Seminarium Gamma

We exhaustively characterize all cases when a certain natural generalization of the Infeld–Rowlands equation admits nontrivial local conservation laws of any order, and give explicit form of these conservation laws modulo trivial ones. The original Infeld–Rowlands equation arises inter alia in the study of solution stability for the Ginzburg–Landau equation. What is more, one of the special cases of our generalization can be seen as an extension of two-dimensional Kuramoto–Sivashinsky equation to dimension three.It turns out that even in the generic case the equation in question, to which we refer to as the generalized Infeld–Rowlands equation, admits an infinite family of local conservation laws parameterized by an arbitrary smooth function of one variable.Furthermore, we prove that the equation under study admits no nontrivial local Hamiltonian structures and no nontrivial local symplectic structures no matter the orders of the structures in question; the method of establishing the said nonexistence results can be readily applied to many other PDEs.For further details please see the article J. Vašíček, Conservation laws and nonexistence of local Hamiltonian structures for generalized Infeld—Rowlands equation, Rep. Math. Phys. 93 (2024), no. 3, 287-300, https://doi.org/10.1016/S0034-4877(24)00038-7.