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This paper is concerned with the investigation of UC and BUC plane partitions based upon the fermion calculus approach. We construct generalized the vertex operators in terms of free charged fermions and neutral fermions and present the interlacing (strict) 2-partitions. Furthermore, it is showed that the generating functions of UC and BUC plane partitions can be written as product forms. [ABSTRACT FROM AUTHOR]

In this paper, we derive the Grammian determinant solutions to the modified two-dimensional Toda lattice, and then we construct the modified two-dimensional Toda lattice with self-consistent sources via the source generation procedure. We show the integrability of the modified two-dimensional Toda lattice with self-consistent sources by presenting its Casoratian and Grammian structure of the N-soliton solution. It is also demonstrated that the commutativity between the source generation procedure and Bäcklund transformation is valid for the two-dimensional Toda lattice. [ABSTRACT FROM AUTHOR]

This paper investigates the Bogoyavlenskii–Kadomtsev–Petviashvili (BKP) equation by using Hirota's direct method and the Kadomtsev–Petviashvili (KP) hierarchy reduction method. Soliton solutions in the Grammian determinant form for the BKP-II equation are obtained and soliton collisions are shown graphically. Lump-soliton solutions for the BKP-I equation are presented in terms of the Grammian determinants. Various evolution processes of the lump-soliton solutions are demonstrated graphically through the study of three kinds of lump-soliton solutions. The fusion of lumps and kink solitons into kink solitons and the fission of kink solitons into lumps and kink solitons are observed in the interactions of lumps and solitons. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the investigation of UC and BUC plane partitions based upon the fermion calculus approach. We construct generalized the vertex operators in terms of free charged fermions and neutral fermions and present the interlacing (strict) 2-partitions. Furthermore, it is showed that the generating functions of UC and BUC plane partitions can be written as product forms.

In this paper, we construct a full-discrete integrable difference equation which is a full-discretisation of the generalised q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of an extended generalised full-discrete q-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the extended full-discrete generalised q-Toda hierarchy are given. [ABSTRACT FROM AUTHOR]

The Camassa-Holm equation, a completely integrable evolution equation, contains rich geometric structures. For the existence of the bi-Hamiltonian structure and the so-called peaked wave solutions, considerable interest has been aroused in the last several decades. Focusing on local geometric properties of the peaked wave solutions for the Camassa-Holm equation, we propose the multisymplectic method to simulate the propagation of the peaked wave in this paper. Based on the multisymplectic theory, we present a multisymplectic formulation of the Camassa-Holm equation and the multisymplectic conservation law. Then, we apply the Euler box scheme to construct the structure-preserving scheme of the multisymplectic form. Numerical results show the merits of the multisymplectic scheme constructed, especially the local conservative properties on the wave form in the propagation process. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of constructing a 1 -parameter family α λ of zero-curvature representations of an equation E , by means of classical symmetry actions on a given zero-curvature representation α . By using the cohomology defined by the horizontal gauge differential of α , we provide an infinitesimal criterion which permits to identify all infinitesimal classical symmetries of E whose flow could be used to embed α into a nontrivial 1 -parameter family α λ of zero-curvature representations of E . The results of the paper are illustrated with some examples. [ABSTRACT FROM AUTHOR]

We found two three-component nonlocal conservation laws for the 3D rdDym equation, the universal hierarchy equation, the modified Veronese web equation, and the Veronese web equation. The aforementioned equations together with Pavlov’s equation are related via Bäcklund transformations. We study correspondences between the nonlocal conservation laws yielded by the Bäcklund transformations. In particular, we prove that the nonlocal conservation laws that depend on one pseudopotential are generated from a local conservation law of the Veronese web equation via appropriate superpositions of the Bäcklund transformations. Also, we prove nontriviality of the conservation laws found in the paper as well as the ones found in Makridin and Pavlov (2017) for the Khokhlov–Zabolotskaya equation and Pavlov’s equation. [ABSTRACT FROM AUTHOR]

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given pde . The exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the pde s under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer–Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones. [ABSTRACT FROM AUTHOR]

This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, u t − u x x t + [ f ( u ) ] x − [ f ( u ) ] x x x + [ g ( u ) + f ″ ( u ) 2 u x 2 ] x = 0 that includes the Camassa–Holm equation as well as the hyperelastic-rod wave equation ( f ( u ) = k u 2 / 2 and g ( u ) = ( 3 − k ) u 2 / 2 ) as special cases. We establish some a local-in-space blow-up criterion (i.e., a criterion involving only the properties of the data u 0 in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation. [ABSTRACT FROM AUTHOR]

The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair A 1 , A 2 , where A 1 is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equivalent to finding a generalized vector field τ such that A 2 = L τ A 1 . We use this result in order to find the Lagrangian representation when A 2 is a homogeneous third-order Hamiltonian operator, although the method that we use can be applied to any other homogeneous Hamiltonian operator. As an example we provide the Lagrangian representation of a WDVV hydrodynamic-type system in 3 components. [ABSTRACT FROM AUTHOR]

