The Cauchy problem of a focusing energy-critical nonlinear Schrödinger equation

In this paper, we combine variational methods and harmonic analysis to discuss the Cauchy problem of a focusing nonlinear Schrödinger equation. We study the global well-posedness, finite time blowup and asymptotic behavior of this problem. By Hamiltonian property, we establish two types of invariant evolution flows. Then from one flow and the stability of classical energy-critical nonlinear Schrödinger equation, we find that the solution exists globally and scattering occurs. Finally, we get a precise blowup criterion of this problem for positive energy initial data via the other flow.     

On a class of nonlinear inhomogeneous Schrödinger equation

In this paper, we study the inhomogeneous Schrödinger equation $$i\varphi_{t}=-\triangle\varphi -|x|^{b}|\varphi|^{p-1}\varphi,\quad x\in \mathbb{R}^{N}.$$ By using variational methods and a refined interpolation inequality, we establish some simple but sharp conditions on the solutions which exist globally or blow up in a finite time. An interesting result is that we obtain the existence of global solution for arbitrarily large data.     

The Cauchy problem for non-autonomous nonlinear Schrödinger equations

In this paper we study the Cauchy problem for cubic nonlinear Schrödinger equation with space- and time-dependent coefficients on ∝^m and \(\mathbb{T}^m \). By an approximation argument we prove that for suitable initial values, the Cauchy problem admits unique local solutions. Global existence is discussed in the cases of m = 1, 2.     

On the inverse spectral problem for the quasi-periodic Schrödinger equation

We study the quasi-periodic Schrödinger equation $$-\psi''(x) + V(x) \psi(x) = E \psi(x), \quad x \in{ \mathbf {R}} $$ in the regime of “small” V. Let $(E_{m}',E''_{m})$ , m∈Z ^ν , be the standard labeled gaps in the spectrum. Our main result says that if $E''_{m} - E'_{m} \le\varepsilon\exp(-\kappa_{0} |m|)$ for all m∈Z ^ν , with ε being small enough, depending on κ ₀>0 and the frequency vector involved, then the Fourier coefficients of V obey $|c(m)| \le \varepsilon^{1/2} \exp(-\frac{\kappa_{0}}{2} |m|)$ for all m∈Z ^ν . On the other hand we prove that if |c(m)|≤εexp(?κ ₀|m|) with ε being small enough, depending on κ ₀>0 and the frequency vector involved, then $E''_{m} - E'_{m} \le2 \varepsilon\exp(-\frac {\kappa_{0}}{2} |m|)$ .     

On bounded solutions of a nonlinear Schrödinger equation on the half-line

We consider the problem on nonzero solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear ordinary differential equation of the second order under zero boundary conditions for the wave function and the condition that the potential is zero at the beginning of the interval and its derivative is zero at infinity. The problem is reduced to the analysis and investigation of solutions of the Cauchy problem for a system of two nonlinear second-order ordinary differential equations with initial conditions depending on two parameters. We show that if the solution of the Cauchy problem for some parameter values can be extended to the entire half-line, then there exists a nonzero solution of the original problem with finitely many zeros.     

Asymptotic solution of the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of finite density type

With the help of the dressing procedure the singular Riemann problem corresponding to the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of finite density type is reduced to a regular Riemann problem. From the asymptotic analysis of the regular Riemann problem we get the principal term of the asymptotic solution of the Cauchy problem in the domain of superpolynomial decrease, which is described in terms of the scattering data corresponding to the initial condition.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 181–195, 1986.The author sincerely thanks A. R. Its for many helpful discussions.     

On the characterization of the nonlinear Schrödinger hierarchy

It is proved that the conserved polynomials of the nonlinear Schrödinger equation have a vanishing residue property analogous to those now known to characterize the Korteweg-de Vries, Modified Korteweg-de Vries and Sine-Gordon hierarchies.     

The inverse problem for a nonlinear Schrödinger equation

Using variational methods, the solution of the inverse problem of finding the refractive index of a nonlinear medium in a multidimensional Schrödinger equation is studied. The correctness of the statement of the problem under consideration is investigated, and a necessary condition that must be satisfied by the solution of this problem is found. Bibliography: 8 titles.     

On the statistical derivation of the Schrödinger equation

The transition from reversible microscopic operator equations to irreversible equations for a deterministic density matrix is considered for examples of simple systems—the hydrogen atom or a free electron in an electromagnetic field. As a result of the transition, a system of a particle and field oscillators is replaced by a continuous medium. The Schrödinger equation for the deterministic wave function also describes the evolution of a continuum but without allowance for dissipative terms. In this sense, there is an analogy between the Schrödinger equation in quantum mechanics and Euler's equation in hydrodynamics. The smallest size of a point of a continuous medium is described by the classical electron radiusr _e . It also determines the effective Thomson cross section for scattering of photons by free electrons. The lengthr _e and the corresponding time interval _e =r _e /c play the role of hidden parameters in quantum mechanics. Two methods of calculating the effective Thomson cross section in terms of the extinction coefficient are considered. The first of them is based on the equation of motion of a free electron in a field with allowance for radiative friction. This equation leads to well-known difficulties. Moreover, the velocity fluctuations calculated on its basis lead to a contradiction with the second law of thermodynamics. The second method is based on the introduction of a constant friction coefficient = _e ^–1 , the presence of which reflects loss of information on smoothing over the volume of a point of the continuous medium. Such a method of calculation leads to the same expression for the effective cross section but makes it possible to avoid the difficulties with the second law of thermodynamics. In the derivation of quantum kinetic equations, the physically infinitesimally small scales are determined by the Compton length _C and the corresponding time interval. The introduction of these scales makes it possible to separate and eliminate small-scale fluctuations, the collision integrals being expressed in terms of the correlation functions of these fluctuations.In memory of Dmitrii Nikolaevich ZubarevMoscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 3–26, October, 1993.     

The elliptic-in-t solutions of the nonlinear Schrödinger equation

Four various anzatzes of the Krichever curves for the elliptic-in-t solutions of the nonlinear Schrödinger equation are considered. An example is given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 188–200, May, 1996.Translated by V. I. Serdobol'skii.     

Explicit breather solution of the nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We present a one-line closed-form expression for the three-parameter breather of the nonlinear Schrödinger equation. This provides an analytic proof of...     

Expanding integrable models of the nonlinear Schrödinger equation

By using a few Lie algebras and the corresponding loop algebras, we establish some isospectral problems whose compatibility conditions give rise to a few various expanding integrable models (including integrable couplings) of the well-known nonlinear Schrödinger equation. The Hamiltonian forms of two of them are generated by making use of the variational identity. Finally, we propose an efficient method for generating a nonlinear integrable coupling of the nonlinear Schrödinger equation.