Weakly irregular quasiperiodic solutions of nonlinear Pfaff systems

We consider the nonlinear quasiperiodic Pfaff system

$$\frac{{\partial x}}{{\partial t_j }} = F^{(j)} (t,x) + G^{(j)} (t,x)(j = 1,...,m).$$

Let K ^(j) be a frequency basis with respect to t _j of the functions F ⁽¹⁾,...,F ^(m), and let L ^(j) be a frequency basis with respect to t _j of the functions G ⁽¹⁾,...,G ^(m). Suppose that the set K ^(j) ∪ L ^(j) of numbers is rationally linearly independent. We obtain necessary and sufficient conditions for the existence of quasiperiodic solutions with frequency bases L ⁽¹⁾,..., L ^(m).

     

Weakly nonlinear discrete multipoint boundary value problems

In this paper we study nonlinear, discrete, multipoint boundary value problems of the form

x(t+1)=A(t)x(t)+?f(t,x(t))     

Weakly localized states for nonlinear Dirac equations

We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor components. Moreover, those solutions admit a variational characterization as least action critical points of a suitable action functional. We also indicate how the content of the present paper allows to extend our previous results for the massive case [5] to more general nonlinearities.     

Weakly nonlinear interaction of two waves in radiative gasdynamics

The interaction of two weakly nonlinear waves in radiative gasdynamics is investigated following the perturbation analysis developed by He and Moodie. Explicit solutions are given and the two waves interactions are graphically shown.     

Periodic solutions of a 2nth-order nonlinear difference equation

In this paper, a 2 nth-order nonlinear difference equation is considered.Using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of periodic solutions.Results obtained complement or improve the existing ones.     

Weakly nonlinear boundary-value problem in a special critical case

We investigate the problem of the determination of conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for finding the generating solution of a weakly nonlinear Noetherian boundary-value problem turns into an identity. We improve the classification of critical cases and construct an iterative algorithm for finding solutions of weakly nonlinear Noetherian boundary-value problems in the special critical case.     

Weakly efficient solutions and pseudoinvexity in multiobjective control problems

In this paper, we introduce new pseudoinvexity conditions on functionals involved in a multiobjective control problem, called W-KT-pseudoinvexity and W-FJ-pseudoinvexity. We prove that all Kuhn-Tucker or Fritz-John points are weakly efficient solutions if and only if these conditions are fulfilled. We relate weakly efficient solutions to optimal solutions of weighting control problems. We generalize recently obtained optimality results of known mathematical programming problems and control problems. We illustrate these results with an example.     

Weakly nonlinear boundary-value problems for operator equations with pulse influence

We consider the problem of finding conditions of solvability and algorithms for construction of solutions of weakly nonlinear boundary-value problems for operator equations (with the Noetherian linear part) with pulse influence at fixed times. The method of investigation is based on passing by methods of the Lyapunov—Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 272–288, February, 1997.     

Weakly nonlinear boundary-value problems for differential systems with pulse influence

The coefficient sufficient conditions for the existence of solutions and the iteration algorithm of constructing these solutions are obtained for weakly nonlinear boundary-value problems for systems of ordinary differential equations with pulse influence in the general case in which the number of boundary conditions does not coincide with the order of the differential system. The equation is derived for generating amplitudes of these boundary-value problems. This equation determines the amplitude of a solution, which can be regarded as generating for the required solution, and gives necessary conditions for the existence of this solution.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 221–225, February, 1993.     

Positive solutions for 2p-order and 2q-order nonlinear ordinary differential systems

In this paper, by using Krasnosel'skii fixed point theorem and under suitable conditions, we present the existence of single and multiple positive solutions to the following systems:

     

Bifurcation of travelling wave solutions for （2＋1）-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.     

Exact solutions for nonlinear diffusion with nonlinear loss

Exact similarity solutions are developed for nonlinear diffusion with nonlinear reactive or irreversible absorptive loss from an instantaneous source. The diffusivity is proportional to a powerm of concentration, with 0 < m 1; and loss rate is also proportional to a power of concentration,n, with 0 n < 1. The solutions are for an arbitrary number of dimensionss > 0 withs=1, 2, 3 in physical applications. All solutions give the slug radius finite, increasing to a maximum, and then decreasing to zero in finite time. Withn < 1, the loss rate at small concentrations is large enough to ensure slug extinction in finite time. The corresponding exact solutions for gain, not loss, are given also. They become independent of initial slug quantity in the limit of infinite time.