Solitary waves and their linear stability in weakly coupled KdV equations

We consider a system of weakly coupled KdV equations developed initially by Gear & Grimshaw to model interactions between long waves. We prove the existence of a variety of solitary wave solutions, some of which are not constrained minimizers. We show that such solutions are always linearly unstable. Moreover, the nature of the instability may be oscillatory and as such provides a rigorous justification for the numerically observed phenomenon of “leapfrogging.”     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

Refined asymptotics for constant scalar curvature metrics with isolated singularities

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, , in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohožaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions. Oblatum 26-II-1997 & 6-II-1998 / Published online: 12 November 1998     

Optimization problems with algebraic solutions: Quadratic fractional programs and ratio games

A mathematical program with a rational objective function may have irrational algebraic solutions even when the data are integral. We suggest that for such problems the optimal solution will be represented as follows: If λ* denotes the optimal value there will be given an intervalI and a polynomialP(λ) such thatI contains λ* and λ* is the unique root ofP(λ) inI. It is shown that with this representation the solutions to convex quadratic fractional programs and ratio games can be obtained in polynomial time.     

Construction of exact solutions in two-field cosmological models

We construct a dark energy model with a phantom scalar field, a standard scalar field, and a polynomial potential inspired by string field theory. We find a two-parameter set of exact solutions of the Friedmann equations. We find a potential satisfying the conditions obtained from the string theory and such that at large times, some of the exact solutions correspond to the state parameter w_DE > −1 while the others correspond to w_DE < −1. We demonstrate that the superpotential method is very effective for seeking new exact solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 47–61, April, 2008.     

Backward SDEs and Cauchy problem for semilinear equations in divergence form

 We extend the definition of solutions of backward stochastic differential equations to the case where the driving process is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic partial differential equations in Sobolev spaces. Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002 Research supported by KBN Grant 0253 P03 2000 19. Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55 Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution     

On Aronsson Equation and Deterministic Optimal Control

When Hamiltonians are nonsmooth, we define viscosity solutions of the Aronsson equation and prove that value functions of the corresponding deterministic optimal control problems are solutions if they are bilateral viscosity solutions of the Hamilton-Jacobi-Bellman equation. We characterize such a property in several ways, in particular it follows that a value function which is an absolute minimizer is a bilateral viscosity solution of the HJB equation and these two properties are often equivalent. We also determine that bilateral solutions of HJB equations are unique among absolute minimizers with prescribed boundary conditions. This research was partially supported by MIUR-Prin project “Metodi di viscosità, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari”.     

Noncommutative unitons

By Uhlenbeck’s results, every harmonic map from the Riemann sphere S² to the unitary group U(n) decomposes into a product of so-called unitons: special maps from S² to the Grassmannians Gr _k(ℂⁿ) ⊂ U(n) satisfying certain systems of first-order differential equations. We construct a noncommutative analogue of this factorization, applicable to those solutions of the noncommutative unitary sigma model that are finite-dimensional perturbations of zero-energy solutions. In particular, we prove that the energy of each such solution is an integer multiple of 8π, give examples of solutions that are not equivalent to Grassmannian solutions, and study the realization of non-Grassmannian zero modes of the Hessian of the energy functional by directions tangent to the moduli space of solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 220–239, February, 2008.     

Stability of translating solutions to mean curvature flow

We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates. The authors are members of SFB 647/B3 “Raum – Zeit – Materie: Singularity Structure, Long-time Behaviour and Dynamics of Solutions of Non-linear Evolution Equations”.     

Monodromy of certain Painlevé–VI transcendents and reflection groups

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ= and 2α=(2μ-1)² with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space. Oblatum 19-III-1999 & 25-XI-1999?Published online: 21 February 2000     

Stationary layered solutions in {\Bbb R}^2 for an Allen–Cahn system with multiple well potential

We study entire solutions on of the elliptic system where is a multiple-well potential. We seek solutions which are “heteroclinic,” in two senses: for each fixed they connect (at ) a pair of constant global minima of , and they connect a pair of distinct one dimensional stationary wave solutions when . These solutions describe the local structure of solutions to a reaction-diffusion system near a smooth phase boundary curve. The existence of these heteroclinic solutions demonstrates an unexpected difference between the scalar and vector valued Allen–Cahn equations, namely that in the vectorial case the transition profiles may vary tangentially along the interface. We also consider entire stationary solutions with a “saddle” geometry, which describe the structure of solutions near a crossing point of smooth interfaces. Received April 15, 1996 / Accepted: November 11, 1996     

Simultaneous Approximation in Positive Characteristic

 Let be the approximation exponent of a power series α (so that when α is algebraic of degree d, then by Dirichlet’s and Liouville’s Theorems). If the characteristic is positive, q is a power of the characteristic, and are related by a fractional linear transformation with polynomial coefficients, then by respective work of Voloch and of de Mathan, there are constants such that has no solution if , and infinitely many solutions if . We will formulate and prove generalizations to simultaneous approximation.     

Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001     

On a class of second order variational problems with constraints

We study the structure of optimal solutions for a class of constrained, second order variational problems on bounded intervals. We show that, for intervals of length greater than some positive constant, the optimal solutions are bounded inC ¹ by a bound independent of the length of the interval. Furthermore, for sufficiently large intervals, the ‘mass’ and ‘energy’ of optimal solutions are almost uniformly distributed.     

On Global Attractors of Multivalued Semiflows Generated by the 3D Bénard System

In this paper we prove the existence of solutions for the 3D Bénard system in the class of functions which are strongly continuous with respect to the second component of the vector (that is, the one corresponding to the parabolic equation). We construct then a multivalued semiflow generated by such solutions and obtain the existence of a global φ −attractor for the weak-strong topology. Moreover, a family of multivalued semiflows is defined on suitable convex bounded subsets of the phase space, proving for them the existence of a global attractor (which is the same for every semiflow of the family) for the weak-strong topology.     

Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains. Communicated by David Kinderlehrer     

Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains

For an arbitrary domain Ω ⊂ ℝⁿ, n=2,3, Ω ≠ ℝⁿ, we prove the existence of weak periodic solutions to the Navier-Stokes equations and of regular solutions if the data are small or satisfy certain symmetry conditions. We also show that the periodic regular solutions are stable. Bibliography: 38 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 142–182.     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M _c such that if solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for . While characterising the possible infinite time blowing-up profile for M  =  M _c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two. This paper is under the Creative Commons licence Attribution-NonCommercial-ShareAlike 2.5.     

Moduli spaces of solutions of a noncommutative sigma model

Using a noncommutative version of the uniton theory, we study the space of those solutions of the noncommutative U(1) sigma model that are representable as finite-dimensional perturbations of the identity operator. The basic integer-valued characteristics of such solutions are their normalized energy e, canonical rank r, and minimum uniton number u, which always satisfy r ≤ e and u ≤ e. Starting with the so-called BPS solutions (u = 1), we completely describe the sets of all solutions with r = 1, 2, e − 1, e (which forces u ≤ 2) and all solutions of small energy (e ≤ 5). The obtained results reveal a simple but nontrivial structure of the moduli spaces and lead to a series of conjectures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 307–327, September, 2008.