Darboux transformation and nonclassical orthogonal polynomials

A Darboux transformation of orderN is introduced for the Schrödinger equation. The relation between this transformation and the factorization method is treated in detail forN=2. It is noted that the potential of the new Schrödinger depends on 2N parameters. A new exactly solvable potential is obtained from the harmonic oscillator potential. The polynomials appearing in the new Schrödinger equation are investigated in detail.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 58–65, April, 1995.     

Equivalence of integral and differential methods of generating accurately solvable potentials of the steady Schrödinger equation

It is proven that the well-known nonlocal (i.e., based on integral transformations) methods of generating accurately solvable potentials of the one-dimensional steady Schrödinger equation are equivalent to multiple use of the local (i.e., based on a differential transformation) method known as the Darboux transformation. New accurately solvable potentials with a hydrogen-like spectrum are obtained, and several functions of the lowest states of the discrete spectrum are presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 95–102, March, 1995.     

Modified Darboux transformations with foreign auxiliary equations

We construct a new type of first-order Darboux transformations for the stationary Schrödinger equation. In contrast to the conventional case, our Darboux transformations support arbitrary (foreign) auxiliary equations. We show that among other applications, our formalism can be used to systematically construct Darboux transformations for Schrödinger equations with energy-dependent potentials, including a recent result (Lin et al., 2007) [16] as a special case.     

Nonrelativistic Green's Function for Systems with Position-Dependent Mass

Given a spatially dependent mass, we obtain the 2-point Green's function for exactly solvable nonrelativistic problems. This is accomplished by mapping the wave equation for these systems into well-known exactly solvable Schrödinger equations with constant mass using point canonical transformation. The one-dimensional oscillator class is considered and examples are given for several mass distributions.     

The Darboux Transformation for the One-Dimensional Stationary Dirac Equation with a Scalar Potential

This paper deals with the Darboux transformation for the Dirac equation with a scalar-type potential. Formulas are derived for the potential difference and for the solutions of the transformed equations. The relationship between the Darboux transforms for Dirac and Schrödinger equations is analyzed. New transparent potentials and a potential with a Coulomb asymptotics are obtained as examples.     

Darboux transformation for the nonsteady Schrödinger equation

The Darboux transformation for a nonsteady one-dimensional Schrödinger equation is introduced; its operator is an N-th order differential operator that converts the solution of an equation with a specified potential to a solution with a new potential constructed from the solutions of the initial equation. A relation is established between this transformation and supersymmetric quantum mechanics. Operators of time-conserved supercharge are introduced; for steady states, they reduce to the well-known operators. Examples of accurately solvable nonsteady potentials of elementary form are given.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 59–65, July, 1995.     

Group-theoretical interpretation of the Korteweg-de Vries type equations

The Korteweg-de Vries equation is studied within the group-theoretical framework. Analogous equations are obtained for which the many-dimensional Schrödinger equation (with nonlocal potential) plays the same role as the one-dimensional Schrödinger equation does in the theory of the Korteweg-de Vries equation.     

Prepotential approach to exact and quasi-exact solvabilities

Exact and quasi-exact solvabilities of the one-dimensional Schrödinger equation are discussed from a unified viewpoint based on the prepotential together with Bethe ansatz equations. This is a constructive approach which gives the potential as well as the eigenfunctions and eigenvalues simultaneously. The novel feature of the present work is the realization that both exact and quasi-exact solvabilities can be solely classified by two integers, the degrees of two polynomials which determine the change of variable and the zeroth order prepotential. Most of the well-known exactly and quasi-exactly solvable models, and many new quasi-exactly solvable ones, can be generated by appropriately choosing the two polynomials. This approach can be easily extended to the constructions of exactly and quasi-exactly solvable Dirac, Pauli, and Fokker-Planck equations.     

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.     

On Quasi-Exact Solvability of the Schrödinger Equation for a Free Particle on the Surface of a Spindle Torus

We show that the Schrödinger equation for a free particle on the surface of a spindle torus is quasi-exactly solvable. Our result complements former ones in an interesting way: it is known that the Schrödinger equation for a free particle on a ring torus is non-solvable, whereas it is exactly solvable for a particle on a horn torus.     

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.     

The Darboux Transformation for the One-Dimensional Stationary Dirac Equation with a Pseudoscalar Potential

This paper consider the Darboux transform for the Dirac equation with a pseudoscalar-type potential. Formulas for the potential difference and for the solutions of the transformed equation are derived. The relationship between the Darboux transforms for Dirac and Schrödinger equations is analyzed. New potentials with the spectrum of a relativistic harmonic oscillator are obtained as examples.     

Periodic Wave Solutions of Generalized Derivative Nonlinear Schrödinger Equation

A Darboux transformation of the generalized derivative nonlinear Schrodinger equation is derived. As an application, some new periodic wave solutions of the generalized derivative nonlinear Schrodinger equation are explicitly given.     

Exact X-wave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential

We find exact solutions of the two- and three-dimensional nonlinear Schrödinger equation with a supporting potential. We focus in the case where the diffraction operator is of the hyperbolic type and both the potential and the solution have the form of an X-wave. Following similar arguments, several additional families of exact solutions can also can be found irrespectively of the type of the diffraction operator (hyperbolic or elliptic) or the dimensionality of the problem. In particular we present two such examples: The one-dimensional nonlinear Schrödinger equation with a stationary and a “breathing” potential and the two-dimensional nonlinear Schrödinger with a Bessel potential.     

Darboux transformation and exactly solvable potentials with quasi-equidistant spectrum

The Darboux transformation of order n introduced in our previous paper is applied to the harmonic-oscillator potential of the one-dimensional Schrödinger equation. New potentials that have a quasi-equidistant spectrum (i.e., an equidistant spectrum with lacunas) and admit of solution of the Schrödinger equation in elementary functions are obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 3–10, August, 1995.     

Methods of generating integrable potentials for the Sochrödinger equation and nonlocal symmetries

Methods of generating exactly integrable potentials for the Schrödinger equation are consolidated within the framework of a simple construction. The Abraham-Moses method is generalized to the case of the nonstationary Schrödinger equation. An algorithm is proposed for solving the Schrödinger equation based on nonlocal symmetry operators.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 19–25, September, 1991.     

Decay estimates for Schrödinger equations

We prove global existence and optimal decay estimates for classical solutions with small initial data for nonlinear nonlocal Schrödinger equations. The Laplacian in the Schrödinger equation can be replaced by an operator corresponding to a non-degenerate quadratic form of arbitrary signature. In particular, the Davey-Stewartson system is included in the the class of equations we discuss.Partially supported by NSF grant DMS-860-2031. Sloan Research Fellow     

Rogue wave solutions for the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

A generalized Darboux transformation for the coupled cubic–quintic nonlinear Schrödinger equation is constructed by the Darboux matrix method. As applications, the Nth-order rogue wave solutions of the coupled cubic–quintic nonlinear Schrödinger equation have been obtained. In particular, the dynamics of the general first- and second-order rogue waves are discussed and illustrated through some figures.     

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.     

Separation of variables in the stationary Schrödinger equation

The problem of separation of variables in the stationary Schrödinger equation is considered for a charge moving in an external electromagnetic field. On the basis of the definition formulated, necessary and sufficient conditions are found for separation of variables in equations of elliptic type to which the stationary Schrödinger equation belongs. Application of general theorems made it possible to enumerate all types of electromagnetic fields and systems of coordinates in which separation of variables in the stationary Schrödinger equation is possible. Systems of ordinary differential equations which the wave function in the separated variables satisfies are written down to explicit form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 45–50, August, 1972.