Deformations of algebraic curves and integrable nonlinear evolution equations

A brief exposition of applications of the methods of algebraic geometry to systems integrable by the IST method with variable spectral parameters is presented. Usually, theta-functional solutions for these systems are generated by some deformations of algebraic curves. The deformations of algebraic curves are also related with theta-functional solutions of Yang-Mills self-duality equations which contain special systems with a variable spectral parameter as a special reduction. Another important situation in which the deformations of algebraic curves naturally occur is the KdV equation with string-like boundary conditions. Most important concrete examples of systems integrable by the IST method with variable spectral parameter having different properties within a framework of the behavior of moduli of underlying curves, analytic properties of the Baker-Akhiezer functions, and the qualitative behavior of the solutions, are vacuum axially symmetric Einstein equations, the Heisenberg cylindrical magnet equation, the deformed Maxwell-Bloch system, and the cylindrical KP equation.Dedicated to the memory of J.-L. Verdier     

Elliptic functions and equations of modular curves

Let be a prime. We show that the space of weight one Eisenstein series defines an embedding into ${mathbb P}^{(p-3)/2}X_1(p)$ for the congruence group that is scheme-theoretically cut out by explicit quadratic equations. Received: 8 November 2000 / Published online: 17 August 2001     

Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind

The nonlinear Schrödinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The ‘derivative’ NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen-Lee-Liu equation, are studied. More precisely, the complex envelope is separated into the absolute value and the phase. The absolute value is solved in terms of a polynomial in elliptic functions while the phase is expressed in terms of elliptic integrals of the third kind. The exact periodicity condition will imply that only a countable set of elliptic function moduli is allowed. This feature contrasts sharply with other periodic solutions of envelope equations, where a continuous range of elliptic function moduli is permitted.     

Lump solutions to nonlinear partial differential equations via Hirota bilinear forms

Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable transformations $u=2{(\ln ?f)}_x$ and $u=2{(\ln ?f)}_{xx}$, where x is one spatial variable. Applications are made for a few generalized KP and BKP equations.     

Variational iteration method for solving cubic nonlinear Schrödinger equation

The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.     

Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order

In this paper, we investigate the existence and uniqueness of the solution to the Cauchy problem for a class of nonlinear wave equations of higher order and prove the existence and nonexistence of global solutions to this problem by a potential well method.     

On asymptotic behavior of solutions to Korteweg-de Vries type equations related to vortex filament with axial flow

We study the global existence and asymptotic behavior in time of solutions to the Korteweg-de Vries type equation called as “Hirota” equation. This equation is a mixture of cubic nonlinear Schrödinger equation and modified Korteweg-de Vries equation. We show the unique existence of the solution for this equation which tends to the given “modified” free profile by using the two asymptotic formulae for some oscillatory integrals.     

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.     

Isogenies of elliptic curves and the Morava stabilizer group

Let S₂ be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over , O the ring of endomorphisms of C, and ? a topological generator of (or if p=2). We show that for p>2 the group Γ⊆O[1/?]^× of quasi-endomorphisms of degree a power of ? is dense in S₂. For p=2, we show that Γ is dense in an index 2 subgroup of S₂.     

Periodic solutions for doubly nonlinear evolution equations

We discuss the existence of periodic solution for the doubly nonlinear evolution equation A(u^′(t))+∂?(u(t))∋f(t) governed by a maximal monotone operator A and a subdifferential operator ∂? in a Hilbert space H. As the corresponding Cauchy problem cannot be expected to be uniquely solvable, the standard approach based on the Poincaré map may genuinely fail. In order to overcome this difficulty, we firstly address some approximate problems relying on a specific approximate periodicity condition. Then, periodic solutions for the original problem are obtained by establishing energy estimates and by performing a limiting procedure. As a by-product, a structural stability analysis is presented for the periodic problem and an application to nonlinear PDEs is provided.     

Asymptotic dynamics of nonlinear Schrödinger equations with many bound states

We consider a nonlinear Schrödinger equation with a bounded localized potential in . The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order n in H¹, and that its ground state component is larger than n^3−ε with ε>0 small. We prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity.     

Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations

Sufficient and necessary conditions for the embeddings between Besov spaces and modulation spaces are obtained. Moreover, using the frequency-uniform decomposition method, we study the Cauchy problem for the generalized BO, KdV and NLS equations, for which the global well-posedness of solutions with the small rough data in certain modulation spaces is shown.     

Nonlinear elliptic equations with subhomogeneous potentials

We prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation

     

Instability of solitary wave solutions for the generalized BO-ZK equation

In this paper we study the generalized BO-ZK equation in two space dimensions

ut+upux+αHuxx+εuxyy=0.     

On the Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations

We investigate some well-posedness issues for the initial value problem (IVP) associated with the system

     

The first integral method to some complex nonlinear partial differential equations

In this paper, the first integral method is used to construct exact solutions of the Hamiltonian amplitude equation and coupled Higgs field equation. The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones.     

Quasi-periodic solutions in a nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equation

iut−uxx+mu+⁴|u|u=0     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

Scattering theory for nonlinear Schrödinger equations with inverse-square potential

We study the long-time behavior of solutions to nonlinear Schrödinger equations with some critical rough potential of a|x|⁻²$a{|x|}^{-2}$ type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with _Pa=−Δ+a|x|⁻²$P_a=-\mathrm{\Delta }+a{|x|}^{-2}$. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schrödinger equation with inverse square potential in energy space H¹(Rⁿ)$H^1({\mathbb{R}}^n)$.