On the breakdown of classical solutions of the planar motion of an elastic string

We prove a result of formation of singularities for the classical solutions of the planar motion of a nonlinear elastic string. In a particular, but physically relevant, case we give a characterization of the global C¹ solutions with positive tension.     

On Schnabl solutions of string field equations

We clarify the relationship between Schnabl’s solution and pure gauge configurations. Both Schnabl’s and pure gauge solutions are obtained by means of an iterative procedure. We show that the pure gauge string field configuration that is used in the construction of a perturbation series for Schnabl’s solution diverges on a large subspace of string configurations, but it can be rendered convergent by adding a compensating term. The additional term ensures the fulfillment of the equations of motion in a weak sense. This compensating term coincides with the term necessary for obtaining an action consistent with Sen’s first conjecture.     

Pure gauge configurations and tachyon solutions for cubic fermionic string field theory equations of motion

Special perturbative pure gauge solutions parameterized by a pair of wedge states are parts of the nontrivial (not purely gauge) tachyon solutions of the cubic fermionic string field theory describing the non-BPS brane true vacuum. We demonstrate explicitly that for the large parameter of the perturbation expansion, these pure gauge configurations are no longer solutions of the equations of motion. We show that this problem is solved by adding an extra term that is just the term needed for the first Sen conjecture to hold.     

On the equations of motion of the folded inextensible string

In a number of excellent, even recently published, books in Theoretical Mechanics the equations of motion of a folded inextensible string, for example to explain the crack of a whip, are derived using either the energy principle, if a system of one degree of freedom is considered, or Lagrange's equations for conservative systems, if a system of more than one degree of freedom is studied. However, it will be shown in this paper that the resulting equations are incorrect because physically a nonconservative system is given. Hence, both methods mentioned above based on energy conservation must not be used. A correct derivation of the equations of motion is given by means of the linear momentum balance.     

Classification of the string solutions of Bethe equations in an XXZ model of arbitrary spin

One investigates the structure of the string solutions of a Heisenberg XXZ model of arbitrary spin. It is discovered that there exist values of the spin for which there exist values of the spin for which there arise constraints on the disposition of the centers of certain string solutions. One finds a class of spins, commensurable in a definite manner with the anisotropy parameter, for which such constraints do not arise. One gives a hypothetical formulation of the classification of the string solutions over a ferromagnetic vacuum in the general case.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 31–36, 1985.     

On classical almost-periodic solutions of nonlinear hyperbolic equations

Translated from Matematicheskie Zametki, Vol. 54, No. 6, pp. 146–148, December, 1993.     

Series solutions of some anharmonic motion equations

Consider the class of nonlinear oscillators of the form

(1)     

Existence of solutions for motion equations for an objective model of viscoelastic medium

We consider the Dirichlet initial-boundary value problem for the equations of motion of incompressible viscoelastic medium with objective rheological constitutive laws of Maxwell and Jeffreys kind. Several types of existence results are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New insight into classical equilibrium solutions of kinetic equations

Summary. The equilibrium solutions to kinetic equations of weak turbulence (weakly nonlinear wave systems) are analyzed in a systematic manner. The study is performed for kinetic equations involving any number of interacting waves of an arbitrary dimension. Conditions for the equilibrium solutions are reduced to generalized Cauchy functional equations defined at specific hypersurfaces. The analysis proves that, among differentiable functions, the formal equilibrium solutions correspond to equipartition of a linear combination of the energy, the components of momentum, and the number of particles. The structure of the set of equilibrium solutions does not depend on the dimension of wave systems, nor does it explicitly depend on the particular dispersion relation. It depends only on the number of wave components in resonance sets, whereas the number of particles does not enter into the equilibrium distribution in interactions of an odd number of harmonics. The most important consequence of this difference is that wave systems with four-wave resonance conserve the number of particles (wave action), whereas systems with three-wave or five-wave resonance violate this law. All the formal solutions, notwithstanding their physical realizability, are proved to be linearly stable with respect to small disturbances. Received October 5, 2000; accepted August 17, 2001 Online publication November 5, 2001     

Radially symmetric global classical solutions of non-linear wave equations

The Cauchy problem for semilinear wave equations u_tt ? Δu + h(|x|)u^p = 0 with radially symmetric smooth ‘large’ data has a unique global classical solution in arbitrary space dimensions if h is non-negative and p any odd integer provided the smooth factor h vanishes with sufficiently high order at the origin and is bounded together with its derivatives.     

Common nonnegative definite solutions of some classical matrix equations

A necessary and sufficient condition for the existence of a Hermitian nonnegative definite solution of system of matrix equations $$A_{1}X=C_{1},\qquad XB_{2}=C_{2}, \qquad A_{3}XA_{3}^{\ast}=C_{3},\qquad A_{4}XA_{4}^{\ast}=C_{4} $$ as well as a representation for this general nonnegative definite solution are derived. As particular cases, the corresponding results on some other systems are also derived.     

Weak solutions and attractors for motion equations for an objective model of viscoelastic medium

We study the boundary value problem for the equations of motion of incompressible viscoelastic medium with an objective constitutive law of Jeffreys kind. We show existence of global weak solutions for any initial data and construct their minimal uniform trajectory attractor and uniform global attractor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)