Painleve Integrability,Consistent Riccati Expansion Solvability and Interaction Solution for the Coupled mKdV-BLMP System

The integrability of the coupled, modified KdV equation and the potential Boiti-Leon-Manna-Pempinelli(mKdVBLMP) system is investigated using the Painleve analysis approach. It is shown that this coupled system possesses the Painleve property in both the principal and secondary branches. Then, the consistent Riccati expansion(CRE)method is applied to the coupled mKdV-BLMP system. As a result, it is CRE solvable for the principal branch while non-CRE solvable for the secondary branch. Finally; starting from the last consistent differential equation in the CRE solvable case, soliton, multiple resonant soliton solutions and soliton-cnoidal wave interaction solutions are constructed explicitly.     

Exact Solutions to Pauli-Villars-Regulated Field Theories

We present a new class of quantum field theories which are exactly solvable. The theories are generated by introducing Pauli-Villars (PV) fermionic and bosonic fields with masses degenerate with the physical positive metric fields. An algorithm is given to compute the spectrum and corresponding eigensolutions. We also give the operator solution for a particular case and use it to illustrate some of the tenets of light-cone quantization. Since the solutions of the solvable theory contain ghost quanta, these theories are unphysical. However, the existence of an exact solution provides an important check on the implementation of PV-regulated discretized light-cone quantization (DLCQ). In the limit of exact mass degeneracy of the ghost and physical fields, the numerical DLCQ solutions are constrained to reduce to the explicit forms we give here. We also discuss how perturbation theory in the difference between the masses of the physical and PV particles could be developed, thus generating physical theories. The existence of explicit solutions of the solvable theory also allows one to study the relationship between the equal-time and light-cone vacua and eigensolutions.     

Consistent Riccati expansion solvable classification and soliton-cnoidal wave interaction solutions for an extended Korteweg-de Vries equation

In this paper, we consider an extended Korteweg-de Vries (KdV) equation. Using the consistent Riccati expansion (CRE) method of Lou, the extended KdV equation is proved to be CRE solvable in only two distinct cases. These two CRE solvable models are the KdV-Lax and KdV-Sawada-Kotera (KdV-SK) equations. In addition, applying the nonauto-Bäcklund transformations which are provided by the CRE method, we present the explicit construction for soliton-cnoidal wave interaction solutions which represent a soliton propagating on a cnoidal periodic wave background in the KdV-Lax and KdV-SK equations, respectively.     

An Alternative Method for Calculating Bound-State of Energy Eigenvalues of Klein-Gordon for Quasi-exactly Solvable Potentials

We obtain the bound-state energy of the Klein-Gordon equation for some examples of quasi-exactly solvable potentials within the framework of asymptotic iteration method （AIM）. The eigenvalues are calculated for type- 1 solutions. The whole quasi-exactly solvable potentials are generated from the defined relation between the vector and scalar potentials.     

Solutions of nonrelativistic Schrödinger equation from relativistic Klein-Gordon equation

The relativistic one-dimensional Klein-Gordon equation can be exactly solved for a certain class of potentials. But the nonrelativistic Schrödinger equation is not necessarily solvable for the same potentials. It may be possible to obtain approximate solutions for the inexact nonrelativistic potential from the relativistic exact solutions by systematically removing relativistic portion. We search for the possibility with the harmonic oscillator potential and the Coulomb potential, both of which can be exactly solvable nonrelativistically and relativistically. Though a rigorous algebraic approach is not deduced yet, it is found that the relativistic exact solutions can be a good starting point for obtaining the nonrelativistic solutions.     

New exact solutions of Bratu Gelfand model in two dimensions using Lie symmetry analysis

We explore new analytical solutions for the two-dimensional nonlinear elliptic Bratu equation. Through the point transformation, the integrable form of Bratu equation was investigated then we obtain the Lie infinitesimals for the new equation. These vectors reduce the integrable equation to solvable ODEs then we use the boundary conditions (BCs) to spin two new exact solutions for Bratu equation in a unit square domain. A three-dimensional plot illustrates some resulting solutions. Comparison with other work has been presented.     

Quantum stationary state of class A Bianchi universe

Padmanabhan derived a differential equation for the stationary state for the class A Bianchi model and obtained some approximate solutions. Here we reduce the differential equation to a standard, well-known, solvable linear differential equation and indicate some exact explicit particular solutions.     

