Abstract

The KdV--Burgers equation \(u_t + uu_x - \nu u_{xx} + \beta u_{xxx} = 0\) models the interplay of nonlinearity, dispersion and dissipation. Through the potential \(u = v_x\), a Lagrangian density is constructed for the resulting potential system. Application of Noether's theorem yields an infinite-dimensional symmetry \(V = f(x,t)\partial_v\), where \(f\) satisfies the linearized equation \(f_t - \nu f_{xx} + \beta f_{xxx} = 0\). This symmetry generates an infinite family of continuity equations of the form \(D_t T + D_x X = 0\) with \(T = f(x,t)\) and \(X = -f_x u + \nu f_x - \beta f_{xx}\). For \(\nu > 0\), these relations constitute linear potential balance laws rather than classical conservation laws, as the density \(T\) depends explicitly on the auxiliary field \(f\) and on time. In the inviscid limit \(\nu = 0\), the family reduces to the classical KdV conservation hierarchy. High-accuracy spectral simulations confirm the validity of these identities up to discretization error.

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