Two-component integrable generalizations of Burgers equations with nondiagonal linearity

Two-component second and third-order Burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries. New symmetry integrable systems with their master symmetries are obtained. Some third order systems are observed to possess conservation laws. Bi-Poisson structures of systems possessing conservation laws are given.

💡 Research Summary

The paper investigates two‑component generalizations of the Burgers equation whose linear part is given by a constant non‑diagonal matrix. By restricting the leading‑order coefficient matrix to the Jordan‑type forms J(1,ε)³ (non‑degenerate, ε≠0) or J(0,1)³ (degenerate), the authors consider (1,1)-homogeneous systems of weight 2 (second order) and weight 3 (third order). The general polynomial ansatz for these systems contains ten undetermined coefficients for the second‑order case and twenty for the third‑order case. Compatibility with a higher symmetry of weight one or two is imposed via the commutation condition (4). The resulting algebraic constraints are solved using the computer algebra package CRACK.

For the second‑order class, only one non‑trivial system survives the selection process, namely equation (7). This system contains the parameter ε and can be written as a linear combination of two simpler triangular systems (8) and (9). Although the three systems share the same linear part, each possesses a distinct symmetry algebra, demonstrating that ε cannot be eliminated by a simple change of variables. The Hopf‑Cole transformation u=∂ₓ ũ, v=ṽ ũ⁻¹ linearizes (7) into a triangular form, linking it directly to the scalar Burgers equation.

In the third‑order class, eight genuinely new systems (equations (10)–(17)) are obtained. Each system is shown to admit a master symmetry M, explicitly displayed for every case. Acting with the adjoint of M on the translation symmetry ∂ₓ generates an infinite hierarchy of higher‑order symmetries, establishing symmetry integrability. Some of the systems (e.g., (11)) are equivalent, after suitable differential substitutions, to the well‑known Sasa‑Satsuma system. Others (e.g., (12), (13), (14)) have more intricate nonlinear terms but still possess master symmetries that generate their full symmetry algebras.

A subset of the third‑order systems exhibits conservation laws. By setting ε=1 and performing the substitution vₓ=z, system (15) is transformed into (18). For this system the authors construct a compatible pair of (generally non‑local) Poisson operators H and K. They verify the Jacobi identities and demonstrate that the flows generated by H and K correspond to the variational derivatives of two conserved densities, thereby providing a bi‑Poisson (or bi‑Hamiltonian) formulation. This structure yields a Lenard‑Magri recursion scheme and guarantees an infinite sequence of conserved quantities.

Overall, the work extends the classification of integrable multi‑component Burgers‑type equations to the previously unstudied case of non‑diagonal linear parts. It delivers (i) a complete list of second‑order and third‑order systems admitting higher symmetries, (ii) explicit master symmetries for all non‑trivial third‑order models, (iii) identification of conservation laws and bi‑Poisson structures for several cases, and (iv) connections to known integrable models such as the Sasa‑Satsuma and Karasu‑Kalkanlı systems. The results enrich the theory of symmetry‑integrable PDEs and open avenues for further analytical, numerical, and physical investigations of these newly discovered non‑diagonal Burgers‑type systems.