Multidimensional nonlinear wave equations with multivalued solutions

We present the theory of breaking waves in nonlinear systems whose dynamics and spatial structure are described by multidimensional nonlinear hyperbolic wave equations. We obtain a general relation between systems of first-order quasilinear equations and nonlinear hyperbolic equations of higher orders, which, in particular, describe electromagnetic waves in a medium with nonlinear polarization of an arbitrary form. We use this approach to construct exact multivalued solutions of such equations and to study their spatial structure and dynamics. The results are generalized to a wide class of multidimensional equations such as d’Alembert equations, nonlinear Klein-Gordon equations, and nonlinear telegraph equations.     

Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. This paper studies a sequential semidefinite programming (SSP) method, which is a generalization of the well-known sequential quadratic programming method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.     

Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases

We consider problems for the nonlinear Boltzmann equation in the framework of two models: a new nonlinear model and the Bhatnagar-Gross-Krook model. The corresponding transformations reduce these problems to nonlinear systems of integral equations. In the framework of the new nonlinear model, we prove the existence of a positive bounded solution of the nonlinear system of integral equations and present examples of functions describing the nonlinearity in this model. The obtained form of the Boltzmann equation in the framework of the Bhatnagar-Gross-Krook model allows analyzing the problem and indicates a method for solving it. We show that there is a qualitative difference between the solutions in the linear and nonlinear cases: the temperature is a bounded function in the nonlinear case, while it increases linearly at infinity in the linear approximation. We establish that in the framework of the new nonlinear model, equations describing the distributions of temperature, concentration, and mean-mass velocity are mutually consistent, which cannot be asserted in the case of the Bhatnagar-Gross-Krook model.     

Sous-solutions d'équations elliptiques dans L1

We study subsolutions for semilinear elliptic boundary value problems in L1. We consider as well nonlinear as linear boundary conditions. The nonlinear functions may be multivated. We characterize in terms of p.d.e. the subsolutions defined by a nonlinear functional analysis argument. Applications are given to obtain existence results for semilinear elliptic boundary value problems and comparison and estimates for nonlinear parabolic boundary value problems.     

New results on the parametric stability of nonlinear systems

In this paper, we derive some new results on the parametric stability of nonlinear systems. Explicitly, we derive a necessary and sufficient condition for a nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. Next, we derive some new results on the parametric stability of discrete-time nonlinear systems. As in the continuous case, we derive a necessary and sufficient condition for a discrete-time nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the discrete-time nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. We illustrate our results with some classical examples from the bifurcation theory.     

一类非线性离散动态系统的分析

时序电路是一类重要的离散动态系统,它在数学控制、计算机、数字通信等工程系统中应用得很普遍。线性时序电路理论已比较成熟。然而,由于非线性带来的本质困难,非线性时序电路理论的某些方面还相当薄弱。即使对一类结构简单的非线性时序电路——非线性移位寄存器,其状态图中圈数与圈长的分析计算仍缺乏系统的理论。而这类寄存器在数字控制、计算机、编译码器、密码机等工程系统中有重要的应用。     

A tau method for nonlinear dynamical systems

In this work we present a new Tau method for the solution of nonlinear systems of differential equations which are linear in the derivative of highest order and polynomial in the remaining. We avoid the linearization of the problem by associating to it a nonlinear algebraic system and combine a forward substitution with the Tau method. We develop an adaptive step by step version of this alternative nonlinear tau method and we apply it to several nonlinear dynamical systems.     

Nonlinear orientational dynamics of a molecular chain

We investigate the nonlinear rotational dynamics of a molecular chain with quadrupole interaction in both the discrete and the continuous cases. Based on a system of nonlinear differential-difference equations, we obtain approximate equations describing the chain excitations and preserving the initial symmetry. We introduce an effective potential and normal coordinates, using which allows decoupling the system into linear and nonlinear parts. As a result of a strong anisotropy of the potential, narrow “valleys” occur in the angle plane. Motion along a valley corresponds to a softer interaction (nonlinear equations). Linear equations describe motion across a valley (hard interaction). We consider cases where the derived nonlinear equations reduce to the sine-Gordon equation. We find integrals of motion and exact solutions of our approximate equations. We uniformly describe the energy interval encompassing the domains of order, of orientational melting, and of rotational motion of the molecules in the chain.     

