3,232 research outputs found

### Cyclic Identities Involving Jacobi Elliptic Functions

We state and discuss numerous mathematical identities involving Jacobi
elliptic functions sn(x,m), cn(x,m), dn(x,m), where m is the elliptic modulus
parameter. In all identities, the arguments of the Jacobi functions are
separated by either 2K(m)/p or 4K(m)/p, where p is an integer and K(m) is the
complete elliptic integral of the first kind. Each p-point identity of rank r
involves a cyclic homogeneous polynomial of degree r (in Jacobi elliptic
functions with p equally spaced arguments) related to other cyclic homogeneous
polynomials of degree r-2 or smaller. Identities corresponding to small values
of p,r are readily established algebraically using standard properties of
Jacobi elliptic functions, whereas identities with higher values of p,r are
easily verified numerically using advanced mathematical software packages.Comment: 14 pages, 0 figure

### Exact Solution of a N-body Problem in One Dimension

Complete energy spectrum is obtained for the quantum mechanical problem of N
one dimensional equal mass particles interacting via potential
$V(x_1,x_2,...,x_N) = g\sum^N_{i < j}{1\over (x_i-x_j)^2} - {\alpha\over
\sqrt{\sum_{i < j} (x_i-x_j)^2}}$ Further, it is shown that scattering
configuration, characterized by initial momenta $p_i (i=1,2,...,N)$ goes over
into a final configuration characterized uniquely by the final momenta $p'_i$
with $p'_i=p_{N+1-i}$.Comment: 8 pages, tex file, no figures, sign in the first term on the right
hand side of eq.3 correcte

### Cyclic Identities Involving Jacobi Elliptic Functions. II

Identities involving cyclic sums of terms composed from Jacobi elliptic
functions evaluated at $p$ equally shifted points on the real axis were
recently found. These identities played a crucial role in discovering linear
superposition solutions of a large number of important nonlinear equations. We
derive four master identities, from which the identities discussed earlier are
derivable as special cases. Master identities are also obtained which lead to
cyclic identities with alternating signs. We discuss an extension of our
results to pure imaginary and complex shifts as well as to the ratio of Jacobi
theta functions.Comment: 38 pages. Modified and includes more new identities. A shorter
version of this will appear in J. Math. Phys. (May 2003

### Some Exact Results for Mid-Band and Zero Band-Gap States of Associated Lame Potentials

Applying certain known theorems about one-dimensional periodic potentials, we
show that the energy spectrum of the associated Lam\'{e} potentials
$a(a+1)m~{\rm sn}^2(x,m)+b(b+1)m~{\rm cn}^2(x,m)/{\rm dn}^2(x,m)$ consists of
a finite number of bound bands followed by a continuum band when both $a$ and
$b$ take integer values. Further, if $a$ and $b$ are unequal integers, we show
that there must exist some zero band-gap states, i.e. doubly degenerate states
with the same number of nodes. More generally, in case $a$ and $b$ are not
integers, but either $a + b$ or $a - b$ is an integer ($a \ne b$), we again
show that several of the band-gaps vanish due to degeneracy of states with the
same number of nodes. Finally, when either $a$ or $b$ is an integer and the
other takes a half-integral value, we obtain exact analytic solutions for
several mid-band states.Comment: 18 pages, 2 figure

### Local Identities Involving Jacobi Elliptic Functions

We derive a number of local identities of arbitrary rank involving Jacobi
elliptic functions and use them to obtain several new results. First, we
present an alternative, simpler derivation of the cyclic identities discovered
by us recently, along with an extension to several new cyclic identities of
arbitrary rank. Second, we obtain a generalization to cyclic identities in
which successive terms have a multiplicative phase factor exp(2i\pi/s), where s
is any integer. Third, we systematize the local identities by deriving four
local ``master identities'' analogous to the master identities for the cyclic
sums discussed by us previously. Fourth, we point out that many of the local
identities can be thought of as exact discretizations of standard nonlinear
differential equations satisfied by the Jacobian elliptic functions. Finally,
we obtain explicit answers for a number of definite integrals and simpler forms
for several indefinite integrals involving Jacobi elliptic functions.Comment: 47 page

### Exact Solutions of the Two-Dimensional Discrete Nonlinear Schr\"odinger Equation with Saturable Nonlinearity

We show that the two-dimensional, nonlinear Schr\"odinger lattice with a
saturable nonlinearity admits periodic and pulse-like exact solutions. We
establish the general formalism for the stability considerations of these
solutions and give examples of stability diagrams. Finally, we show that the
effective Peierls-Nabarro barrier for the pulse-like soliton solution is zero

### Soliton Lattice and Single Soliton Solutions of the Associated Lam\'e and Lam\'e Potentials

We obtain the exact nontopological soliton lattice solutions of the
Associated Lam\'e equation in different parameter regimes and compute the
corresponding energy for each of these solutions. We show that in specific
limits these solutions give rise to nontopological (pulse-like) single
solitons, as well as to different types of topological (kink-like) single
soliton solutions of the Associated Lam\'e equation. Following Manton, we also
compute, as an illustration, the asymptotic interaction energy between these
soliton solutions in one particular case. Finally, in specific limits, we
deduce the soliton lattices, as well as the topological single soliton
solutions of the Lam\'e equation, and also the sine-Gordon soliton solution.Comment: 23 pages, 5 figures. Submitted to J. Math. Phy

### New Shape Invariant Potentials in Supersymmetric Quantum Mechanics

Quantum mechanical potentials satisfying the property of shape invariance are
well known to be algebraically solvable. Using a scaling ansatz for the change
of parameters, we obtain a large class of new shape invariant potentials which
are reflectionless and possess an infinite number of bound states. They can be
viewed as q-deformations of the single soliton solution corresponding to the
Rosen-Morse potential. Explicit expressions for energy eigenvalues,
eigenfunctions and transmission coefficients are given. Included in our
potentials as a special case is the self-similar potential recently discussed
by Shabat and Spiridonov.Comment: 8pages, Te

### Exact Moving and Stationary Solutions of a Generalized Discrete Nonlinear Schrodinger Equation

We obtain exact moving and stationary, spatially periodic and localized
solutions of a generalized discrete nonlinear Schr\"odinger equation. More
specifically, we find two different moving periodic wave solutions and a
localized moving pulse solution. We also address the problem of finding exact
stationary solutions and, for a particular case of the model when stationary
solutions can be expressed through the Jacobi elliptic functions, we present a
two-point map from which all possible stationary solutions can be found.
Numerically we demonstrate the generic stability of the stationary pulse
solutions and also the robustness of moving pulses in long-term dynamics.Comment: 22 pages, 7 figures, to appear in J. Phys.

### New classes of quasi-solvable potentials, their exactly-solvable limit and related orthogonal polynomials

We have generated, using an sl(2,R) formalism, several new classes of
quasi-solvable elliptic potentials, which in the appropriate limit go over to
the exactly solvable forms. We have obtained exact solutions of the
corresponding spectral problems for some real values of the potential
parameters. We have also given explicit expressions of the families of
associated orthogonal polynomials in the energy variable.Comment: 14 pages, 5 tables, LaTeX2

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