Related papers: Negative Even Grade mKdV Hierarchy and its Soliton…
In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and…
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include both higher order effects (KdV2) and an uneven river bottom. Although this equation is…
We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV…
We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of…
This paper aims to find new explicit solutions including multi-soliton, multi-positon, multi-negaton, and multi-periodic for a coupled Volterra lattice system which is an integrable discrete version of the coupled KdV equation. The…
In this paper, we report an interesting integrable equation that has both solitons and kink solutions. The integrable equation we study is $(\frac{-u_{xx}}{u})_{t}=2uu_{x}$, which actually comes from the negative KdV hierarchy and could be…
The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the…
The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink…
We show that a fifth order KdV-type equation admits several real as well as complex parity-time reversal or PT-invariant solutions with linear superposition of quadratic functions involving Jacobi elliptic functions of the form ${\rm…
We study the plus and minus type discrete mKdV equation. Some different symmetry conditions associated with two Lax pairs are introduced to derive the matrix Riemann-Hilbert problem with zero. By virtue of regularization of the…
We discuss a differential integrable hierarchy, which we call the (N, M)$--th KdV hierarchy, whose Lax operator is obtained by properly adding $M$ pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. This new hierarchy…
Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of…
In this paper, we are interested in solvable complete Lie algebras, over the field $\K=\R$ or $\mathbb{C}$, which admit a symplectic structure. Specifically, important classes are studied, and a description of complete Lie Algebra with the…
We give a representation--theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new…
In this paper, we prove that the nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity, 29, 915-946 (2016)] is gauge equivalent to a spin-like model. From the gauge equivalence, one…
The construction of a solution of the perturbed KdV equation encounters obstacles to asymptotic integrability beyond the first order, when the zero-order approximation is a multiple-soliton wave. In the standard analysis, the obstacles lead…
This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as…
A hierarchy of integrable hamiltonian nonlinear ODEs is associated with any decomposition of the Lie algebra of Laurent series with coefficients being elements of a semi-simple Lie algebra into a sum of the subalgebra consisting of the…
The lattice Gelfand-Dickey hierarchy is a lattice version of the Gelfand-Dickey hierarchy. A special case is the lattice KdV hierarchy. Inspired by recent work of Buryak and Rossi, we propose an extension of the lattice Gelfand-Dickey…
We discuss some algebraic aspects of the integrable vector non-linear Schr\"{o}dinger hierarchies (GNLS$_{r}$). These are hierarchies of zero-curvature equations constructed from affine Kac-Moody algebras $\hat{sl}_{r+1}$. Using the…