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Register a new userSymmetry classification of third-order nonlinear evolution equations
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We give a Lie-algebraic classification of third order quasilinear equations which admit non-trivial Lie point symmetries.
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Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no equations invariant under algebras which are semi-direct sums of Levi factor and radical. Making use of these results we prove that there are three, nine, thirty-eight, fifty-two inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively. Finally, we perform a complete group classification of the most general linear third-order evolution equation.
We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of equations is understood as 3D-consistency. The latter is a possibility to consistently impose equations of the same type on all the faces of a three-dimensional cube. This allows to set these equations also on multidimensional lattices Z^N. We classify integrable equations with complex fields x, and Q affine-linear with respect to all arguments. The method is based on analysis of singular solutions.
Supersymmetrization of a nonlinear evolution equation in which the bosonic equation is independent of the fermionic variable and the system is linear in fermionic field goes by the name B-supersymmetrization. This special type of supersymmetrization plays a role in superstring theory. We provide B-supersymmetric extension of a number of quasilinear and fully nonlinear evolution equations and find that the supersymmetric system follows from the usual action principle while the bosonic and fermionic equations are individually non Lagrangian in the field variable. We point out that B-supersymmetrization can also be realized using a generalized Noetherian symmetry such that the resulting set of Lagrangian symmetries coincides with symmetries of the bosonic field equations. This observation provides a basis to associate the bosonic and fermionic fields with the terms of bright and dark solitons. The interpretation sought by us has its origin in the classic work of Bateman who introduced a reverse-time system with negative friction to bring the linear dissipative systems within the framework of variational principle.
We derive a straightforward variational method to construct embedded soliton solutions of the third-order nonlinear Schodinger equation and analytically demonstrate that these solitons exist as a continuous family. We argue that a particular embedded soliton when perturbed may always relax to the adjacent one so as to make it fully stable.
We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known two-dimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges.
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