Nontrivial Time Periodic Solutions Research Articles

Periodic pattern formation in the coupled chemotaxis-(Navier–)Stokes system with mixed nonhomogeneous boundary conditions

AbstractWe consider the coupled chemotaxis-fluid model for periodic pattern formation on two- and three-dimensional domains with mixed nonhomogeneous boundary value conditions, and prove the existence of nontrivial time periodic solutions. It is worth noticing that this system admits more than one periodic solution. In fact, it is not difficult to verify that (0, c, 0, 0) is a time periodic solution. Our purpose is to obtain a time periodic solution with nonconstant bacterial density.

Self-excited vibrations for damped and delayed higher dimensional wave equations

In the article [12] it is shown that time delay induces self-excited vibrations in a one dimensional damped wave equation. Here we generalize this result for higher spatial dimensions. We prove the existence of branches of nontrivial time periodic solutions for spatial dimensions $ d\ge 2 $. For $ d> 2 $, the bifurcating periodic solutions have a fixed spatial frequency vector, which is the solution of a certain Diophantine equation. The case $ d = 2 $ must be treated separately from the others. In particular, it is shown that an arbitrary number of symmetry breaking orbitally distinct time periodic solutions exist, provided $ d $ is big enough, with respect to the symmetric group action. The direction of bifurcation is also obtained.

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin–Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t=0. We use our method to study global paths of nontrivial time-periodic solutions connecting stationary and traveling waves of the Benjamin–Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached. By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODEs governing the evolution of solitons using the ansatz suggested by the numerical simulations.

Existence of time periodic solutions for one-dimensional Newtonian filtration equation with multiple delays

In this paper, we study one-dimensional Newtonian filtration equation including unbounded sources with multiple delays. The existence of nonnegative non-trivial time periodic solutions will be established by the Leray-Schauder fixed point theorem based on some suitable Lyapunov functionals and some a priori estimates for all possible periodic solutions.

Global paths of time-periodic solutions of the Benjamin–Ono equation connecting pairs of traveling waves

We classify all bifurcations from traveling waves to non-trivial time-periodic solutions of the Benjamin-Ono equation that are predicted by linearization. We use a spectrally accurate numerical continuation method to study several paths of non-trivial solutions beyond the realm of linear theory. These paths are found to either re-connect with a different traveling wave or to blow up. In the latter case, as the bifurcation parameter approaches a critical value, the amplitude of the initial condition grows without bound and the period approaches zero. We then prove a theorem that gives the mapping from one bifurcation to its counterpart on the other side of the path and exhibits exact formulas for the time-periodic solutions on this path. The Fourier coefficients of these solutions are power sums of a finite number of particle positions whose elementary symmetric functions execute simple orbits (circles or epicycles) in the unit disk of the complex plane. We also find examples of interior bifurcations from these paths of already non-trivial solutions, but we do not attempt to analyze their analytic structure.

Exact Solutions to Degenerate Conservation Laws

We consider large variation solutions to systems of conservation laws, for which the Glimm--Lax theory of decay breaks down. We identify and isolate geometric nonlinearities which are distinct from the usual genuine nonlinearity of each wave field by describing some degenerate systems in which all nonlinearity is geometric and is manifested in the coupling of the different wave families. We then construct exact explicit solutions to these equations and examine properties of these solutions. We find a wide variety of phenomena, depending on the form of the nonlinearity. The most striking of these include strong nonlinear instability of solutions and nontrivial time-periodic solutions. We also find solutions which grow or decay exponentially and oscillating solutions which correspond to rotations byan irrational angle. These oscillating, periodic, and exponential solutions can all appear in a single system with small initial data, demonstrating sensitive dependence on initial conditions.

Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz

AbstractA new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation uu ‐ uxx + g(u) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g(u)/u→∞ as |u|→∞. The proof uses a modified form (PS)c of the Palais‐Smale condition (PS).

Free vibrations for a semilinear wave equation

Abstract : Existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness, monotonicity, and superlinearity assumptions. The forced vibration case is also treated.