Albert Erkip

Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity

We study the initial-value problem for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided.

Normal solvability of elliptic boundary value problems on asymptotically flat manifolds

Abstract Normal solvability is shown for a class of boundary value problems on Riemannian manifolds with noncompact boundary using a concept of weighted pseudodifferential operators and weighted Sobolev spaces together with Lopatinski-Shapiro type boundary conditions. An essential step is to show that the standard normal derivative defined in terms of the Riemannian metric is in fact a weighted pseudodifferential operator of the considered class provided the metric is compatible with the symbols.

Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

Abstract We study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finite-time blow-up and as well as global existence of solutions of the problem.

The Cauchy Problem for a One Dimensional Nonlinear Elastic Peridynamic Model

This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar. The issues of local well-posedness and smoothness of the solutions are discussed. The existence of a global solution is proved first in the sublinear case and then for nonlinearities of degree at most three. The conditions for finite-time blow-up of solutions are established.

The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials

This paper is concerned with the analysis of the Cauchy problem of a general class of two-dimensional nonlinear nonlocal wave equations governing anti-plane shear motions in nonlocal elasticity. The nonlocal nature of the problem is reflected by a convolution integral in the space variables. The Fourier transform of the convolution kernel is nonnegative and satisfies a certain growth condition at infinity. For initial data in L2 Sobolev spaces, conditions for global existence or finite time blow-up of the solutions of the Cauchy problem are established.

Instability and stability properties of traveling waves for the double dispersion equation

Abstract In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation u t t − u x x + a u x x x x − b u x x t t = − ( | u | p − 1 u ) x x for p > 1 , a > b > 0 . The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms u x x x x and u x x t t . We obtain an explicit condition in terms of a , b and p on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation ( b = 0 ), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide analytical as well as numerical results on the variation of the stability region of wave velocities with a , b and p and then state explicitly the conditions under which the traveling waves are orbitally stable.

Parameter optimization for the Gaussian model of protein folding

Computational models of protein folding and ligand docking are large and complex. Few systematic methods have yet been developed to optimize the parameters in such models. We describe here an iterative parameter optimization strategy that is based on minimizing a structural error measure by descent in parameter space. At the start, we know the ‘correct’ native structure that we want the model to produce, and an initial set of parameters representing the relative strengths of interactions between the amino acids. The parameters are changed systematically until the model native structure converges as closely as possible to the correct native structure. As a test, we apply this parameter optimization method to the recently developed Gaussian model of protein folding: each amino acid is represented as a bead and all bonds, covalent and noncovalent, are represented by Hookes law springs. We show that even though the Gaussian model has continuous degrees of freedom, parameters can be chosen to cause its ground state to be identical to that of Go-type lattice models, for which the global ground states are known. Parameters for a more realistic protein model can also be obtained to produce structures close to the real native structures in the protein database.

Derivation of the Camassa-Holm equations for elastic waves

Abstract In this paper we provide a formal derivation of both the Camassa–Holm equation and the fractional Camassa–Holm equation for the propagation of small-but-finite amplitude long waves in a nonlocally and nonlinearly elastic medium. We first show that the equation of motion for the nonlocally and nonlinearly elastic medium reduces to the improved Boussinesq equation for a particular choice of the kernel function appearing in the integral-type constitutive relation. We then derive the Camassa–Holm equation from the improved Boussinesq equation using an asymptotic expansion valid as nonlinearity and dispersion parameters that tend to zero independently. Our approach follows mainly the standard techniques used widely in the literature to derive the Camassa–Holm equation for shallow-water waves. The case where the Fourier transform of the kernel function has fractional powers is also considered and the fractional Camassa–Holm equation is derived using the asymptotic expansion technique.

Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations

In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.