We derive a set of bilinear functional equations of Hirota type for the partition functions of the sl(2) related integrable statistical models defined on a random lattice. These equations are obtained as deformations of the Hirota equations for the KP integrable hierarchy , which are satisfied by the partition function of the ensemble of planar graphs.

and Applied Analysis 3 Theorem 5. If u(x, t) = f(x)g(t), function f(x) = xh(x), where λ > −1 and h(x) has the generalized Taylor series expansion h(x) = ∑∞ n=0 a n (x − x 0 ) αn, (i) β < λ + 1 and α arbitrary, or (ii) β ≥ λ+1, α arbitrary, and a n = 0 for n = 0, 1, . . . , m− 1, wherem − 1 < β ≤ m, then the generalized differential transform (8) becomes U α,β (k, h) = 1 Γ (αk + 1) Γ (βh + 1) [D...

In this paper, we consider multi-component generalizations of the Hirota–Satsuma coupled Korteweg–de Vries (KdV) equation. By introducing a Lax pair, we present a matrix generalization of the Hirota–Satsuma coupled KdV equation, which is shown to be reduced to a vector Hirota–Satsuma coupled KdV equation. By using Hirota's bilinear method, we find a few soliton solutions to the vector Hirota–Sa...

Novel explicit rogue wave solutions of the coupled Hirota equations are obtained by using the Darboux transformation. In contrast to the fundamental Peregrine solitons and dark rogue waves, we present an interesting rogue-wave pair that involves four zero-amplitude holes for the coupled Hirota equations. It is significant that the corresponding expressions of the rogue-wave pair solutions conta...

We explore when Hirota bilinear equations possess linear subspaces of solutions. First, we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to Hirota bilinear equations. Second, we show that multivariate polynomials whose zeros form a vector space can generate the desired Hirota bilinear equations with given linear subs...

In this paper, an efficient algorithm of logarithmic transformation to Hirota bilinear form of the KdV-type bilinear equation is established. In the algorithm, some properties of Hirota operator and logarithmic transformation are successfully applied, which helps to prove that the linear terms of the nonlinear partial differential equation play a crucial role in finding the Hirota bilinear form...