Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.     

Solution of the radial Schrödinger equation by a modified “shooting” method

Based on a comparative analysis of various modifications of the shooting method, used for solving the radial Schrödinger equation, a new algorithm is proposed, which allows to increase considerably its solution accuracy. This is achieved, first, by rising the order of approximation of the Schrödinger equation fromh ⁴ (in the traditional Numerov-Cooley method) up toh ⁶ and, second, by using the variable integration steph. The computer program has been written in Fortran IV to execute this algorithm for calculating vibrational wave functions and related quantities for diatomic molecules.     

Novel method for solution of coupled radial Schrödinger equations

One of the major problems in numerical solution of coupled differential equations is the maintenance of linear independence for different sets of solution vectors. A novel method for solution of radial Schrödinger equations is suggested. It consists of rearrangement of coupled equations in a way that is appropriate to avoid usual numerical instabilities associated with components of the wave function in their classically forbidden regions. Applications of the new method for nuclear structure calculations within the hyperspherical harmonics approach are given.     

Interaction of coupled higher order nonlinear Schrödinger equation solitons

The novel inelastic collision properties of two-soliton interaction for an n-component coupled higher order nonlinear Schr?dinger equation are studied. Some interesting features of three soliton interactions, related to the integrability of the n-component coupled higher order nonlinear Schr?dinger equation are also discussed. Received 17 April 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: abhijit@iitg.ernet.in RID="b" ID="b"e-mail: sasanka@iitg.ernet.in RID="c" ID="c"e-mail: sudipta@iitg.ernet.in     

Lagrangian form of Schrödinger equation

Lagrangian formulation of quantum mechanical Schrödinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein–Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrödinger equation.     

A numerical method for fractional Schrödinger equation of Lennard-Jones potential

Depending on fractional analysis, we find a numerical algorithm to solve the time-independent fractional Schrödinger equation in case of Lennard-Jones potential in one dimension. We apply the algorithm for multiple values of the fractional parameter of the space-dependent fractional Schrödinger equation and multiple values of the system's energy to find the wave function and the probability in these cases.     

Exact solution of N-dimensional radial Schrödinger equation for the fourth-order inverse-power potential

Radial Schrödinger equation in N-dimensional Hilbert space with the potential V(r)=ar^-1+br^-2+cr^-3+dr^-4 is solved exactly by power series method via a suitable ansatz to the wave function with parameters those also exist in the potential function possibly for the first time. Exact analytical expressions for the energy spectra and potential parameters are obtained in terms of linear combinations of known parameters of radial quantum number n, angular momentum quantum number l, and the spatial dimensions N. Expansion coefficients of the wave function ansatz are generated through the two-term recursion relation for odd/even solutions.     

Lagrangian formulation of the Schrödinger equation

We recast the Schrödinger equation in a new Lagrangian formulation. The equation is —i?dψ (x,t)/dt = Lψ (x,t), whereL is the Lagrangian operator. Expressions forL and ford/dt — ⊥ are derived in terms of coordinate and momentum operators.     

A new approach to solve the one-dimensional Schrödinger equation using a wavefunction potential

A new approach to find exact solutions to one–dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non–vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non–vanishing Bohm potentials. New examples of this kind are found and discussed.     

Time dependent Schrödinger equation on arbitrary structured grids: Application to photoionization

A systematic technique for conservatively discretizing the time dependent Schrödinger equation on an arbitrary structured grid is given. Spatial differencing is carried out by finite volumes, and temporal differencing is carried out semi-implicitly. It is shown that the resulting algorithm conserves probability to within a round-off error regardless of the grid geometry. The algorithm is efficient for both serial and parallel computation. The conservative nature of the algorithm, and its phase accuracy, are demonstrated for a bound state, and for a free state in an electromagnetic field. The ionization rate for a hydrogen atom in a strong electromagnetic field is computed, and compared with the rate from tunneling theory. The regime of validity of tunneling theory is clarified.     

New methods to solve the resonant nonlinear Schrödinger’s equation with time-dependent coefficients

In this study, the generalized \(\tan (\phi /2)\)-expansion method and He’s semi-inverse variational method (HSIVM) are applied to seek the exact solitary wave solutions for the resonant nonlinear Schrödinger equation with time-dependent coefficients. Using these methods, we investigate exact solutions for the nonlinear resonant Schrödinger equation with time-dependent coefficients two forms of nonlinearity, including power and dual-power law nonlinearity. Moreover, many new analytical exact solutions are obtained which are expressed by hyperbolic solutions, trigonometric solutions, and rational solutions. In addition, we obtained the bright soliton by HSIVM. These methods are powerful, efficient and those can be used as an alternative to establishing new solutions of different types of differential equations in mathematical physics and engineering.     

Properties of weakly collapsing solutions to the nonlinear Schrödinger equation

It is shown that one of the conditions for a weakly collapsing solution with zero energy produces an infinite number of functionals I _N identically vanishing on the regular solutions to the corresponding differential equation. On the parameter plane {A, C₁}, there are at least two singular lines. Along one of these lines (A/C₁=1/6), are located weakly collapsing solutions with zero energy. It is assumed that, along the second line (A/C₁=α_c), another family of weakly collapsing solutions with zero energy is located. In the domain of large values of the parameters C₁, α=A/C₁, there exists a domain of an intermediate asymptotic form, where the amplitude of oscillations of the function U grows in a large domain relative to the ξ coordinate.     

Non-standard finite-difference time-domain method for solving the Schrödinger equation

The literature on chaos has highlighted several chaotic systems with special features. In this work, a novel chaotic jerk system with non-hyperbolic equilibrium is proposed. The dynamics of this new system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents. In addition, we investigate the time-delay effects on the proposed system. Realisation of such a system is presented to verify its feasibility.     

Statistical mechanics of the nonlinear Schrödinger equation

We investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian $$H(\phi ) = \int_\Omega {[\frac{1}{2}|\nabla \phi |^2 - (1/p) |\phi |^p ] dx}$$ is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, whenΩ is the circle and theL ² norm of the field (which is conserved by the dynamics) is bounded byN, the Gibbs measureυ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only ifp andN are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, asN and the temperature are varied.     

A canonical dilation of the Schrödinger equation

In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.     

Energy minimization related to the nonlinear Schrödinger equation

In this the window of the Sobolev gradient technique to the problem of minimizing a Schrödinger functional associated with a nonlinear Schrödinger equation. We show that gradients act in a suitably chosen Sobolev space (Sobolev gradients) can be used in finite-difference and finite-element settings in a computationally efficient way to find minimum energy states of Schrödinger functionals.