Variational Operators,Symplectic Operators,and the Cohomology of Scalar Evolution Equations

For a scalar evolution equation u_t = K(t, x, u, u_x, . . . , u_2m+1) with m ≥ 1, the cohomology space H^1,2() is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for u_t = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H^1,2() and subsequently can be naturally identified with the space of variational operators. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The variational operator (or symplectic) nature of the potential form of a bi-Hamiltonian evolution equation is also presented in order to generate examples of interest.     

Synthesis of Layered Carbonitrides from Biotic Molecules for Photoredox Transformations

The construction of layered covalent carbon nitride polymers based on tri-s-triazine units has been achieved by using nucleobases (adenine, guanine, cytosine, thymine and uracil) and urea to establish a two-dimensional semiconducting structure that allows band-gap engineering applications. This biomolecule-derived binary carbon nitride polymer enables the generation of energized charge carrier with light-irradiation to induce photoredox reactions for stable hydrogen production and heterogeneous organosynthesis of C−O, C−C, C−N and N−N bonds, which may enrich discussion on chemical reactions in prebiotic conditions by taking account of the photoredox function of conjugated carbonitride semiconductors that have long been considered to be stable HCN-derived organic macromolecules in space.     

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Rectangular matrix solutions of the defocusing nonlinear Schrödinger equation (dNLS) are studied in quarter-plane and semi-strip. Evolution of the corresponding Weyl–Titchmarsh (Weyl) function is described in terms of the initial Weyl function and boundary conditions. In the next step, the initial Weyl function is recovered (for the quarter-plane case) from the long-time asymptotics of the wave function considered at the boundary. Thus, it is shown that the evolution of the Weyl function is uniquely defined by the boundary conditions. Moreover, a procedure to recover solutions of dNLS (uniquely defined by the boundary conditions) is given. In a somewhat different way, the same boundary value problem is also dealt with in a semi-strip (for the case of a quasi-analytic initial condition).