Darboux transformations for the time-dependent nonhomogeneous Burgers equation in (1+1) dimensions

Abstract

We extend the formalism of nth order Darboux transformations to the time-dependent nonhomogeneous Burgers equation (NBE) in (1+1) dimensions. Similar to the Schrödinger case, our Darboux transformation retains the form of the NBE, while changing the nonhomogeneous term. The transformed solution of the NBE and the corresponding transformed nonhomogeneity are given in closed form. Furthermore, properties of the transformation are discussed and an application is given.

Dirichlet problem on the half-line for a forced Burgers equation with time-variable coefficients and exactly solvable models

“…Moreover, for real x and complex time t there is a possibility to develop singular Burgers solutions from non-singular initial data. For more details on the Burgers singularities one can see [33,34] .…”

“…Burgers problem (48) reduces to the heat problem (30) , with 0 < τ < 2/ γ , and therefore (48) has solution where ϕ m ( η, τ ) is given by ( 33) and η( x, t ), τ ( t ) are as defined in (43) . According to (34) heat function ϕ m ( η, τ ) has moving zero η = χ m (τ ) , and it follows that for x > 0, t > 0, Burgers solution U m ( x, t ) has moving singularity described by…”

Exact solutions of forced Burgers equations with time variable coefficients

“…is a known integrable model and one can see for example [17,23] In what follows, using the discussion in previous section, we will write explicitly some special solutions of BE (25), and note that, in the limit case x 0 ! 0, these solutions Uðx; tÞ approach the solutions Vðx; tÞ of the standard BE.…”

A new type of Darboux transformations for the one-dimensional Burgers equation with forcing

“…Since the Schrödinger equation is obtained from its Burgers counterpart by application of the Cole-Hopf linearization [23,24], the Darboux transformation is adaptable to the Burgers scenario. While the latter method is known and has been applied, see for example [25], the purpose of the present work is to construct a different Darboux transformation to the Burgers equation. This transformation was first introduced in [26] for coupled Korteweg-de Vries equations, and in its final form adapted to a Schrödinger equation for a class of energy-dependent potentials [1].…”