Integrable variant Blaszak-Szum lattice equation

TL;DR

This work develops a novel integrable variant of the Blaszak-Szum lattice by introducing trigonometric-type bilinear operators and obtaining a Gram-determinant solution via Hirota's method. It delivers a comprehensive solution landscape including one- and two-soliton interactions (elastic collisions), lump solutions via a Bäcklund transformation and Schur polynomials, and breather families such as Akhmediev and Kuznetsov-Ma, along with numerical three-periodic waves computed by Gauss-Newton. The combination of analytic constructions and numerical demonstrations broadens the repertoire of discrete integrable systems, revealing rich lump and breather structures and suggesting deeper connections to classical polynomials in their rational limits. Overall, the paper provides a unified framework for generating and analyzing diverse localized and periodic structures in a new variant BS lattice, with potential implications for related 2D Toda-derived lattices and their applications.

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 Bäcklund transformation for the variant Blaszak-Szum lattice equation is established. By virtue of this Bäcklund 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.

Figures (14)

• Figure 1: One-soliton with parameters: $p_1=1+\mathrm{i}$, $q_1=p_1^*$, $c_{11}=1$, $\eta_{0,1}=\xi_{0,1}=0$, $z=0$ and $t=0$.
• Figure 2: Two-soliton solution with parameters: $p_1=1+\mathrm{i}$, $p_2=1-\frac{1}{4}\mathrm{i}$, $\eta_{0,i}=\xi_{0,i}=0$ and $z=0$.
• Figure 3: lump solutions with the parameters: (a) $\lambda_1=1+4\mathrm{i}$; (b) $\lambda_1=2+\mathrm{i}$; (c) $\lambda_1=1+\mathrm{i}$.
• Figure 4: One-lump solution $w(n)$ with the parameters: (a) $a=1$, $b=1$; (b) $a=-1$, $b=1$.
• Figure 5: Fundamental two-lump solutions with the parameters: $\lambda_1=1+\mathrm{i}$, $\lambda_2=2+2\mathrm{i}$.

Theorems & Definitions (8)

• Theorem 1
• proof
• Proposition 1
• Remark 1
• Proposition 2
• Remark 2
• Remark 3
• Remark 4