We investigate the relation between the Korteweg de Vries and modified Korteweg de Vries equations (KdV and mKdV), and find a new algebro-analytic mechanism, similar to the Lax L-A pair, which involves a family of first-order operators Qλ depending on a spectral parameter λ, instead of the third-order operator A. In our framework, any generalized eigenfunction of the Schrödinger operator L, who...

Complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory. In this article, we explicitly construct such eigenfunctions for a large class of integrable equations including the KdV, NLS and mKdV hierarchies. We establish the striking result that the linearization operators of all equations in the...

An elementary yet remarkable similarity between the Cole-Hopf transformation relating the Burgers and heat equation and Miura's transformation connecting the KdV and mKdV equations is studied in detail. 1. Introduction Our aim in this note is to display the close similarity between the well-known Cole{Hopf transformation relating the Burgers and the heat equation, and the celebrated Miura trans...

For a particular class of integral operators K we show that the quantity φ := log det (I + K)log det (/ K) satisfies both the integrated mKdV hierarchy and the Sinh-Gordon hierarchy. This proves a conjecture of Zamolodchikov.

A bi-Hamiltonian hierarchy of complex vector soliton equations is derived from geometric flows of non-stretching curves in the Lie groups G = SO(N + 1), SU(N) ⊂ U(N), generalizing previous work on integrable curve flows in Riemannian symmetric spaces G/SO(N). The derivation uses a parallel frame and connection along the curves, involving the Klein geometry of the group G. This is shown to yield...

We investigate bi–Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non–stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric–induced nonlinear connection (N–connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a ...

We establish the nonlinear stability of $N$-soliton solutions modified Korteweg-de Vries (mKdV) equation. The are global mKdV behaving at (positive and negative) time infinity as sums $1$-solitons with speeds $0<c_1<\cdots< c_N$.The proof relies on variational characterization $N$-solitons. show that $N$-solitons realize local minimum $(N+1)$-th conserved quantity subject to fixed constraints $...

The construction of Miura and B\"acklund transformations for $A_n$ mKdV KdV hierarchies are presented in terms gauge acting upon the zero curvature representation. As well known $sl(2)$ case, we derive relate equations motion two hierarchies. Moreover, Miura-gauge transformation is not unique, instead, it shown to be connected a set generators labeled by exponents generalized gauge-B\"acklund $...

The dynamics of the highly nonlinear fifth order KdV-type equation is discussed in the framework of the Lagrangian and Hamiltonian formalisms. The symmetries of the Lagrangian produce three commuting conserved quantities that are found to be recursively related to one-another for a certain specific value of the power of nonlinearity. The above cited recursion relations are obeyed with a second ...

Methods in Riemann–Finsler geometry are applied to investigate bi–Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non–stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N–connections), Sas...