On solitary wave solutions of the compound Burgers–Korteweg–de Vries equation

The goal of this note is to construct a class of traveling solitary wave solutions for the compound Burgers–Korteweg–de Vries equation by means of a hyperbolic ansatz. A computational error in a previous work has been clarified.     

On solutions of the Schrödinger equation for some molecular potentials: wave function ansatz

Making an ansatz to the wave function, the exact solutions of the D-dimensional radial Schrödinger equation with some molecular potentials, such as pseudoharmonic and modified Kratzer, are obtained. Restrictions on the parameters of the given potential, δ and ν are also given, where η depends on a linear combination of the angular momentum quantum number ? and the spatial dimensions D and δ is a parameter in the ansatz to the wave function. On inserting D = 3, we find that the bound state eigensolutions recover their standard analytical forms in literature.     

Scaled Schrödinger equation and the exact wave function

We propose the scaled Schr?dinger equation and the related principles, and construct a general method of calculating the exact wave functions of atoms and molecules in analytical forms. The nuclear and electron singularity problems no longer occur. Test applications to hydrogen atom, helium atom, and hydrogen molecule are satisfactory, implying a high potentiality of the proposed method.     

On the Schrödinger equation and the eigenvalue problem

If _k is thek ^th eigenvalue for the Dirichlet boundary problem on a bounded domain in ⁿ , H. Weyl's asymptotic formula asserts that , hence . We prove that for any domain and for all . A simple proof for the upper bound of the number of eigenvalues less than or equal to - for the operator –V(x) defined on ⁿ (n3) in terms of is also provided.Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1     

Fisher information and the complex nature of the Schrödinger wave equation

We show that the minimum Fisher information (MFI) approach to estimating the probability law p(x) on particle position x, over the class of all two-component laws p(x), yields the complex Schrödinger wave equation. Complexity, in particular, traces from an efficiency scenario (demanded by MFI) where the two components of p(x) are so separated that their informations add.     

On the Schrödinger equation for the supersymmetric FRW model

We consider a time-dependent Schrödinger equation for the Friedmann–Robertson–Walker (FRW) model. We show that for this purpose it is possible to include an additional action invariant under reparametrization of time. The last one does not change the equations of motion for the minisuperspace model, but changes only the constraint. The same procedure is applied to the supersymmetric case.     

On the power series solution of the Schrödinger equation

A numerical method for solving the Schrödinger equation for a one-dimensional potential expressed as a function which increases in both directions away from its minimum is proposed. The basic assumption relies on the asymptotic properties of the solution. We exemplify the method calculating energies and expectation values for the quartic anharmonic oscillator.     

The inhomogeneous Schrödinger equation for the off-shell wave function

A method has been developed for calculating the off-shell wave function by solving the inhomogeneous Schrödinger equation with allowance for the nuclear and Coulomb interactions. The off-shell wave function makes it possible to construct the off-shell scattering amplitude in order to solve the problems for three or more particles. An important application of the method is the Trojan Horse calculations of nuclear reactions that are important in nuclear astrophysics. Specific calculations are performed for neutron and proton scattering on the ⁷Be nucleus. The Woods-Saxon potential is used and the spin-orbital interaction is taken into account.     

On the nonlinear Schrödinger limit of the Korteweg-de Vries equation

Using the multiscale approach of Zakharov and Kuznetsov it is shown that the nonlinear Schrödinger periodic scattering data is related to the Korteweg-de Vries periodic scattering data via an average over the Korteweg-de Vries carrier oscillation. This allows a complete elucidation of the physical meaning of the nonlinear Schrödinger scattering data, conservation laws, theta function solutions and reality constraint.     

Rogue wave solutions of the vector Lakshmanan–Porsezian–Daniel equation

We use a nonrecursive Darboux transformation method to obtain a special hierarchy of rogue wave solutions of the vector Lakshmanan–Porsezian–Daniel equation, which can govern the propagation of ultrashort optical pulses in a long-haul telecommunication fiber. In terms of the exact rational solutions, we demonstrate several interesting rogue wave dynamics such as rogue wave doublets, quartets and sextets. The modulation instability responsible for the excitation of rogue waves from an unstable continuous background in such a complex nonlinear system is also discussed.     

On an isomonodromy deformation equation without the Painlevé property

We show that the fourth-order nonlinear ODE which controls the pole dynamics in the general solution of equation P _I ² compatible with the KdV equation exhibits two remarkable properties: (1) it governs the isomonodromy deformations of a 2 × 2 matrix linear ODE with polynomial coefficients, and (2) it does not possess the Painlevé property. We also study the properties of the Riemann-Hilbert problem associated to this ODE and find its large-t asymptotic solution for physically interesting initial data.     

