On some new forms of lattice integrable equations

Abstract

Abstract:Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon -type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings. PACS

On the Soliton Solutions of a Family of Tzitzeica Equations

“…There are many methods for deriving the soliton solutions; we have demonstrated two of the most used: the dressing method and the Hirota method [3,6,17]. Both methods give the same results both for the kinks and for the breathers.…”

Special Solutions of Bi-Riccati Delay-Differential Equations

“…1 We will refer to our rational-exponential solutions as soliton-type solutions principally for their relation to Hirota's method: these are not soliton solutions in the usual sense, since the equations we are dealing with depend on a single variable and we do not impose reality conditions on our complex solutions. Alternatively, our solutions could be viewed as complex soliton solutions of the differential-difference equation obtained by separating the shifts and derivatives (i.e., inverting the traveling wave reductions (1.5) and (1.8)) as in [2]. The existence of N -soliton solutions for N ≥ 3 is strong indicator of integrability [9].…”

Constructing soliton solutions and super-bilinear form of lattice supersymmetric KdV equation

“…The discrete version of potential KdV can be obtained directly by discretizing the bilinear forms imposing gauge invariance [22] (i.e.invariance with respect to a multiplication with an exponential exp (an + bt) for any a and b) [26], [27], [28], [29], [30]. When we discretise time t we put t → m and for the time-derivative we consider ∂ t f (n, t) → 1 h (f (n, m + h) − f (n, m)) where h is the discretisation step (we can put t = mh to have step 1 as well).…”