Rotational symmetry of self-similar solutions to the Ricci flow

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman’s first paper.     

Global solutions and self-similar solutions of semilinear wave equation

We prove existence, uniqueness and regularity results for the global solutions of the semilinear wave equations. In particular, we show existence of regular self-similar solutions. Also, we build some finite-energy asymptotically self-similar solutions. Received: 20 September 1999; in final form: 10 May 2000 / Published online: 25 June 2001     

Asymptotically self-similar solutions of the damped wave equation

We consider the damped hyperbolic equation

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Transonic shock and rarefaction wave interactions of two-dimensional Riemann problems for the self-similar nonlinear wave system

This paper addresses the self-similar transonic irrotational flow in gas dynamics in two space dimensions.We consider a configuration that the incident shock becomes a transonic shock as it enters the sonic circle, interacts with the rarefaction wave downstream, and then becomes sonic. The rarefaction wave further downstream becomes sonic (degenerate) creating an unknown boundary for the governing system. We present the Riemann data for this configuration, provide the characteristic decomposition of the system, and formulate the boundary value problem for this configuration. The numerical results are presented, and a method to establish the existence result is briefly discussed.     

A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation

We present a systematic study of local solutions of the ODE of the form near t=0. Such ODEs occur in the study of self-similar radial solutions of some second order PDEs. A general theorem of existence and uniqueness is established. It is shown that there is a dichotomy between the cases γ>0 and γ<0, where γ=∂f/∂x^′ at t=0. As an application, we study the singular behavior of self-similar radial solutions of a nonlinear wave equation with superlinear damping near an incoming light cone. A smoothing effect is observed as the incoming waves are focused at the tip of the cone.     

Existence of the self-similar solutions in the heat flow of harmonic maps

It is proved that the heat equations of harmonic maps have self-similar solutions satisfying certain energy condition.     

Solitary wave and shock wave solutions to a second order wave equation of Korteweg-de Vries type

This paper obtains the solitary wave as well as the shock wave solutions to the second order wave equation of Korteweg-de Vries type that was first proposed in 2002. The ansatz method is used to retrieve these solutions. The domain restrictions as well as the parameter regimes are all identified in the process of obtaining the solution.     

The reflection of a shock wave from a centre or axis of symmetry at adiabatic exponents from 1.2 to 3

The self-similar problem of the reflection of a shock wave from a centre or axis of symmetry for adiabatic exponents from 1.2 to 3 with a maximum step of 0.1 is solved. The distributions of the main parameters behind the reflected shock wave are obtained.     

Instability of the self-similar flow into a cavity

In this paper we have investigated the instability of the self-similar flow behind the boundary of a collapsing cavity. The similarity solutions for the flow into a cavity in a fluid obeying a gas lawp = K_ρ ^γ, K = constant and 7 ≥ γ > 1 has been solved by Hunter, who finds that for the same value of γ there are two self-similar flows, one with accelerating cavity boundary and other with constant velocity cavity boundary. We find here that the first of these two flows is unstable. We arrive at this result only by studying the propagation of disturbances in the neighbourhood of the singular point.     

Isothermal flow behind a shock wave propagating in the solar wind

A Parker-type blast wave, which is headed by a strong shock, driven out by a propelling contact surface, moving into an ambient solar wind having a strictly inverse square law radial decay in density, is studied. Assuming the self-similar flow behind the shock to be isothermal, approximate analytical and exact numerical solutions are obtained. There is a good agreement between the approximate analytical and exact numerical solutions. It is observed that the mathematical singularity in density at the contact surface is removed for the isothermal flow.     

Traveling wave solutions in a diffusive system with two preys and one predator

This paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator. By constructing upper and lower solutions, the existence of nontrivial traveling wave solutions is established. The asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles. Applying the theory of asymptotic spreading, the nonexistence of traveling wave solutions is proved.     

Shrinking self-similar solutions of a nonlinear diffusion equation with nondivergence form

In this paper we study the shrinking self-similar solutions of the nonlinear diffusion equation with nondivergence form

     

Stability of a supersonic flow past a wedge with adjoint weak neutrally stable shock wave

We study the classical problem of a supersonic stationary flow of a nonviscous nonheat-conducting gas in local thermodynamic equilibrium past an infinite plane wedge. Under the Lopatinski? condition on the shock wave (neutral stability), we prove the well-posedness of the linearized mixed problem (the main solution is a weak shock wave), obtain a representation of the classical solution, where, in this case (in contrast to the case of the uniform Lopatinski? condition—an absolutely stable shock wave), plane waves additionally appear in the representation. If the initial data have compact support, the solution reaches the given regime in infinite time.     

Limit cycles for a perturbation of a quadratic center with symmetry

To estimate the number of limit cycles appearing under a perturbation of a quadratic system that has a center with symmetry, we use the method of generalized Dulac functions. To this end, we reduce the perturbed system to a Liénard system with a small parameter, for which we construct a Dulac function depending on the parameter. This permits one to estimate the number of limit cycles in the perturbed system for all sufficiently small parameter values. We find the Dulac function by solving a linear programming problem. The suggested method is used to analyze four specific perturbed systems that globally have exactly three limit cycles [i.e., the limit cycle distribution 3 or (3, 0)] and two systems that have the limit cycle distribution (3, 1) (i.e., one nest around each of the two foci).     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

Rotational symmetry and properties of the ancient solutions of Ricci flow on surfaces

We give a simple proof for the rotational symmetry of ancient solutions of Ricci flow on surfaces. As a consequence we obtain a simple proof of some results of Daskalopoulos, Hamilton and Sesum on the a priori estimates for the ancient solutions of Ricci flow on surfaces. We also give a simple proof for the solution to be a Rosenau solution under some mild conditions on the solutions of Ricci flow on surfaces.     

Local well-posedness of a multidimensional shock wave for the steady supersonic isothermal flow

In this paper, we prove the local existence, uniqueness and stability of a supersonic shock for the supersonic isothermal incoming flow past a curved cone. Major difficulties include constructing an appropriate solution and treating the Neumann boundary conditions and local stability condition.     

Bifurcations of travelling wave solutions in a new integrable equation with peakon and compactons

Degasperis and Procesi applied the method of asymptotic integrability and obtain Degasperis–Procesi equation. They showed that it has peakon solutions, which has a discontinuous first derivative at the wave peak, but they did not explain the reason that the peakon solution arises. In this paper, we study these non-smooth solutions of the generalized Degasperis–Procesi equation u_t − u_txx + (b + 1)uu_x = bu_xu_xx + uu_xxx, show the reason that the non-smooth travelling wave arise and investigate global dynamical behavior and obtain the parameter condition under which peakon, compacton and another travelling wave solutions engender. Under some parameter condition, this equation has infinitely many compacton solutions. Finally, we give some explicit expression of peakon and compacton solutions.