In mathematical physics, the NovikovâÂÂVeselov equation (or VeselovâÂÂNovikov equation) is a nonlinear partial differential equation. It is a two-dimensional analogue of the well-known KortewegâÂÂde Vries equation (KdV equation), which can model shallow water waves. A key feature of the NovikovâÂÂVeselov equation is its integrability, meaning it can be solved exactly through a method known as the inverse scattering transform.
The equation's integrability is linked to the two-dimensional stationary Schrödinger equation, just as the KdV equation's integrability is linked to the one-dimensional Schrödinger equation. This property distinguishes it from other two-dimensional analogues of the KdV equation, such as the KadomtsevâÂÂPetviashvili equation. The equation is named after Soviet mathematicians Sergei P. Novikov and A.P. Veselov, who introduced it in .
The NovikovâÂÂVeselov equation is most commonly written as
where and the following standard notation of complex analysis is used: is the real part,
The function is generally considered to be real-valued. The function is an auxiliary function defined via up to a holomorphic summand, is a real parameter corresponding to the energy level of the related 2-dimensional Schrödinger equation
When the functions and in the NovikovâÂÂVeselov equation depend only on one spatial variable, e.g. , , then the equation is reduced to the classical KortewegâÂÂde Vries equation. If in the NovikovâÂÂVeselov equation , then the equation reduces to another (2+1)-dimensional analogue of the KdV equation, the KadomtsevâÂÂPetviashvili equation (to KP-I and KP-II, respectively) .
The inverse scattering transform method for solving nonlinear partial differential equations (PDEs) begins with the discovery of C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura , who demonstrated that the KortewegâÂÂde Vries equation can be integrated via the inverse scattering problem for the 1-dimensional stationary Schrödinger equation. The algebraic nature of this discovery was revealed by Lax who showed that the KortewegâÂÂde Vries equation can be written in the following operator form (the so-called Lax pair):
where , and is a commutator. Equation () is a compatibility condition for the equations
for all values of .
Afterwards, a representation of the form () was found for many other physically interesting nonlinear equations, like the KadomtsevâÂÂPetviashvili equation, sine-Gordon equation, nonlinear Schrödinger equation and others. This led to an extensive development of the theory of inverse scattering transform for integrating nonlinear partial differential equations.
When trying to generalize representation () to two dimensions, one obtains that it holds only for trivial cases (operators , , have constant coefficients or operator is a differential operator of order not larger than 1 with respect to one of the variables). However, S.V. Manakov showed that in the two-dimensional case it is more correct to consider the following representation (further called the Manakov L-A-B triple):
or, equivalently, to search for the condition of compatibility of the equations
at one fixed value of parameter .
Representation () for the 2-dimensional Schrödinger operator was found by S.P. Novikov and A.P. Veselov in . The authors also constructed a hierarchy of evolution equations integrable via the inverse scattering transform for the 2-dimensional Schrödinger equation at fixed energy. This set of evolution equations (which is sometimes called the hierarchy of the NovikovâÂÂVeselov equations) contains, in particular, the equation ().
The dispersionless version of the NovikovâÂÂVeselov equation was derived in a model of nonlinear geometrical optics .
The behavior of solutions to the NovikovâÂÂVeselov equation depends essentially on the regularity of the scattering data for this solution. If the scattering data are regular, then the solution vanishes uniformly with time. If the scattering data have singularities, then the solution may develop solitons. For example, the scattering data of the GrinevichâÂÂZakharov soliton solutions of the NovikovâÂÂVeselov equation have singular points.
Solitons are traditionally a key object of study in the theory of nonlinear integrable equations. The solitons of the NovikovâÂÂVeselov equation at positive energy are transparent potentials, similarly to the one-dimensional case (in which solitons are reflectionless potentials). However, unlike the one-dimensional case where there exist well-known exponentially decaying solitons, the NovikovâÂÂVeselov equation (at least at non-zero energy) does not possess exponentially localized solitons .