Elliptic solutions of nonlinear integrable equations and related topics

The theory of elliptic solitons for the Kadomtsev-Petviashvili (KP) equation and the dynamics of the corresponding Calogero-Moser system is integrated. It is found that all the elliptic solutions for the KP equation manifest themselves in terms of Riemann theta functions which are associated with algebraic curves admitting a realization in the form of a covering of the initial elliptic curve with some special properties. These curves are given in the paper by explicit formulae. We further give applications of the elliptic Baker-Akhiezer function to generalized elliptic genera of manifolds and to algebraic 2-valued formal groups.Dedicated to the memory of J.-L. Verdier     

Deformations of nonassociative algebras and integrable differential equations

A new class of nonassociative algebras related to integrable PDE's and ODE's is introduced. These algebras can be regarded as a noncommutative generalization of Jordan algebras. Their deformations are investigated. Relationships between such algebras and graded Lie algebras are established.     

On thec-spectral sequence for systems of evolution equations

The groups E ₁ ^2,n–1 ( ) of Vinogradov'sC-spectral sequence for determined systems of evolution equations are considered. Presentation of these groups useful in practical computations is obtained. The group E ₁ ^2,1 ( ) is calculated for a system of Schrödinger type equations.     

Graded differential equations and their deformations: A computational theory for recursion operators

An algebraic model for nonlinear partial differential equations (PDE) in the category ofn-graded modules is constructed. Based on the notion of the graded Frölicher-Nijenhuis bracket, cohomological invariants H ^* (A) are related to each object (A, ) of the theory. Within this framework, H ⁰ (A) generalizes the Lie algebra of symmetries for PDE's, while H ¹ (A) are identified with equivalence classes of infinitesimal deformations. It is shown that elements of a certain part of H ¹ (A) can be interpreted as recursion operators for the object (A, ), i.e. operators giving rise to infinite series of symmetries. Explicit formulas for computing recursion operators are deduced. The general theory is illustrated by a particular example of a graded differential equation, i.e. the Super KdV equation.Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia.     

Numerical solutions of nonlinear evolution equations using variational iteration method

The variational iteration method is used to solve three kinds of nonlinear partial differential equations, coupled nonlinear reaction diffusion equations, Hirota–Satsuma coupled KdV system and Drinefel’d–Sokolov–Wilson equations. Numerical solutions obtained by the variational iteration method are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. He's variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomial in Adomian method. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Weierstrass weight of the hyperosculating points of generalized Fermat curves

Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F?S$ is the set of fixed points of the non-trivial elements of the group H, then F is exactly the set of hyperosculating points of the standard embedding $S?{\mathbb{P}}^n$. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.     

On the number of solutions of NLS equations with magnetics fields in expanding domains

In this paper we look for multiple weak solutions u:Ωλ→C for the complex equation in Ωλ=λΩ. The set Ω⊂RN is a smooth bounded domain, λ>0 is a parameter, A is a regular magnetic field and f is a superlinear function with subcritical growth. Our main result relates, for large values of λ, the number of solutions with the topology of the set Ω. In the proof we apply minimax methods and Ljusternik-Schnirelmann theory.     

Parabolic equations of Von Karman type on Kähler manifolds

We study a parabolic version of a system of Von Karman type on a compact Kähler manifold of arbitrary dimension. We provide local in time regular solutions, which can be extended to global bounded ones if the data of the problem are small.     

Parabolic equations of Von Karman type on Kähler manifolds, II

We complete the study of a parabolic version of a system of Von Karman type on a compact Kähler manifold of complex dimension m. We consider a family of problems k(P). We prove existence of local in time solutions when k=0. When −m?k<0, we define a notion of weak solution, and give some uniqueness and existence results.     

Dispersive blow-up for nonlinear Schrödinger equations revisited

The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.     

Integration on moduli spaces of stable curves through localization

We introduce a new method of calculating intersections on , using localization of equivariant cohomology. As an application, we give a proof of Mirzakhani's recursion relation for calculating intersections of mixed ψ and κ₁ classes.     

Evolution equations for shapes and geometries

The first object of this paper is to introduce a new evolution equation for the characteristic function of the boundary Γ of a Lipschitzian domain Ω in the N-dimensional Euclidean space under the influence of a smooth time-dependent velocity field. The originality of this equation is that the evolution takes place in an L^p-space with respect to the (N − 1)-Hausdorff measure. A second more speculative objective is to discuss how that equation can be relaxed to rougher velocity fields via some weak formulation. A candidate is presented and some of the technical difficulties and open issues are discussed. Continuity results in several metric topologies are also presented. The paper also specializes the results on the evolution of the oriented distance function to initial sets with zero N-dimensional Lebesgue measure.     

Aleksandrov reflection and nonlinear evolution equations,I: The n-sphere and n-ball

We consider the (degenerate) parabolic equationu _t =G(u + ug, t) on then-sphereS ⁿ . This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, whereu is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate ¦u(·,t)¦ <C, whereC depends on the initial conditionu(·, 0) but not ont, nor on the nonlinear functionG. We also prove analogous results for the equationu _t =G(u +cu, ¦x¦,t) on then-ballB ⁿ , wherec ₂(B ⁿ ).     

Sweeping algebraic curves for singular solutions

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented.     

Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e2iωtφ,0,0)$(e^{2i\omega t}\phi ,0,0)$, (0,e2iωtφ,0)$(0,e^{2i\omega t}\phi ,0)$, (0,0,e2iωtφ)$(0,0,e^{2i\omega t}\phi )$, where φ is a ground state of the scalar nonlinear Schrödinger equation.     

Approximate analytical solution for the Zakharov–Kuznetsov equations with fully nonlinear dispersion

In this paper, variational iteration method (VIM) is used to obtain numerical and analytical solutions for the Zakharov–Kuznetsov equations with fully nonlinear dispersion. Comparisons with exact solution show that the VIM is a powerful method for the solution of nonlinear equations.     

Algebraic curves and the Gauss map of algebraic minimal surfaces

In this paper, we first establish a second main theorem for algebraic curves into the n-dimensional projective space. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in Rm with finite total curvature, as well as the uniqueness problem.     

Reduction and quotient equations for differential equations with symmetries

The method of reduction previously known in the theory of Hamiltonian systems with symmetries is developed in order to obtain exact group-invariant solutions of systems of partial differential equations. This method leads to representations of quotient equations which are very convenient for the systematic analysis of invariant solutions of boundary value problems. In the case of partially invariant solutions, necessary and sufficient conditions of their invariance with respect to subalgebras of symmetry algebras are given. The concept of partial symmetries of differential equations is considered.     

STABILITY ANALYSIS OF NONLINEAR GALERKIN ＆ GALERKIN METHODS FOR NONLINEAR EVOLUTION EQUATIONS

Abstract. Ogr object in this artlcle is to describe tbe Galerkln scheme and nonlin-eax Galerkin scheme for the approximation of nonlinear evolution equations, and tostudy the stability of these schemes. Spatial discretizatlon can be pedormed by eitherGalerkln spectral method or nonlinear Galerldn spectral method; time discretizatlort isdone hy Euler sin.heine wklch is explicit or implicit in the nonlinear terms. According tothe stability analysis of the above schemes, the stability of nonllneex Galerkln methodis better than that of Galexkln method.