With the aid of symbolic computation, the sinh-Gordon equation expansion method is extended to seek Jacobi elliptic function solutions of (2+1)-dimensional long wave-short wave resonance interaction equation, which describe the long and short waves propagation at an angle to each other in a two-layer fluid. As a result, new Jacobi elliptic function solutions are obtained. When the modulus m of ...

We introduce 3N × 3N Lax pair with spectral parameter for Calogero-Inozemtsev model. The one degree of freedom case appears to have 2 × 2 Lax representation. We derive it from the elliptic Gaudin model via some reduction procedure and prove algebraic integrability. This Lax pair provides elliptic linear problem for the Painlevé VI equation in elliptic form.

In this paper a new Weierstrass semi-rational expansion method is developed via the Weierstrass elliptic function ℘(ξ; g2, g3). With the aid of Maple, we choose the coupled water wave equation and the generalized Hirota-Satsuma coupled KdV equation to illustrate the method. As a consequence, it is shown that the method is powerful to obtain many types of new doubly periodic solutions in terms o...

We start from an interpretation of the BC2-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding elliptic beta integral and transformation formula in a new way, ...

In this work we present a general regularity result for small perturbation solutions of elliptic equations. Our approach was motivated by the analysis of flat level sets in Ginzburg-Landau phase transitions models, which were considered in Savin (2003). When dealing with uniformly elliptic equations of the form (1), the classical approach to regularity is to differentiate the equation with resp...

In this review we explain interrelations between the Elliptic Calogero-Moser model, integrable Elliptic Euler-Arnold top, monodromy preserving equations and the KnizhnikZamolodchikov-Bernard equation on a torus. Dedicated to 70-th birthday of Francesco Calogero

In this note we establish multiple solutions for a semilinear elliptic equation with superlinear nonlinearility without assuming any symmetry.

We find a new class of algebraic geometric solutions of Heun’s equation with the accessory parameter belonging to a hyperelliptic curve. Dependence of these solutions from the accessory parameter as well as their relation to Heun’s polynomials is studied. Methods of calculating the algebraic genus of the curve, and its branching points, are suggested. Monodromy group is considered. Numerous exa...

The bidomain equations represent the most complete description of cardiac electrical activity. However, the equations prove computationally burdensome as the resulting system of equations has two entries per spatial node. This paper examines the computational performance obtained by decoupling the bidomain equations into two separate systems of equations, an elliptic equation for the extracellu...

In this article, we give the extended trial equation method for solving nonlinear partial differential equations with higher order nonlinearity. By use of this method, the exact travel-ing wave solutions including soliton solution, singular soliton solutions, rational function solution and elliptic integral function solution to one-dimensional general improved KdV (GIKdV) equation and Rðm; nÞ e...