It is well known that in some cases the spectral parameter has a group interpretation. We discuss in detail the case of Gauss–Codazzi equations for isothermic surfaces immersed in E 3 . The algebra of Lie point symmetries is 4-dimensional and all these symmetries are also symmetries of the Gauss–Weingarten equations (which can be considered as s o ( 3 ) -valued non-parametric linear problem). In order to obtain a non-removable spectral parameter one has to consider s o ( 4 , 1 ) -valued linear problem which has a 3-dimensional algebra of Lie point symmetries. The missing symmetry introduces a non-removable parameter. In the second part of the paper we extend these results on the case of isothermic immersions in arbitrary multidimensional Euclidean spaces. In order to simplify calculations the problem was formulated in terms of a Clifford algebra. [ABSTRACT FROM AUTHOR]

In this paper we first give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in R + 4 are Hamiltonian systems. Then we study the integrability of these Hamiltonian systems in the Liouville sense. Finally, we investigate the global dynamics of the completely integrable Lotka–Volterra Hamiltonian systems in R + 4 . As an application of the invariant subsets of these systems, we obtain topological classifications of the 3-submanifolds in R + 4 defined by the hypersurfaces a x y + b z w + c x 2 y + d x y 2 + e z 2 w + f z w 2 = h , where a , b , c , d , e , f , w and h are real constants. [ABSTRACT FROM AUTHOR]

The double ramification hierarchy is a new integrable hierarchy of Hamiltonian PDEs introduced recently by the first author. It is associated to an arbitrary given cohomological field theory. In this paper we study the double ramification hierarchy associated to the cohomological field theory formed by Witten's r -spin classes. Using the formula for the product of the top Chern class of the Hodge bundle with Witten's class, found by the second author, we present an effective method for a computation of the double ramification hierarchy. We do explicit computations for r = 3 , 4 , 5 and prove that the double ramification hierarchy is Miura equivalent to the corresponding Dubrovin–Zhang hierarchy. As an application, this result together with a recent work of the first author with Paolo Rossi gives a quantization of the r -th Gelfand–Dickey hierarchy for r = 3 , 4 , 5 . [ABSTRACT FROM AUTHOR]

We define a matrix first Painlevé hierarchy and a matrix second Painlevé ( P II ) hierarchy. For our matrix P II hierarchy we also give auto-Bäcklund transformations and consider the iteration of solutions. This is the first paper to define matrix Painlevé hierarchies and to give auto-Bäcklund transformations for a matrix Painlevé hierarchy. We also consider, amongst other results, the derivation of sequences of special integrals and autonomous limits. Until now it has been unknown how to connect the known matrix P II equation to the obvious candidates for related completely integrable matrix partial differential equations. Our matrix P II hierarchy is placed firmly within the context of a matrix modified Korteweg–de Vries (mKdV) hierarchy. In deriving our matrix P II hierarchy we make use of the Hamiltonian structure of this matrix mKdV hierarchy. We thus see once again the importance for Painlevé hierarchies of the integrability structures of related completely integrable equations. [ABSTRACT FROM AUTHOR]

Consider a complex Hamiltonian system and an integral curve. In this paper, we give an effective and efficient procedure to put the variational equation of any order along the integral curve in reduced form provided that the previous one is in reduced form with an abelian Lie algebra. Thus, we obtain an effective way to check the Morales–Ramis–Simó criterion for testing meromorphic Liouville integrability of Hamiltonian systems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dissipative Camassa–Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated. [ABSTRACT FROM AUTHOR]

We consider the Benjamin–Ono equation on the torus with an additional damping term on the smallest Fourier modes (cos and sin ). We first prove global well-posedness of this equation in L r , 0 2 (T) . Then, we describe the weak limit points of the trajectories in L r , 0 2 (T) when time goes to infinity, and show that these weak limit points are strong limit points. Finally, we prove the boundedness of higher-order Sobolev norms for this equation. Our key tool is the Birkhoff map for the Benjamin–Ono equation, that we use as an adapted nonlinear Fourier transform. [ABSTRACT FROM AUTHOR]

In this work, we develop an efficient numerical scheme based on the method of lines (MOL) to investigate the interesting phenomenon of collisions and reflections of optical solitons. The established scheme is of second order in space and of fourth order in time with an explicit nature. We deduce stability restrictions using the von Neumann stability analysis. We consider a (2 + 1) -dimensional system of a coupled nonlinear Schrödinger equation as a general model of nonlinear Schrödinger-type equations. We consider several numerical experiments to demonstrate the robustness of the scheme in capturing many scenarios of collisions and reflections of the optical solitons related to nonlinear Schrödinger-type equations. We verify the scheme accuracy through computing the conserved invariants and comparing the present results with some existing ones in the literature. [ABSTRACT FROM AUTHOR]