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.     

Blow-up of the solution of a nonlinear system of equations with positive energy

We consider the Dirichlet problem for a nonlinear system of equations, continuing our study of nonlinear hyperbolic equations and systems of equations with an arbitrarily large positive energy. We use a modified Levine method to prove the blow-up.     

Galilei-invariant higher-order equations of burgers and korteweg-de vries types

We describe nonlinear Galilei-invariant higher-order equations of Burgers and Korteweg-de Vries types. We study symmetry properties of these equations and construct new nonlinear extensions for the Galilei algebra AG(1, 1).     

Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity

We consider nonlinear autonomous operator differential equations with pseudomonotone dependence between the defining parameters of the problemand study the dynamics of all weak solutions on the positive time semiaxis. We prove the existence of a trajectory and a global attractor and study their structure. As a possible application, we consider the class of high-order nonlinear parabolic equations.     

Representations for the solutions and coefficients of second-order differential equations

New representations are given for the solutions and coefficients of second-order evolution differential equations in the linear and nonlinear cases. The formulas for linear equations have wide arbitrariness which can be used in identification problems. We study the questions of running-wave type for nonlinear one-dimensional equations.     

Homogenization of linear and nonlinear transport equations

We study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields. Two types of homogenized equations are derived. For general n-dimensional linear and nonlinear problems, we derive homogenized equations by introducing additional independent variables to represent the small scales. For the two-dimensional linear transport equations, we derive effective equations for the averaged quantities. Such equations take the form of either a degenerate non-local diffusion equation with memory or a higher order hyperbolic equation. To study the nonlinear transport equations we introduce the concept of two-scale Young measure and extend DiPerna's method to prove that it reduces to a family of Dirac measures.     

A method of directly defining the inverse mapping for solutions of coupled systems of nonlinear differential equations

Recently, Liao introduced a new method for finding analytical solutions to nonlinear differential equations. In this paper, we extend this idea to nonlinear systems. We study the system of nonlinear differential equations that governs nonlinear convective heat transfer at a porous flat plate and find functions that approximate the solutions by extending Liao’s Method of Directly Defining the Inverse Mapping (MDDiM).     

Preservation of stability under discretization of systems of ordinary differential equations

We study the stability preservation problem while passing from ordinary differential to difference equations. Using the method of Lyapunov functions, we determine the conditions under which the asymptotic stability of the zero solutions to systems of differential equations implies that the zero solutions to the corresponding difference systems are asymptotically stable as well. We prove a theorem on the stability of perturbed systems, estimate the duration of transition processes for some class of systems of nonlinear difference equations, and study the conditions of the stability of complex systems in nonlinear approximation.     

A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations

We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations. At last, we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.     

Analysis of Local Dynamics of Difference and Close to Them Differential-Difference Equations

We study the local dynamics of one class of nonlinear difference equations which is important for applications. Using perturbation theory methods, we construct sets of singularly perturbed differential-difference equations that are close (in a sense) to initial difference equations. For the problem on the stability of the zero equilibrium state and for certain infinite-dimensional critical cases, we propose a method that allows us to construct analogs of normal forms. We mean special nonlinear boundary value problems without small parameters, whose nonlocal dynamics describes the structure of solutions to initial equations in a small neighborhood of the equilibrium state. We show that dynamic properties of difference and close to them differential-difference equations considerably differ.     

Global boundary value problems for nonlinear dynamic equations on time scales

We consider nonlinear boundary value problems for dynamic equations on time scales. We study nonlinear dynamic equations subject to global boundary conditions. Criteria are provided for the solvability of such problems. In the case of weak nonlinearities, we also examine the dependence of the solution on parameters.     

Propagation of infinitesimal thermo-mechanical waves during the finite-deformation loading of a viscoelastic material: general theory

We study the theory of propagation of infinitesimal thermo-mechanical waves in a special class of nonlinear viscoelastic materials under homogeneous and inhomogeneous finite static and time-varying deformations. These results are based on a thermodynamically consistent finite-deformation nonlinear viscoelastic model that reduces to a general linear viscoelastic model of integral form. On a thermo-mechanically deforming body, we impose a thermo-mechanical perturbation history and obtain the equations to solve for the perturbation parameters from the constitutive model and the balance laws. We use these equations to study the characteristics of different perturbations. We develop the special equations for both time-harmonic and time-damping plane waves for homogeneous pre-loads.     

Convergence in variation of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary measures

We study convergence in variation of probability solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary solutions. We obtain sufficient conditions for the exponential convergence of solutions to the stationary solution in case of coefficients that can have an arbitrary growth at infinity and depend on the solutions through convolutions with unbounded discontinuous kernels. In addition, we study a more difficult case where the nonlinear equation has several stationary solutions and convergence to a stationary solution depends on initial data. Finally, we obtain sufficient conditions for solvability of nonlinear Fokker–Planck–Kolmogorov equations.     

Correspondence between frame shrinkage and high-order nonlinear diffusion

Nonlinear diffusion filtering and wavelet/frame shrinkage are two popular methods for signal and image denoising. The relationship between these two methods has been studied recently. In this paper we investigate the correspondence between frame shrinkage and nonlinear diffusion.We show that the frame shrinkage of Ron-Shen?s continuous-linear-spline-based tight frame is associated with a fourth-order nonlinear diffusion equation. We derive high-order nonlinear diffusion equations associated with general tight frame shrinkages. These high-order nonlinear diffusion equations are different from the high-order diffusion equations studied in the literature. We also construct two sets of tight frame filter banks which result in the sixth- and eighth-order nonlinear diffusion equations.The correspondence between frame shrinkage and diffusion filtering is useful to design diffusion-inspired shrinkage functions with competitive performance. On the other hand, the study of such a correspondence leads to a new type of diffusion equations and helps to design frame-inspired diffusivity functions. The denoising results with diffusion-inspired shrinkages provided in this paper are promising.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation

The modified method of simplest equation is powerful tool for obtaining exact and approximate solutions of nonlinear PDEs. These solutions are constructed on the basis of solutions of more simple equations called simplest equations. In this paper we study the role of the simplest equation for the application of the modified method of simplest equation. We follow the idea that each function constructed as polynomial of a solution of a simplest equation is a solution of a class of nonlinear PDEs. We discuss three simplest equations: the equations of Bernoulli and Riccati and the elliptic equation. The applied algorithm is as follows. First a polynomial function is constructed on the basis of a simplest equation. Then we find nonlinear ODEs that have the constructed function as a particular solution. Finally we obtain nonlinear PDEs that by means of the traveling-wave ansatz can be reduced to the above ODEs. By means of this algorithm we make a first step towards identification of the above-mentioned classes of nonlinear PDEs.     

Sharp criteria of global existence and blow-up for a type of nonlinear parabolic equations

We study a type of nonlinear parabolic equations. In terms of the variational characterization of the corresponding nonlinear elliptic equations and the invariant flow arguments, we establish the sharp criteria for global existence and blow-up. Furthermore, we also get the instability of the steady states and the global existence with small initial data.     

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.     

Global existence of small analytic solutions to schrodinger equations with quadratic nonlinearity

We study the nonlinear Schriidinger equations