Bäcklund Transformations for Liouville Equations with Exponential Nonlinearity

This work aims to obtain new transformations and auto-Bäcklund transformations for generalized Liouville equations with exponential nonlinearity having a factor depending on the first derivatives. This paper discusses the construction of Bäcklund transformations for nonlinear partial second-order derivatives of the soliton type with logarithmic nonlinearity and hyperbolic linear parts. The construction of transformations is based on the method proposed by Clairin for second-order equations of the Monge–Ampere type. For the equations studied in the article, using the Bäcklund transformations, new equations are found, which make it possible to find solutions to the original nonlinear equations and reveal the internal connections between various integrable equations.

Comment on ‘Study of lump solutions to an extended Calogero-Bogoyavlenskii-Schiff equation involving three fourth-order terms’ (2020 Phys. Scr. 95 095207)

Of current interest, in nonlinear optics, fluid dynamics and plasma physics, the paper commented (i.e., Phys. Scr. 95, 095207, 2020) has investigated a (2+1)-dimensional extended Calogero-Bogoyavlenskii-Schiff system. Hereby, we make the issue raised in that paper more complete. Using the Hirota method and symbolic computation, we construct three sets of the bilinear auto-Bäcklund transformations for that system, along with some analytic solutions. As for the amplitude of the relevant wave in nonlinear optics, fluid dynamics or plasma physics, our results depend on the coefficients in that system.

Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

We demonstrate the way to derive the second Painlevé equation P2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of P2 while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra JMat(N,N) with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of P2, whereas the VN algebra produces a different JP-system that serves as a generalization of the Sokolov’s form of a vectorial NLS.