Mathematical Modelling of Natural Phenomena

Research Article

Bifurcation of Nonlinear Conservation Laws from the Classical Energy Conservation Law for Internal Gravity Waves in Cylindrical Wave Field

N.H. Ibragimova1 and R.N. Ibragimova2 c1

a1 Department of Mathematics and Science Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden

a2 Department of Mathematics University of Texas at Brownsville, TX 78520, USA


New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth’s rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, θ), b(t, r) and an arbitrary function c(t, θ, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, θ) and b(t, r).

(Online publication September 17 2013)

Key Words:

  • Kelvin internal waves;
  • conservation laws

Mathematics Subject Classification:

  • 76U05;
  • 76B55;
  • 35Q80


c1 Corresponding author. E-mail: