Instability Of Travelling Wave Solutions Of Several Kinds Of Nonlinear Evolution Equation

With the developing of modern science and technology, many chemical, physical and biological phenomena have shown vibration properties and perturbation has spread to the finite speed. And the travelling wave solutions with the shap of u(x,t) =φ(x—ct) demonstrate just two properties. Thus in the research of mathematical model of chemistry, physics and biology, the existence, uniqueness, stability of reaction-diffusion equation have been concerned by a lot of researchers.Stability theory studies the behavior of differential equations when time goes infinity. In natural sciences, engineering and technology, environment ecology, scocial economy, ect. It has a wide range of applications. As a system one of the most important states is its equilibrium state, but if it is not durable equilibrim, it does not matter much. In this paper we discuss the instability of travelling wave solutions of a single nonlinear evolution equation.For some evolution equations with practical backgroud, the previous researchers have only discussed the existence of the travelling wave solutions, and the stability of the travelling wave solutions is less studied. In this paper several kinds of nonlinear evolution equation on the instability of travelling wave solutions are discussed in detail, and the main methods are spectral analysis and semigroup theory described in [16]. Firstly, we linearized the systems at travelling wave solution. Then we consider the linear part. It will be divided into two parts: one is linear operator with constant coefficients; the other with variable coefficents. The purpose is facilitate to use the Fourier transform and spectrum analysis. The key of spectrum analysis is that if the spectrum of the operator is all located on the left half-plane, travelling wave solutions are stable, and if one of the spectrum of the operator is in the right half-plane, travelling wave solutions are instable. In this paper, we consider not only reaction-diffusion equations, but also dissipative, frequency dispersion evolution equations with instable nonlinear terms, and we have got the consequence that their travelling wave solutions are nonlinear instability in Hillbert space. In many application areas, it is crucial of the stability of the solutions. How about the stability of these unstable travelling wave solutions in the weighted spaces, in the L~p spaces, or in other spaces? These are topics in the future.