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Conservation laws for third-order variant Boussinesq system. (English) Zbl 1193.35104

Summary: The conservation laws for the variant Boussinesq system are derived by an interesting method of increasing the order of partial differential equations. The variant Boussinesq system is a third-order system of two partial differential equations. The transformations \(u\rightarrow U_x, v\rightarrow V_x\) are used to convert the variant Boussinesq system to a fourth order system in \(U,V\) variables. It is interesting that a standard Lagrangian exists for the fourth-order system. