear Schrödinger equations

This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations

iφt+Δφ−²|x|φ+μ|φ|^p−1φ+γ|φ|^q−1φ=0,     

Evans functions and bifurcations of standing wave fronts of a nonlinear system of reaction diffusion equations

Consider the following nonlinear system of reaction diffusion equations arising from mathematical neuroscience $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u]-w,~ \frac{\partial w}{\partial t}=\varepsilon(u-\gamma w).$ Also consider the nonlinear scalar reaction diffusion equation $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u].$ In these model equations, $\alpha>0$, $\beta>0$, $\gamma>0$, $\varepsilon>0$ and $\theta>0$ are positive constants, such that $0<2\theta<\beta$. In the model equations, $u=u(x,t)$ represents the membrane potential of a neuron at position $x$ and time $t$, $w=w(x,t)$ represents the leaking current, a slow process that controls the excitation.\\indent The main purpose of this paper is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing wave fronts) and Evans functions (complex analytic functions) to establish the existence, stability, instability and bifurcations of standing wave fronts of the nonlinear system of reaction diffusion equations and to establish the existence and stability of the standing wave fronts of the nonlinear scalar reaction diffusion equation.     

On some applications of transformation groups to a class of nonlinear dispersive equations

We consider a special class of nonlinear dispersive equations and search for classical and potential symmetries. We classify them in both cases. Reduced equations and wide classes of solutions of physical interest are obtained. Some equations of this special class are linearized via a symmetry approach, and an equivalence algebra is obtained.     

Note on exact solutions for a class of nonlinear diffusion equations

An effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms. Further, a new auxiliary equation ansatz is derived. Consequently, new exact traveling wave trigonometric function, solitary-like and Weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized Riccati equation expansion method.     

Existence results for a coupled system of nonlinear neutral fractional differential equations

Applying the monotone iterative method, we investigate the existence of solutions for a coupled system of nonlinear neutral fractional differential equations, which involves Riemann–Liouville derivatives of different fractional orders. As an application, an example is presented to illustrate the main results.     

Maximum principles and bounds for a class of fourth order nonlinear elliptic equations

We deduce maximum principles for a class of fourth order nonlinear elliptic equations by using auxiliary functions containing the square of the second gradient of the solution of such equations. A priori bounds on various quantities of interest are obtained.     

A new control method and its robustness for a class of nonlinear systems

For a class of time-varying nonlinear systems described by the equation , the precalculating control is not available if the input matrixg(x,t) is not invertible. With Lyapunov's second method, a stabilizing controller which makes the system practically stable is constructed in this paper. It is shown that the implementation of this scheme depends on some so-called posi-invertibility conditions forg(x,t). In case the system is partly stable, the method, named part-calculating control, can simplify the on-line computations. Without the assumption that the nominal system is asymptotically stable, the method is applied to the problems of control for the corresponding uncertain system that satisfies the matching condition. When the matching condition is not satisfied, the mismatching control problem is also studied with Lyapunov's second method.This work was supported by the Science Fund of the Chinese Academy of Science.     

First integrals and reduction of a class of nonlinear higher order ordinary differential equations

We propose a method for constructing first integrals of higher order ordinary differential equations. In particular third, fourth and fifth order equations of the form are considered. The relation of the proposed method to local and nonlocal symmetries are discussed.     

Global solutions and blow-up for a class of strongly damped wave equations systems

The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied. The existence of global solutions for this problem is proved by using potential well method, and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function. Meanwhile, blow-up of solutions in the unstable set is also obtained.     

Parallel computations and numerical simulations for nonlinear systems of Volterra integro-differential equations

We investigate thalamo-cortical systems that are modeled by nonlinear Volterra integro-differential equations of convolution type. We divide the systems into smaller subsystems in such a way that each of them is solved separately by a processor working independently of other processors results of which are shared only once in the process of computations. We solve the subsystems concurrently in a parallel computing environment and present results of numerical experiments, which show savings in the run time and therefore efficiency of our approach. For our numerical simulations, we apply different numbers np of processors and each case shows that the run time decreases with increasing np. The optimal speed-up is obtained with np = N, where N is the (moderate) number of equations in the thalamo-cortical model.     

Instability of standing waves for a class of nonlinear Schrödinger equations

This paper discusses a class of nonlinear Schrödinger equations with different power nonlinearities. We first establish the existence of standing wave associated with the ground states by variational calculus. Then by the potential well argument and the concavity method, we get a sharp condition for blowup and global existence to the solutions of the Cauchy problem and answer such a problem: how small are the initial data, the global solutions exist? At last we prove the instability of standing wave by combing those results.     

Liouville-Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations

A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y^″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given.     

Exact multiplicity and global structure of solutions for a class of semilinear elliptic equations

In this paper, we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation. We also obtain a precise global bifurcation diagram of the solution set. As a result, an open problem presented by C.-H. Hsu and Y.-W. Shih [C.-H. Hsu, Y.-W. Shih, Solutions of semilinear elliptic equations with asymptotic linear nonlinearity, Nonlinear Anal. 50 (2002) 275-283] is completely solved. Our argument is mainly based on bifurcation theory and continuation method.     

Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow

The paper studies the existence, both locally and globally in time, stability, decay estimates and blowup of solutions to the Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. Under the assumption that the nonlinear term of the equations is of polynomial growth order, say α, it proves that when α>1, the Cauchy problem admits a unique local solution, which is stable and can be continued to a global solution under rather mild conditions; when α?5 and the initial data is small enough, the Cauchy problem admits a unique global solution and its norm in L1,p(R) decays at the rate for 2<p?10. And if the initial energy is negative, then under a suitable condition on the nonlinear term, the local solutions of the Cauchy problem blow up in finite time.     

Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility

In this paper, we show that for a class of nonlinear partial differential equations with arbitrary order the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition. The nonlinear wave equation and the Boussinesq equation all serve as examples illustrating this fact.     

Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models

In this paper we consider a class of evolution equations up to fifth-order containing many arbitrary smooth functions from the point of view of nonlinear self-adjointness. The studied class includes many important equations modeling different phenomena. In particular, some of the considered equations were studied previously by other researchers from the point of view of quasi self-adjointness or strictly self-adjointness. Therefore we find new local conservation laws for these equations invoking the obtained results on nonlinearly self-adjointness and the conservation theorem proposed by Nail Ibragimov.