Integrable variant Blaszak-Szum lattice equation

TL;DR

This paper introduces a new integrable variant of the Blaszak-Szum lattice equation, deriving explicit soliton, lump, breather, and periodic solutions using Hirota's method, Bäcklund transformations, and numerical techniques.

Contribution

It presents a novel integrable variant of the Blaszak-Szum lattice equation with explicit solutions and transformations, expanding the understanding of its soliton and breather dynamics.

Findings

Derived Gram-type determinant solutions for the equation

Constructed one- and two-soliton solutions with asymptotic analysis

Obtained multi-lump, breather, and periodic solutions through transformations and numerical methods

Abstract

We derive a novel variant of the Blaszak-Szum lattice equation by introducing a new class of trigonometric-type bilinear operators. By employing Hirota's bilinear method, we obtain the Gram-type determinant solution of the variant Blaszak-Szum lattice equation. One-soliton and two-soliton solutions are constructed, with a detailed analysis of the asymptotic behaviors of the two-soliton solution. A B\"acklund transformation for the variant Blaszak-Szum lattice equation is established. By virtue of this B\"acklund transformation, multi-lump solutions of the equation are further constructed. Rational solutions are derived by introducing two differential operators applied to the determinant elements; in particular, lump solutions derived via these differential operators can be formulated in terms of Schur polynomials. Through parameter variation, three types of breather solutions are obtained, including the Akhmediev breather, Kuznetsov-Ma breather, and a general breather that propagates along arbitrary oblique trajectories. Finally, numerical three-periodic wave solutions to the variant Blaszak-Szum lattice equation are computed using the Gauss-Newton method.