Lattice Schwarzian Boussinesq equation and two-component systems

Symmetry, Integrability and Geometry: Methods and Applications, 2011

Integrable multi-component lattice equations of the Boussinesq family have been known for some time. Recently some new equations of this type were found using the Consistency-Around-the-Cube approach. Here we investigate one of these models, B-2, and in particular the consequences of a nonzero deformation parameter b 0 > 0, which allows special kinds of solitons in the parameter range -b 0 /3 < k < b 0 .

Physica D: Nonlinear Phenomena, 2018

The Boussinesq equations are known since the end of the XIX st century. However, the proliferation of various Boussinesq-type systems started only in the second half of the XX st century. Today they come under various flavours depending on the goals of the modeller. At the beginning of the XXI st century an effort to classify such systems, at least for even bottoms, was undertaken and developed according to both different physical regimes and mathematical properties, with special emphasis, in this last sense, on the existence of symmetry groups and their connection to conserved quantities. Of particular interest are those systems admitting a symplectic structure, with the subsequent preservation of the total energy represented by the Hamiltonian. In the present paper a family of Boussinesq-type systems with multi-symplectic structure is introduced. Some properties of the new systems are analyzed: their relation with already known Boussinesq models, the identification of those systems with additional Hamiltonian structure as well as other mathematical features like well-posedness and existence of different types of solitary-wave solutions. The consistency of multi-symplectic systems with the full Euler equations is also discussed.

Abstract and Applied Analysis, 2014

We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether's approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.

Applied Mathematics Letters, 2010

The conservation laws for the variant Boussinesq system are derived by an interesting method of increasing the order of partial differential equations. The variant Boussinesq system is a third-order system of two partial differential equations. The transformations u → U x , v → V x are used to convert the variant Boussinesq system to a fourth order system in U, V variables. It is interesting that a standard Lagrangian exists for the fourthorder system. Noether's approach is then used to derive the conservation laws. Finally, the conservation laws are expressed in the variables u, v and they constitute the conservation laws for the third-order variant Boussinesq system. Infinitely many nonlocal conserved quantities are found for the variant Boussinesq system.

Nuclear Physics B, 2019

In this paper, we formulate a "Grassmann extension" scheme for constructing noncommutative (Grassmann) extensions of Yang-Baxter maps together with their associated systems P∆Es, based on the ideas presented in [15]. Using this scheme, we first construct a Grassmann extension of a Yang-Baxter map which constitutes a lift of a lattice Boussinesq system. The Grassmann-extended Yang-Baxter map can be squeezed down to a novel, integrable, Grassmann lattice Boussinesq system, and we derive its 3D-consistent limit. We show that some systems retain their 3D-consistency property in their Grassmann extension.

International Journal of Engineering Science, 2006

We apply the symmetry method based on the Fréchet derivative of the differential operators to deduce the Lie symmetries of the following variant of the Boussinesq equations u t þ a 1 ðtÞv x þ b 1 ðtÞuu x þ c 1 ðtÞu xx ¼ 0 v t þ a 2 ðtÞuv x þ b 2 ðtÞvu x þ c 2 ðtÞv xx þ pðtÞu xxx ¼ 0 where a i (t), b i (t), c i (t), i = 1, 2 and p(t) are arbitrary functions of t. For each infinitesimal generator in the optimal system of subalgebras we study the reduced ODE and, among other solutions, furnish some nontrivial exact solutions in terms of hyperbolic functions.

Using new methods of analysis of integrable systems,based on a general geometric approach to nonlinear PDE,we discuss the Dispersionless Boussinesq Equation, which is equivalent to the Benney-Lax equation,being a system of equations of hydrodynamical type. The results include: a description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. Highly interesting are the appearences of the Noether and Inverse Noether operators ,leading to multiple infinite hierarchies of these operators as well as recursion operators.

Journal of Nonlinear Mathematical Physics, 1998

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f (u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.

Symmetry Integrability and Geometry-methods and Applications, 2008

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov . This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

Nonlinear Systems and Their Remarkable Mathematical Structures, 2021

We present a comprehensive review of the discrete Boussinesq equations based on their three-component forms on an elementary quadrilateral. These equations were originally found by Nijhoff et al using the direct linearization method and later generalized by Hietarinta using a search method based on multidimensional consistency. We derive from these three-component equations their two-and onecomponent variants. From the one-component form we derive two different semicontinuous limits as well as their fully continuous limits, which turn out to be PDE's for the regular, modified and Schwarzian Boussinesq equations. Several kinds of Lax pairs are also provided. Finally we give their Hirota bilinear forms and multi-soliton solutions in terms of Casoratians.