Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNonlinear eigenvalue problems for generalized Painleve equations
105
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Eigenvalue problems for linear differential equations, such as time-independent Schrodinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that give rise to the separatrix play the role of eigenvalues. Previously studied examples of nonlinear differential equations that possess discrete eigenvalue spectra are the first-order equation $y(x)=cos[pi xy(x)]$ and the first, second, and fourth Painleve transcendents. It is shown here that the differential equations for the first and second Painleve transcendents can be generalized to large classes of nonlinear differential equations, all of which have discrete eigenvalue spectra. The large-eigenvalue behavior is studied in detail, both analytically and numerically, and remarkable new features, such as hyperfine splitting of eigenvalues, are described quantitatively.
rate research
Read More
We present a variational approach which shows that the wave functions belonging to quantum systems in different potential landscapes, are pairwise linked to each other through a generalized continuity equation. This equation contains a source term proportional to the potential difference. In case the potential landscapes are related by a linear symmetry transformation in a finite domain of the embedding space, the derived continuity equation leads to generalized currents which are divergence free within this spatial domain. In a single spatial dimension these generalized currents are invariant. In contrast to the standard continuity equation, originating from the abelian $U(1)$-phase symmetry of the standard Lagrangian, the generalized continuity equations derived here, are based on a non-abelian $SU(2)$-transformation of a Super-Lagrangian. Our approach not only provides a rigorous theoretical framework to study quantum mechanical systems in potential landscapes possessing local symmetries, but it also reveals a general duality between quantum states corresponding to different Schr{o}dinger problems.
An overview is given on recent developments in the affine Weyl group approach to Painleve equations and discrete Painleve equations, based on the joint work with Y. Yamada and K. Kajiwara.
This paper is a natural continuation of the previous paper cite{TyuVo13} where generalized oscillator representations for Calogero Hamiltonians with potential $V(x)=alpha/x^2$, $alphageq-1/4$, were constructed. In this paper, we present generalized oscillator representations for all generalized Calogero Hamiltonians with potential $V(x)=g_{1}/x^2+g_{2}x^2$, $g_{1}geq-1/4$, $g_{2}>0$. These representations are generally highly nonunique, but there exists an optimum representation for each Hamiltonian, representation that explicitly determines the ground state and the ground-state energy. For generalized Calogero Hamiltonians with coupling constants $g_1<-1/4$ or $g_2<0$, generalized oscillator representations do not exist in agreement with the fact that the respective Hamiltonians are not bounded from below.
The well-known Greens function method has been recently generalized to nonlinear second order differential equations. In this paper we study possibilities of exact Greens function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Greens function can be represented in terms of the homogeneous solution. Specific examples and a numerical error analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg-de Vries equations, we are forced to introduce higher order Green functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.
In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