Exact Analytic Study of the PT-Symmetry-Breaking Mechanism

We employ the exactly solvable Scarf II potential to illustrate how the spontaneous breakdown of PT symmetry is realized, and how it influences the discrete energy spectrum, the solutions and their discussion in the supersymmetric and algebraic framework. Differences with respect to other types of solvable potentials are also pointed out.     

Similarity solutions for a class of Fractional Reaction-Diffusion equation

This work studies exact solvability of a class of fractional reaction-diffusion equation with the Riemann-Liouville fractional derivatives on the half-line in terms of the similarity solutions. We derived the conditions for the equation to possess scaling symmetry even with the fractional derivatives. Relations among the scaling exponents are determined, and the appropriate similarity variable introduced. With the similarity variable we reduced the differential partial differential equation to a fractional ordinary differential equation. Exactly solvable systems are then identified by matching the resulted ordinary differential equation with the known exactly solvable fractional ones. Several examples involving the three-parameter Mittag-Leffler function (Kilbas-Saigo function) are presented. The models discussed here turn out to correspond to superdiffusive systems.     

An approach to one-dimensional elliptic quasi-exactly solvable models

One-dimensional Jacobian elliptic quasi-exactly solvable second-order differential equations are obtained by introducing the generalized third master functions. It is shown that the solutions of these differential equations are generating functions for a new set of polynomials in terms of energy with factorization property. The roots of these polynomials are the same as the eigenvalues of the differential equations. Some one-dimensional elliptic quasi-exactly quantum solvable models are obtained from these differential equations.      

Supersymmetry in quantum mechanics

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable. In this lecture I review the theoretical formulation of supersymmetric quantum mechanics and discuss many of its applications. I show that the well-known exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials and shape invariance. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Further, it is pointed out that the connection between the solutions of the Dirac equation and the Schrödinger equation is exactly same as between the solutions of the MKdV and the KdV equations.     

Interaction Solutions for (1+1)-Dimensional Higher-Order Broer-Kaup System

The (1+1)-dimensional higher-order Broer-Kaup (HBK) system is studied by consistent tanh expansion (CTE) method in this paper. It is proved that the HBK system is CTE solvable, and some exact interaction solutions among different nonlinear excitations such as solitons, rational waves, periodic waves, corresponding images are explicitly given.     

A Search on Dirac Equation

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.     

Universal solutions to the simplex equations

We construct a class of solutions to the p-simplex equation in terms of solutions to the Yang-Baxter equation, for every p ≥ slant 3. This may make the construction of solvable p-dimensional lattice models possible.     

Quasi-periodic and Non-periodic Waves in (2+1)-Dimensional Nonlinear Systems

New exact quasi-periodic and non-periodic solutions for the (2 1)-dimensional nonlinear systems are studied by means of the multi-linear variable separation approach (MLVSA) and the Jacobi elliptic functions with the space-time-dependent modulus. Though the result is valid for all the MLVSA solvable models, it is explicitly shown for the long-wave and short-wave interaction model.     

An affinity theorem for integral imaging equations

It is shown that any integral imaging equation whose solutions are known can be used to generate a class of solvable integral equations via an affinity transformation.     

Elementary bäcklund transformations,nonlinear superposition formulae and algebraic construction of solutions for the nonlinear evolution equations solvable by the Zakharov-Shabat spectral transform

Novel nonlinear superposition formulae are given for solutions of the class of nonlinear evolution equations solvable via the spectral transform associated with the Zakharov-Shabat spectral problem, and an algebraic procedure is provided to construct solutions of these equations.     

Chirped and chirp-free self-similar cnoidal and solitary wave solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients

By means of the similarity transformation connecting with the solvable stationary cubic-quintic nonlinear Schrödinger equation (CQNLSE), we construct explicit chirped and chirp-free self-similar cnoidal wave and solitary wave solutions of the generalized CQNLSE with spatially inhomogeneous group velocity dispersion (GVD) and amplification or attenuation. As an example, we investigate their propagation dynamics in a soliton control system.     

A new solvable many-body problem of goldfish type

A new solvable many-body problem of gold?sh type is introduced and the behavior of its solutions is tersely discussed.