Rafael Hernández Heredero

Classification of Fully Nonlinear Integrable Evolution Equations of Third Order

Abstract A fully nonlinear family of evolution equations is classified. Nine new integrable equations are found, and all of them admit a differential substitution into the Korteweg-de Vries or Krichever-Novikov equations. One of the equations contains hyperelliptic functions, but it is transformable into the Krichever-Novikov equation by a differential substitution that only involves elliptic functions.

Multiscale Analysis of Discrete Nonlinear Evolution Equations: The Reduction of the dNLS

Abstract In this paper we consider multiple lattices and functions defined on them. We introduce some slow varying conditions and define a multiscale analysis on the lattice, i.e. a way to express the variation of a function in one lattice in terms of an asymptotic expansion with respect to the other. We apply these results to the case of the multiscale expansion of the differential-difference Nonlinear Schrödinger equation.

Multiscale expansion on the lattice and integrability of partial difference equations

We conjecture an integrability and linearizability test for dispersive Z^2-lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscale expansion give a partial differential equation of the form of the nonlinear Schrodinger (NLS) equation. If the starting lattice equation is integrable then the resulting NLS equation turns out to be integrable, while if the starting equation is linearizable we get a linear Schrodinger equation. On the other hand, if we start with a non-integrable lattice equation we may obtain a non-integrable NLS equation. This conjecture is confirmed by many examples.

Multiscale Expansion and Integrability Properties of the Lattice Potential KdV Equation

Abstract We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schrödinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schrödinger equation.

NONLOCAL SYMMETRIES, COMPACTON EQUATIONS, AND INTEGRABILITY

We review the theory of nonlocal symmetries of nonlinear partial differential equations and, as examples, we present infinite-dimensional Lie algebras of nonlocal symmetries of the Fokas–Qiao and Kaup–Kupershmidt equations. Then, we consider nonlocal symmetries of a family which contains the Korteweg–de Vries (KdV) and (a subclass of) the Rosenau–Hyman compacton-bearing K(m, n) equations. We find that the only member of the family which possesses nonlocal symmetries (of a kind specified in Sec. 3 below) is precisely the KdV equation. We take this fact as an indication that the K(m, n) equations are not integrable in general, and we use the formal symmetry approach of Shabat to check this claim: we prove that the only integrable cases of the full K(m, n) family are the KdV and modified KdV equations.

Formal recursion operators of integrable nonevolutionary equations and Lagrangian systems

We derive the general structure of the space of formal recursion operators of nonevolutionary equations~

Симметрии дискретного нелинейного уравнения Шредингера@@@Symmetries of the Discrete Nonlinear Schrödinger Equation

q_{tt}=f(q,q_{x},q_t,q_{xx},q_{xt},q_{xxx},q_{xxxx})

Geometric Integrability of the Camassa–Holm Equation. II

. This allows us to classify integrable Lagrangian systems with a higher order Lagrangian of the form~

Интегрируемые квазилинейные уравнения@@@Integrable Quasilinear Equations

\mathscr{L}=\frac12 L_2(q_{xx}, q_x, q)\,q_t^2 + L_1(q_{xx}, q_x, q)\, q_{t} + L_0(q_{xx}, q_x, q)

Formal recursion operators of integrable PDEs of the form Utt = F(U,Ux,Ut,...)

. The key technique relays on exploiting a homogeneity of the determining equations of formal recursion operators. This technique allows us to extend the main results to more general equations~