Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line

It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a 2×2 matrix Riemann–Hilbert problem. This problem is specified by the spectral functions {a(k),b(k)} which are defined in terms of the initial condition q(x,0)=q ₀(x), and by the spectral functions {A(k),B(k)} which are defined in terms of the specified boundary condition q(0,t)=g ₀(t) and the unknown boundary value q _x (0,t)=g ₁(t). Furthermore, it has been shown that given q ₀ and g ₀, the function g ₁ can be characterized through the solution of a certain 'global relation' coupling q ₀, g ₀, g ₁, and (t,k), where satisfies the t-part ofthe associated Lax pair evaluated at x=0. We show here that, by using a Gelfand–Levitan–Marchenko triangular representation of , the global relation can be explicitly solved for g ₁.     

The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.     

Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

On singular Lagrangian underlying the Schrödinger equation

We analyze the properties that manifest Hamiltonian nature of the Schrödinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian formulation). It is used to show that any solution of the Schrödinger equation with time independent potential can be presented in the form , where the real field ?(t,xi) is some solution of nonsingular Lagrangian theory being specified below. Preservation of probability turns out to be the energy conservation law for the field ?. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics.     

A simple and direct periodic solution of the Korteweg-de vries equations

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrödinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw _t (x, t, λ)=φ _t (x + c, x, t, λ) withq _t (x, t). The functionφ(x, x ₀,t, λ) obeys the Schrödinger equation and the boundary conditionsφ(x ₀,x ₀,t, λ)=0,φ _x (x ₀,x ₀;t, λ)=1. The shiftingc is equal to the period. We differentiatew _t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq _t (x, t) by an expression of the KdV hiearchy in the relation betweenq _t (x, t) andw _t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.     

Integrable Nonlinear Evolution Equations on a Finite Interval

Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n−N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂ⁿ_xq. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving q_t+q_xxx and q_t−q_xxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively. Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II equations. We first show that the unknown boundary values at each end (for example, q_x(0,t) and q_x(L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution.     

An invariant measure for the equationu tt −u xx +u 3=0

Numerical studies of the initial boundary-value problem of the semilinear wave equationu _tt –u _xx +u ³=0 subject to periodic boundary conditionsu(t, 0)=u(t, 2),u _t (t, 0)=u _t (t, 2) and initial conditionsu(0,x)=u ₀(x),u _t(0,x)=v ₀(x), whereu ₀(x) andv ₀(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial stateu ₀(x),v ₀(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above returning property.     

The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem uniquely determined in terms of the initial datum q(x,0), as well as of the boundary values {q(0, t),qx(0, t),qxx(0, t)}. For the Dirichlet problem, it is necessary to characterize the unknown boundary values qx(0, t) and qxx(0, t) in terms of the given data q(x, 0) and q(0, t). It is shown here that in the particular case of a vanishing initial datum and of a sine wave as Dirichlet datum, qx(0, t) and qxx(0, t) can be computed explicitly at least up to third order in a perturbative expansion and that at least up to this order, these functions are asymptotically periodic for large t.     

Quantum theory of single events: Localized De Broglie wavelets,Schrödinger waves,and classical trajectories

For an arbitrary potential V with classical trajectoriesx=g(t), we construct localized oscillating three-dimensional wave lumps (x, t,g) representing a single quantum particle. The crest of the envelope of the ripple follows the classical orbitg(t), slightly modified due to the potential V, and (x, t,g) satisfies the Schrödinger equation. The field energy, momentum, and angular momentum calculated as integrals over all space are equal to the particle energy, momentum, and angular momentum. The relation to coherent states and to Schrödinger waves is also discussed.     

Inverse scattering with non-overdetermined data

Let A(β,α,k) be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain D⊂R³. The unit vector α is the direction of the incident plane wave, the unit vector β is the direction of the scattered wave, k>0 is the wave number. The governing equation for the waves is [∇²+k²−q(x)]u=0 in R³. For a suitable class M of potentials it is proved that if Aq₁(−β,β,k)=Aq₂(−β,β,k),∀β∈S², ∀k∈(k₀,k₁), and q₁, q₂∈M, then q₁=q₂. This is a uniqueness theorem for the solution to the inverse scattering problem with backscattering data. It is also proved for this class of potentials that if , ∀k∈(k₀,k₁), and q₁, q₂∈M, then q₁=q₂. Here is an arbitrarily small open subset of S², and |k₀−k₁|>0 is arbitrarily small.     

How to Find Separation Coordinates for the Hamilton–Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems

The method of separation of variables applied to the natural Hamilton–Jacobi equation (u/q _i )²+V(q)=E consists of finding new curvilinear coordinates x _i (q) in which the transformed equation admits a complete separated solution u(x)=u _(i)(x _i ;). For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmic solution, which we present here. It is based on recursive use of the Bertrand–Darboux equations, which are linear second order partial differential equations with undetermined coefficients. The result applies to the Helmholtz (stationary Schrödinger) equation as well.     

Ground-state properties of the one dimensional electron liquid

We present a theory of the pair distribution function g(z) and many-body effective electron-electron interaction for the one dimensional (1D) electron liquid. Our approach involves the solution of a zero-energy scattering Schrödinger equation for where we implemented the Fermi hypernetted-chain approximation including the elementary diagram corrections. We present numerical results for g(z) and the static structure factor S(k) and obtain good agreement with data from diffusion Monte Carlo studies of the 1D system. We calculate the correlation energy and charge excitation spectrum over an extensive range of electron density. Furthermore, we obtain the static correlations in good qualitative agreement with those calculated for the Luttinger liquid model with long-range interactions.     

The low energy scattering for slowly decreasing potentials

For the radial Schrödinger equation with a potentialq(x) decreasing at infinity asq ₀ q ^–, (0, 2), the low energy asymptotics of spectral and scattering data is found. In particular, it is shown that forq ₀>0 the spectral function vanishes exponentially as the energyk ² tends to zero. On the contrary, there is always a zero-energy resonance forq ₀<0. These results determine the local asymptotics of solutions of the time-dependent Schrödinger equation for large timest. Specifically, for positive potentials its solutions decay as exp(–₀ t ^(2–)/(2+), ₀>0,t. In the case (1, 2) it is shown that for ±q ₀>0 the phase shift tends to ± ask0 and its asymptotics is evaluated.     

Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(?2iβx) (or asymptotically periodic, u(x, 0) =α exp(?2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: u_x(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0.     

Kinetic Theory and Lax Equations for Shock Clustering and Burgers Turbulence

We study shock statistics in the scalar conservation law ? _t u+? _x f(u)=0, x∈?, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈?. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Structural, electronic and optical calculations of CaxZn1−xO alloys: A first principles study

First principles density functional calculations, using full potential linearized augmented plane wave (FP-LAPW) method, have been performed in order to investigate the structural, electronic and optical properties of Ca_xZn_1−xO alloy in B1 (NaCl) phase. Dependence of structural parameters as well as the band gap values on the composition x have been analyzed in the range 0?x?1. Calculated electronic structure and the density of states of these alloys are discussed in terms of the contribution of Zn d, O p and Ca p and d states. Furthermore, optical properties such as complex dielectric constants ε(ω), refractive index including extinction coefficient k(ω), normal-incidence reflectivity R(ω), absorption coefficient α(ω) and optical conductivity σ(ω) are calculated and discussed in the incident photon energy range 0-45 eV.     

Scaling and memory effect in volatility return interval of the Chinese stock market

We investigate the probability distribution of the volatility return intervals τ for the Chinese stock market. We rescale both the probability distribution P_q(τ) and the volatility return intervals τ as to obtain a uniform scaling curve for different threshold value q. The scaling curve can be well fitted by the stretched exponential function , which suggests memory exists in τ. To demonstrate the memory effect, we investigate the conditional probability distribution P_q(τ|τ₀), the mean conditional interval 〈τ|τ₀〉 and the cumulative probability distribution of the cluster size of τ. The results show clear clustering effect. We further investigate the persistence probability distribution P_±(t) and find that P₋(t) decays by a power law with the exponent far different from the value 0.5 for the random walk, which further confirms long memory exists in τ. The scaling and long memory effect of τ for the Chinese stock market are similar to those obtained from the United States and the Japanese financial markets.     

Relativistic quantum theory for charged spinless particles in external vector fields

Guided by a diagonalized form of the classical field-energy we construct a time-dependent canonical pair of Schrödinger fields _t (x) and _t (x) which diagonalizes the field-HamiltonianH _t . These Schrödinger fields in general belong to inequivalent representations of the canonical commutation relations for differentt's.The Heisenberg field is constructed by solving the Heisenberg equation of motion and its time-evolution turns out to be governed by a unitary operator, i.e. the Heisenberg fields at different times are unitarily equivalent.Scattering theory (including eventual incoming and/or outgoing bound-states) is finally constructed.