Comparison of quasilinear and WKB approximations

It is shown that the quasilinearization method (QLM) sums the WKB series. The method approaches solution of the Riccati equation (obtained by casting the Schrödinger equation in a nonlinear form) by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of a smallness parameter. Each pth QLM iterate is expressible in a closed integral form. Its expansion in powers of reproduces the structure of the WKB series generating an infinite number of the WKB terms. Coefficients of the first 2^p terms of the expansion are exact while coefficients of a similar number of the next terms are approximate. The quantization condition in any QLM iteration, including the first, leads to exact energies for many well known physical potentials such as the Coulomb, harmonic oscillator, Pöschl–Teller, Hulthen, Hyleraas, Morse, Eckart, etc.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential: Excited states

The quasilinearization method (QLM) is used to approximate analytically, both the ground state and the excited state solutions of the Schrödinger equation for arbitrary potentials. The procedure of approximation was demonstrated on examples of a few often used physical potentials such as the quartic anharmonic oscillator, the Yukawa and the spiked harmonic oscillator potentials. The accurate analytic expressions for the ground and excited state energies and wave functions were presented. These high-precision approximate analytic representations are obtained by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first QLM iteration. In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist. The choice of zero iteration is based on general features of solutions near the boundaries. In order to estimate the accuracy of the QLM solutions, the exact numerical solutions were found as well. The first QLM iterate given by analytic expression allows to estimate analytically the role of different parameters and the influence of their variation on different characteristics of the relevant quantum systems.     

Accurate analytic presentation of solution for the spiked harmonic oscillator problem

High precision approximate analytic expressions of the ground state energies and wave functions for the spiked harmonic oscillator are found by first casting the correspondent Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of parameters. The accuracy ranging between 10⁻³ and 10⁻⁷ for the energies and, correspondingly, 10⁻² and 10⁻⁷ for the wave functions in the regions, where they are not extremely small is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems.     

Analytical computation of amplification of coupling in relativistic equations with Yukawa potential

The approximate analytic solutions to the Klein-Gordon and Dirac equations with the Yukawa potential were derived by using the quasilinearization method (QLM). The accurate analytic expressions for the ground state energies and wave functions were presented. These high-precision approximate analytic representations are obtained by first casting the proper relativistic equation into a nonlinear Riccati form and then solving that nonlinear equation in the first QLM iteration. The choice of zero iteration is based on general features of the exact solutions near the origin and infinity. To estimate the accuracy of the QLM solutions, the exact numerical solutions were found, as well. The analytical QLM solutions are found to be extremely accurate for a small exponent parameter w of the Yukawa potential. The reasonable accuracy is kept for the medium values of w. When w approaches the critical values, the precision of the QLM results falls down markedly. However, the approximate analytic QLM solution to the Dirac equation corresponding to the maximum relativistic effect turned out to be very accurate even for w close to the exact critical , whereas the QLM calculations yield . This effect of “amplification” in compare with the Schrödinger equation critical parameter was investigated earlier [S. De Leo, P. Rotelli, Phys. Rev. D 69 (2004) 034006]. In this work, it was found that the “amplification” for the Klein-Gordon equation became all the more evident. The exact numerical value is , whereas the QLM approximation yields .     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential

High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10⁻⁴ and 10⁻⁸ for the energies and, correspondingly, 10⁻² and 10⁻⁴ for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.     

Quasilinearization method: Nonperturbative approach to physical problems

The general properties of the quasilinearization method (QLM), particularly its fast quadratic convergence, monotonicity, and numerical stability, are analyzed and illustrated on different physical problems. The method approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of linear ones and is not based on the existence of a small parameter. It is shown that QLM gives excellent results when applied to different nonlinear differential equations in physics, such as Blasius, Lane-Emden, and Thomas-Fermi equations, as well as in computation of ground and excited bound-state energies and wave functions in quantum mechanics (where it can be applied by casting the Schrödinger equation in the nonlinear Riccati form) for a variety of potentials most of which are not treatable with the help of perturbation theory. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and the first few iterations already yield extremely precise results. The QLM approximations, unlike the asymptotic series in perturbation theory and 1/N expansions, are not divergent at higher orders. The method sums many orders of perturbation theory as well as of the WKB expansion. It provides final and accurate answers for large and infinite values of the coupling constants and is able to handle even supersingular potentials for which each term of the perturbation series is infinite and the perturbation expansion does not exist.     

Adiabatic theory,Liapunov exponents,and rotation number for quadratic Hamiltonians

We consider the adiabatic problem for general time-dependent quadratic Hamiltonians and develop a method quite different from WKB. In particular, we apply our results to the Schrödinger equation in a strip. We show that there exists a first regular step (avoiding resonance problems) providing one adiabatic invariant, bounds on the Liapunov exponents, and estimates on the rotation number at any order of the perturbation theory. The further step is shown to be equivalent to a quantum adiabatic problem, which, by the usual adiabatic techniques, provides the other possible adiabatic invariants. In the special case of the Schrödinger equation our method is simpler and more powerful than the WKB techniques.     

A new application of Riccati equation to some nonlinear evolution equations

By means of symbolic computation, a new application of Riccati equation is presented to obtain novel exact solutions of some nonlinear evolution equations, such as nonlinear Klein-Gordon equation, generalized Pochhammer-Chree equation and nonlinear Schrödinger equation. Comparing with the existing tanh methods and the proposed modifications, we obtain the exact solutions in the form as a non-integer power polynomial of tanh (or tan) functions by using this method, and the availability of symbolic computation is demonstrated.     

Higher-order integrable evolution equation and its soliton solutions

We consider an extended nonlinear Schrödinger equation with higher-order odd and even terms with independent variable coefficients. We demonstrate its integrability, provide its Lax pair, and, applying the Darboux transformation, present its first and second order soliton solutions. The equation and its solutions have two free parameters. Setting one of these parameters to zero admits two limiting cases: the Hirota equation on the one hand and the Lakshmanan–Porsezian–Daniel (LPD) equation on the other hand. When both parameters are zero, the limit is the nonlinear Schrödinger equation.     

An alternative derivation of the pulse solutions of the KdV hierarchy

An alternative approach issues from the Appelle transformation of the Schrödinger equation. One solves the inverse problem for the transformed equation, a general solution of which is a quadratic form of two independent solutions of the primary Schrödinger equation. If the potential in the Schrödinger equation obeys one equation of the KdV hierarchy, the time derivative of this form is a linear combination of the form and its space derivative. The coefficients in the combination depend on the potential and the energy parameter of the Schrödinger equation only. This relation also determines the time dependence of the spectral data which along with the solution of the inverse problem gives the solution of the KdV equations as usual.     

非线性与立方非线性Schrodinger方程的多级包络周期解

本文基于Jacobi椭圆函数和Lamé方程,应用摄动法研究了非线性与立方非线性Schrodinger方程,获得了其新的多级包络周期解。这些解在极限条件下可以退化为各种形式的包络孤波解。     

非线性与立方非线性Schrodinger方程的多级包络周期解

本文基于Jacobi椭圆函数和Lamé方程,应用摄动法研究了非线性与立方非线性Schrodinger方程,获得了其新的多级包络周期解。这些解在极限条件下可以退化为各种形式的包络孤波解。     

The one-dimensional Schrödinger equation and a periodic potential

After recalling basic facts from the Titchmarsh-Weyl theory we derive and investigate the linear matrix equation, which holds for functions related to the spectral matrix of the one-dimensional periodic Schrödinger equation. The Weyl's solutions of the Schrödinger equation are used, when we solve this equation and associated nonlinear equations of the Milne's type. Two distinct trace formulae reconstructing the potential follow simply from the transformed and modified Milne's equations. Necessary spectral data of the inverse problem are determined by an infinite system of nonlinear first-order ordinary differential equations. Nonuniqueness of the solution of the inverse problem is confirmed on the other hand by writing a broad variety of the isospectral Darboux transformations.     

A solution spectrum of the nonlinear schrödinger equation

The soliton solutions of the form=A/coshkx and=B tanhkx of the nonlinear Schrödinger equation have been considered with respect to many problems. In this paper, it is shown that the nonlinear Schrödinger equation also possesses a solution manifold that generalizes the above soliton functions and provides a discrete spectrum of eigenfunctions and eigenvalues. With the help of a slight modification of these eigenfunctions, it is possible to extend them to the relativistic case, where they become solutions of a nonlinear Klein-Gordon equation associated with a discrete mass spectrum.     

Bessel solitary waves in strongly nonlocal nonlinear media with distributed parameters

Bessel solitary wave solutions to a two-dimensional strongly nonlocal nonlinear Schrödinger equation with distributed coefficients are obtained. Bessel solitary wave solutions have unique characteristics compared with Gaussian solitary wave solutions, Laguerre-Gaussian solitary wave solutions, and Hermite-Gaussian solitary wave solutions. The generalized two-dimensional nonlocal nonlinear Schrödinger equation with distributed coefficients is investigated for the first time to our knowledge.     

具有色散系数的(2+1)维非线性Schrödinger方程的有理解和空间孤子

非线性Schrödinger方程是物理学中具有广泛应用的非线性模型之一. 本文采用相似变换, 将具有色散系数的(2+1)维非线性Schrödinger方程简化成熟知的Schrödinger方程, 进而得到原方程的有理解和一些空间孤子.     

A general method for the solution of nonlinear soliton and kink Schrödinger equations

We present a method by which one-dimensional nonlinear soliton and kink Schrödinger equations can be solved in closed form. The hermitean nonlinear soliton operator may contain up to second derivatives of the wave function and the vanishing condition must hold. The method is applied to solve known nonlinear Schrödinger equations for one-soliton and one-kink solutions and, by inverting the procedure, to derive new operators with wave packet solutions of algebraic and arbitrary shapes. One of them is equivalent to the Derivative Nonlinear Schrödinger equation.     

Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method

This paper presents the coupled version of a previous work on nonlinear Schrödinger equation [23]. It focuses on the construction of approximate solutions of nonlinear Schrödinger equations. In this paper, we applied the differential transformation method (DTM) to solving coupled Schrödinger equations. The obtained results show that the technique suggested here is accurate and easy to apply.     

变系数非线性Schrödinger方程的孤子解及其相互作用

在光孤子通信和Bose-Einstein凝聚体动力学研究中,求解广义非线性Schrödinger方程是一个重要的研究方向.稳定的孤子模式具有潜在的应用,可为实验技术的实现提供依据.本文引进一种相似变换,将变系数非线性Schrödinger方程转化成非线性Schrödinger方程,并利用已知解深入研究变系数非线性Schrödinger方程解的单孤子解、两孤子解和连续波背景下的孤子解.同时通过选择不同的具体参数,给出它们的图像分析和相应的讨论. 关键词： 非线性Schrö dinger方程 相似变换 变系数 孤子解     

Using the complex WKB method for studying the evolution of initial pulses obeying the nonlinear Schrödinger equation

Using the complex WKB method, we found an asymptotic solution of the associated Zakharov-Shabat problem in the limit of a small coefficient h 0 multiplying the derivative of the potential that has a single hump. The obtained formulas can be used in describing the evolution of optical pulses of such shape obeying the nonlinear Schrödinger equation (NSE). Several examples are considered.V. V. Kuibyshev State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 19–25, May, 1993.