Soliton and rogue wave solutions of two-component nonlinear Schr?dinger equation coupled to the Boussinesq equation

The nonlinear Schr?dinger(NLS) equation and Boussinesq equation are two very important integrable equations.They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright–bright, bright–dark, and dark–dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright–bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright–bright or bright–dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems.     

Separation of variables in the wave equation. Sets of the type (1.1) and Schrödinger equation

All complete sets of symmetry operators of the wave equation in Minkowski space containing one isotropic first-order operator of the form /x⁰+/x³ and two second-order operators are obtained. Privileged coordinates corresponding to these sets are given. The variables are separated in the wave equation and in the free Schrödinger equation in three-dimensional spacetime.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 44–48, February, 1991.     

Can the Bogoliubov–de Gennes equation be interpreted as a ‘one-particle’ wave equation?

The solutions of the Bogoliubov–de Gennes (BdG) equation are usually interpreted as the excitations from the superconducting ground state. This viewpoint is not easily applied to a strongly coupled heterojunction since the ground state changes across the interface and it is not clear how the ground state should be connected across the heterointerface. In this paper, we present a different viewpoint that does not suffer from this conceptual drawback. We show that the BdG equation can be viewed as a ‘one-particle’ wave equation whose eigenstates (including the negative energy states) can be filled up systematically to describe the superconducting state, in much the same way that we fill the eigenstates of the Schrödinger equation to describe normal conductors. The only difference is that we need to start from a special vacuum | V〉, consisting of a full band of down-spin electrons, instead of the usual vacuum devoid of all particles. Any quantity of interest, A (such as the charge density or the current density), can be interpreted as the sum of a ‘vacuum contribution’A_{V AC}due to the vacuum | V〉 and a one-particle contribution A_BdGdue to the filled eigenstates of the BdG equation. This picture is easily applied even to strongly coupled heterojunctions since the vacuum | V〉 is the same on both sides of a heterointerface. As such, we believe it puts the scattering theory of transport for superconductors on a firmer conceptual basis.     

On wave and rheidity properties of the Earth’s crust

The properties of the Earth’s solid crust have been studied on the assumption that this crust has a block structure. According to the rotation model, the motion of such a medium (geomedium) follows the angular momentum conservation law and can be described in the scope of the classical elasticity theory with a symmetric stress tensor. A geomedium motion is characterized by two types of rotation waves with shortand long-range actions. The first type includes slow solitons with velocities of 0 ≤ V_sol ≤ c0, max = 1–10 cm s^–1; the second type, fast excitons with V₀ ≤ V_ex ≤ V_S–V_P. The exciton minimal velocity (V₀ = 0) depends on the energy of the collective excitation of all seismically active belt blocks proportional to the Earth’s pole vibration frequency (the Chandler vibration frequency). The exciton maximal velocity depends on the velocities of S (V_S ≈ 4 km s^–1) and/or P (V_P ≈ 8 km s^–1) seismic (acoustic) waves. According to the rotation model, a geomedium is characterized by the property physically close to the corpuscular–wave interaction between blocks that compose this medium. The possible collective wave motion of geomedium blocks can be responsible for the geomedium rheidity property, i.e., a superplastic volume flow. A superplastic motion of a quantum fluid can be the physical analog of the geomedium rheid motion.     

The classification of the single travelling wave solutions to the generalized Pochhammer–Chree equation

By the complete discrimination system for the polynomial, we invest the classifications of single travelling wave solutions to the generalized Pochhammer–Chree (PC) equation with p?=?1/2 and p?=?3/2.     

On the gauge equivalent structure of the modified nonlinear Schrödinger equation

By introducing the concept of differential equations with “given curvature condition”, we show that the modified nonlinear Schrödinger equations for κ=1 and −1 are, respectively, gauge equivalent to the modified HF model and the modified M-HF model. As a consequence, soliton solutions to the modified HF model are constructed.     

Classification of single travelling wave solutions to the generalized Zakharov–Kuznetsov equation

By the complete discrimination system for the polynomial method, the classification of single travelling wave solutions to the generalized Zakharov–Kuznetsov equation with p?=?2 was obtained.     

The periodic wave solutions for the generalized Nizhnik－Novikov－Veselov equation

The periodic wave solutions expressed by Jacobi elliptic functions for the generalized Nizhnik－Novikov－Veselov equation are obtained using the F-expansion method proposed recently. In the limiting cases, the solitary wave solutions and other types of travelling wave solutions for the system are obtained.