Conservation laws for linear equations on braided linear spaces

Abstract

L m ϭx m ‫ץ‬ x , L n ϭy n ‫ץ‬ y , n,mϾ1, ͑54͒also produces the solution from the solution of the wave equation ͑48͒. According to Proposition 5.1 we obtain the currents connected with deformed conformalwhich fulfill the conservation law:The operators L m and L n add to the set of conserved currents the following expressions:J m ϭ⌽Ј⌫ ͑ ‫ץ,ץ‬ ឈ † ͒L m ⌽, J n ϭ⌽Ј⌫ ͑ ‫ץ,ץ‬ ឈ † ͒L n ⌽, n,mϾ1. ͑57͒ B. The conserved currents for scalar wave equation on q-Minkowski spaceLet us now pass to the scalar wave equation on q-Minkowski space. We have studied this equation in Ref. 16 within the framework given by Azcárraga, Kulish, and Rodenas. 17,18 Now we 4171

Conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces

“…The following propositions hold for equation (1 -3) and are the extension of results derived in [11,12,13]. …”

“…In the previous papers we discussed the conservation laws for linear equations with constant coefficients for discrete differential calculus [10,11] as well as for a wider category of equations acting on noncommutative spaces namely on quantum Minkowski spaces [12,15] and the braided linear ones [13,14,16]. Now we would like to present our results for an extended class of equations with variable coefficients of the form:…”

“…The Leibnitz rule of the form (5) is also characteristic for noncommutative spaces. We have checked [12,13] that for quantum Minkowski spaces and for braided linear spaces (including the q-Minkowski space) it is given by the formula (5) with transformation operators determined by their multiplicity property:…”

“…In the previous papers [10,11,12,13] we have extended this procedure to the linear equations on discrete and noncommutative spaces including quantum Minkowski [22,23,24] and braided linear spaces [17,18,19,20]. It appears however that we can consider in a similar way a wide class of equations with variable coefficients built within framework of any differential calculus with the Leibnitz rule for partial derivatives deformed via the transformation operator which is multiplicative and invertible.…”

“…The equations we shall investigate have the variable coefficients which fulfill the corresponding restriction (3) including the conjugated derivatives. The possibility of derivation the conservation laws for some equations with variable coefficients on noncommutative spaces was indicated earlier [12,13] but the proposed conditions for coefficients were too strong to construct usefull examples. In constrast the new condition (as we show explicitly in the applications) appears to be identical with the nonlinear equations of some of the integrable models for which we derive the conserved currents via the auxiliary linear equations with variable coefficients.…”