Small solutions of nonlinear differential equations near branching points

We construct parametric families of small branching solutions to nonlinear differential equations of the nth order near branching points. We use methods of the analytical theory of branching solutions of nonlinear equations and the theory of differential equations with a regular singular point. We illustrate the general existence theorems with an example of a nonlinear differential equation in a certain magnetic insulation problem.     

Coherent States for Systems of L 2-Supercritical Nonlinear Schrödinger Equations

We consider the propagation of wave packets for a nonlinear Schrödinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, at leading order, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.     

Exactly Solvable Model for Nonlinear Pulse Propagation in Optical Fibers

The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation, which was first derived by means of bi-Hamiltonian methods in [ 1 ]. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher-order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative NLS equation; (c) We analyze traveling-wave solutions.     

一类模糊非线性方程组及解法

模糊非线性方程组 ,在模糊控制和现实生活中很普遍 .本文考虑一类模糊非线性方程组的性质 ,然后给出一种解法 .首先把模糊非线性方程组转变成非线性规划 ,再用非线性规划中的方法或软件来解 .     

Reduced order observer design for nonlinear systems

This work is a geometric study of reduced order observer design for nonlinear systems. Our reduced order observer design is applicable for Lyapunov stable nonlinear systems with a linear output equation and is a generalization of Luenberger’s reduced order observer design for linear systems. We establish the error convergence for the reduced order estimator for nonlinear systems using the center manifold theory for flows. We illustrate our reduced order observer construction for nonlinear systems with a physical example, namely a nonlinear pendulum without friction.     

Separation of Variables in a Nonlinear Wave Equation with a Variable Wave Speed

We develop a generalized conditional symmetry approach for the functional separation of variables in a nonlinear wave equation with a nonlinear wave speed. We use it to obtain a number of new (1+1)-dimensional nonlinear wave equations with variable wave speeds admitting a functionally separable solution. As a consequence, we obtain exact solutions of the resulting equations.     

Fourth-order problems with nonlinear boundary conditions

We develop a new method of lower and upper solutions for a fourth-order nonlinear boundary value problem where the differential equation has dependence on all lower-order derivatives. Our boundary conditions are nonlinear. We will assume the functions that define the nonlinear boundary conditions are either monotone or nonmonotone. As a result we obtain existence principles which improve recent results in the literature.     

Some solvability problems for the Boltzmann equation in the framework of the Shakhov model

We consider the nonlinear Boltzmann equation in the framework of the Shakhov model for the classical problem of gas flow in a plane layer. The problem reduces to a system of nonlinear integral equations. The nonlinearity of the studied system can be partially simplified by passing to a new argument depending on the solution of the problem itself. We prove the existence theorem for a unique solution of the linear system and the existence theorem for a positive solution of the nonlinear Urysohn equation. We determine the temperature jumps on the lower and upper walls in the linear and nonlinear cases, and it turns out that the difference between them is rather small.     

Successive approximations to the solutions to nonlinear equations with a vector parameter in a nonregular case

We consider a nonlinear operator equation with a Fredholm linear operator in the principal part. The nonlinear part of the equation depends on the functionals defined on an open set in a normed vector space. We propose a method of successive asymptotic approximations to branching solutions. The method is used for studying the nonlinear boundary value problem describing the oscillations of a satellite in the plane of its elliptic orbit.     

Blow-up solutions of second-order differential inequalities with a nonlinear memory term

We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain.     

Admissible equivalence transformations for linearization of nonlinear wave type equations

In the present paper, we consider a general family of two‐dimensional wave equations, which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones.