This paper is devoted to the derivation of a macroscopic diffusion equation (Fourier’s law) describing the transport of energy in an anharmonic chain. More precisely, we study here the so-called FPU-β chain, which is a very simple model for a one-dimensional crystal in which atoms are coupled to their nearest neighbors by a harmonic potential, weakly perturbed by a quartic potential. The starti...

D−dimensional central and complex potentials of a Coulomb plus quartic-polynomial form are considered in a PT −symmetrized radial Schrödinger equation. Arbitrarily large finite multiplets of bound states are shown obtainable in an elementary form. Relations between their energies and couplings are determined by a finite-dimensional secular equation. The Bender’s and Boettcher’s one-dimensional ...

In this paper we prove that the Diophantine equation x − 4cxy + (6c+ 2)xy + 4cxy + y = 1, where c ≥ 3 is an integer, has only the trivial solutions (±1, 0), (0,±1). Using the method of Tzanakis, we show that solving this quartic Thue equation reduces to solving the system of Pellian equations (2c+ 1)U − 2cV 2 = 1, (c− 2)U − cZ = −2, and we prove that all solutions of this system are given by (U...

We show that the quartic generalised KdV equation ut + uxxx + (u )x = 0 is globally wellposed for data in the critical (scale-invariant) space Ḣ −1/6 x (R) with small norm (and locally wellposed for large norm), improving a result of Gruenrock [8]. As an application we obtain scattering results in H x(R) ∩ Ḣ −1/6 x (R) for the radiation component of a perturbed soliton for this equation, improv...

In this paper we consider a nonlinear system of PDEs coupling the viscous Cahn–Hilliard–Oono equation with dynamic boundary conditions enjoying similar structure on boundary. After proving well-posedness corresponding initial value problem, study an associated optimal control problem related to tracking-type cost functional, first-order necessary optimality. The controls enter in form distribut...

in the present paper a solution of the generalizedquadratic functional equation$$f(kx+ y)+f(kx+sigma(y))=2k^{2}f(x)+2f(y),phantom{+} x,yin{e}$$ isgiven where $sigma$ is an involution of the normed space $e$ and$k$ is a fixed positive integer. furthermore we investigate thehyers-ulam-rassias stability of the functional equation. thehyers-ulam stability on unbounded domains is also studied.applic...

We consider the stochastic quantization of a quartic double-well energy functional in semiclassical regime and derive optimal asymptotics for exponentially small splitting ground state energy. Our result provides an infinite-dimensional version some sharp tunneling estimates known finite dimensions Witten Laplacians degree zero. From point view it proves that L2 spectral gap one-dimensional All...

The stability problem of functional equations originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms. Let G1, · be a group and let G2, ∗, d be a metric group with the metric d ·, · . Given > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d h x · y , h x ∗ h y < δ for all x, y ∈ G1, then there exists a homomorphism H...

We consider the singularly perturbed Allen-Cahn equation on a strictly convex plane domain. We show that when the perturbation parameter tends to zero there are solutions having a transition layer that tends to a straight line segment. This segment can be characterized as the shortest path intersecting the boundary orthogonally at two points.