Integrable variant Blaszak-Szum lattice equation

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.

💡 Research Summary

This paper presents a comprehensive study on a novel integrable variant of the Blaszak-Szum (BS) lattice equation, achieved by introducing a new class of trigonometric-type bilinear operators. The primary motivation stems from recent work by Liu et al., who defined operators sin(δD_z) and cos(δD_z), leading to new discrete integrable systems. Recognizing that the BS lattice shares an algebraic origin with the Toda and Leznov lattices, the authors apply a transformation (D_t, D_y, D_z, D_n) → i(D_t, D_y, D_z, D_n) to the original bilinear BS equations. This transforms hyperbolic functions into trigonometric ones, yielding a new set of bilinear equations (15)-(16). Through a dependent variable transformation (17), these bilinear equations are converted into the novel “variant BS lattice equation” (18)-(20), a system of real-valued equations for real variables.

A central result is Theorem 1, which establishes that the bilinear equations admit a Gram-type determinant solution (25). The proof demonstrates that if the basis functions ϕ_j(n) and ψ_k(n) satisfy specific differential-difference relations (26)-(27), the determinant τ(n) automatically satisfies the bilinear equations. This provides a powerful and general framework for constructing solutions.

The paper then systematically constructs and analyzes various types of solutions. Starting from exponential forms for ϕ_j and ψ_k, one-soliton and two-soliton solutions are derived. The one-soliton solution for u(n) is shown to be a dark soliton. A detailed asymptotic analysis of the two-soliton solution proves that their interaction is purely elastic, with shape and velocity preserved upon collision.

Furthermore, the authors establish a Bäcklund transformation (BT) for the variant BS system (Proposition 1). Leveraging this BT and a nonlinear superposition formula (52), they construct multi-lump solutions. Beginning with the trivial solution τ_0(n)=1, a first-order lump solution τ_12(n) is generated. The morphology of the resulting lump solution for u(n) is shown to depend critically on the parameter ratio |a/b| in λ = a+ib, classifying them into bright lumps (one global maximum), dark lumps (two global maxima), and fundamental lumps (two maxima and two minima). High-order lump solutions are generated via a determinant formula (59). Rational solutions are also expressed in terms of Schur polynomials, offering an algebraic perspective.

By employing parameter variations in the general solution scheme, three distinct types of breather solutions are obtained: the spatially localized Kuznetsov-Ma breather, the temporally localized Akhmediev breather, and a general breather propagating along an arbitrary oblique trajectory. This showcases the model’s capacity to describe different localized oscillatory phenomena.

Finally, acknowledging the complexity of finding periodic wave solutions analytically, the paper employs numerical methods. Specifically, the Gauss-Newton iterative method is used to compute numerical three-periodic wave solutions for the variant BS lattice equation, bridging analytical integrability theory with computational techniques.

In summary, this work delivers a complete package: the derivation of a new integrable system via innovative bilinear operators, the provision of a general determinant solution, and a thorough exploration of its rich solution space—encompassing solitons, lumps, breathers, and periodic waves—through both analytical and numerical means, significantly advancing the study of discrete integrable systems.