We show that the Ablowitz-Ladik equation, which is an integrable form of the discretized nonlinear Schrödinger equation, has rogue wave solutions in the form of the rational solutions. We show that there is a hierarchy of rational solutions and we derive the two lowest-order ones using the Hirota technique. More generally, we present rational solutions for the discrete Hirota equation which inc...

This research explores a (2 + 1)-D generalized Camassa–Holm–Kadomtsev–Petviashvili model. We use probable transformation to build bilinear formulation the model by Hirota technique. derive single lump waves, multi-soliton solutions from this form. present various dynamical properties of such as one, two, three and four solitons. The double periodic breather line rogue wave, interaction between ...

This talk deals with a remarkable integrable discretization of the SO(3) Euler top introduced in [1] by R. Hirota and K. Kimura. A class of implicit discretizations of the Euler top sharing the integrals of motion with the continuous system has been presented and studied in [2]. The Hirota-Kimura discretization of the Euler top leads to an explicit map. Its integrability is proven by finding tw...

R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. According to a proposal by T. Ratiu, discretizations of the Hirota-Kimura...

We introduce a class of reductions the two-component KP hierarchy, which includes Hirota-Ohta system hierarchy. The description reduced hierarchies is based on Hirota bilinear identity and an extra relation characterising reduction. derive reduction conditions in terms Lax operator higher linear operators as well basic equations.

We construct certain higher order smooth positon and breather solutions of an extended nonlinear Schrödinger equation with the cubic quartic nonlinearity. utilize generalized Darboux transformation method to aforementioned solutions. The three well-known equations, namely equation, Hirota are sub-cases considered equation. which we more general. analyze how constituent equations get modified by...

Under the well-known bilinear method of Hirota, specific expression for N-soliton solutions (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada(gCDGKS) equation in fluid mechanics is given. By defining a novel restrictive condition on solutions, resonant Y-type and X-type soliton are generated. new proposed constraint, combined with velocity resonance module method, mixed solitons l...

In this paper, the two-dimensional generalized nonlinear Schrödinger equations are introduced with Lax pair. The existence of pair defines integrability for partial differential equation, so integrable. Related to development was understanding that certain coherent structures called solitons play a basic role in phenomena as fluid mechanics, optics relativity, and lattice dynamics. Via Hirota b...