Related papers: Integrable evolution equations with constant separ…
Two integrable systems are constructed in a 2 + 1-dimensional space. Every of these systems involve two evolutions with negative numbers.
Four new integrable evolutions equations with operator Lax pairs are found for an octonion variable. The method uses a scaling ansatz to set up a general polynomial form for the evolution equation and the Lax pair, using KdV and mKdV…
A general form of the fifth-order nonlinear evolution equation is considered. Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type…
We prove that arbitrary (nonpolynomial) scalar evolution equations of order $m\ge 7$, that are integrable in the sense of admitting the canonical conserved densities $\ro^{(1)}$, $\ro^{(2)}$, and $\ro^{(3)}$ introduced in [MSS,1991], are…
We consider second-order evolution equations in an abstract setting with intermittently delayed/ not-delayed damping. We give sufficient conditions for asymptotic and exponential stability, improving and generalising our previous results…
The "separant" of the evolution equation u_t=F, where F is some differentiable function of the derivatives of u up to order m is the partial derivative \partial F}/{\partial u_m} where u_m={\partial^m u}/{\partial x}^m. We apply the formal…
The question of complete integrability of evolution equations associated to $n\times n$ first order isospectral operators is investigated using the inverse scattering method. It is shown that for $n>2$, e.g. for the three-wave interaction,…
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown…
Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws is discussed. In the generic case, nonlocal conservation…
A list of forty third-order exactly integrable two-field evolutionary systems is presented. Differential substitutions connecting various systems from the list are found. It is proved that all the systems can be obtained from only two of…
V.V. Sokolov's modifying symmetry approach is applied to anisotropic evolution equations of the third order on the n-dimensional sphere. The main result is a complete classification of such equations. Auto-B\"acklund transformations are…
We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)-dimensional evolution equations of order $n$ that admit an $N$-shock type solution with $N\leq n+1$. To this end we develop a refinement of the technique…
To an algebraic variety equipped with an involution, we associate a cycle class in the modulo two Chow group of its fixed locus. This association is functorial with respect to proper morphisms having a degree and preserving the involutions.…
Obstacles to integrability in perturbed evolution equations are overcome by allowing higher-order terms in the expansion of the solution to depend explicitly on time and position. With a special expansion algorithm, obstacles vanish…
We consider a projective transformation and establish the invariants for this transformation group up to order seven. We use the obtained invariants to construct a class of nonlinear evolution equations and identify some symmetry-integrable…
This paper is concerned with the initial-boundary value problem for an evolutionary variational inequality complying with three intrinsic properties: complete irreversibility, unilateral equilibrium of an energy and an energy conservation…
We prove that, for $m\ge 7$, scalar evolution equations of the form $u_t=F(x,t,u,...,u_m)$ which admit a nontrivial conserved density of order $m+1$ are linear in $u_m$. The existence of such conserved densities is a necesary condition for…
We solve the classification problem for integrable lattices of the form $u_{,t}=f(u_{-2},\dots,u_2)$ under the additional assumption of invariance with respect to the group of linear-fractional transformations. The obtained list contains 5…
We consider $u_t=u^{\alpha} u_{xxx}+n(u)u_xu_{xx}+m(u)u_x^3+ r(u)u_{xx} +p(u)u_x^2 + q(u)u_x+s(u)$ with $\alpha=0$ and $\alpha=3$, for those functional forms of $m, n, p, q, r, s$ for which the equation is integrable in the sense of an…
In our article "A tree of linearisable second-order evolution equations by generalised hodograph transformations" [J. Nonlin. Math. Phys. {\bf 8} (2001), 342-362] we presented a tree of linearisable (C-integrable) second-order evolution…