In this paper we show new exact solutions for a type of generalized sine-Gordon equation which is obtained by constructing a Lagrange function for a dynamical coupled system of oscillators. We convert it into a nonlinear system by perturbing the potential energy from a point of view of an approach proposed by Fermi [1].

In this paper, based on the generalized Jacobi elliptic function expansion method,we obtain abundant new complex doubly periodic solutions of the double Sine-Gordon equation (DSGE), which are degenerated to solitary wave solutions and triangle function solutions in the limit cases,showing that this new method is more powerful to seek exact solutions of nonlinear partial differential equations i...

A single, tractable, special case of the problem of continuous mixtures of beta distributions over their parameters is considered. This is the uniform mixture of generalized arc-sine distributions which, curiously, turns out to be linked by transformation to the Cauchy distribution. AMS 2000 subject classification. 62E15.

We introduce certain integrals of a product of the Bernoulli polynomials and logarithms of Milnor’s multiple sine functions. It is shown that all the integrals are expressed by the Mordell-Tornheim zeta values at positive integers and that the converse is also true. Moreover, we apply the theory of the integral to obtain various new results for the Mordell-Tornheim zeta values.

We study some systems of non-linear PDE's (Eqs. 1.1 below) which can be regarded either as generalizations of the sine-Gordon equation or as two-dimensional versions of the Toda lattice equations. We show that these systems have an infinite number of non-trivial conservation laws and an infinite number of symmetries. The second result is deduced from the first by a variant of the Hamiltonian fo...

The discrete cosine and sine transforms are generalized to a triangular fragment of the honeycomb lattice. The two-variable orbit functions of the Weyl group A2, discretized simultaneously on the weight and root lattices, induce the family of the extended Weyl orbit functions. The periodicity and von Neumann and Dirichlet boundary properties of the extended Weyl orbit functions are detailed. Th...

This study introduces a subclass Sqs* of starlike functions associated with the q-analogue sine function defined in symmetric unit disk. article comprises investigation sharp coefficient bounds, and upper bound third-order Hankel determinant for this class. It also includes findings Zalcman generalized conjectures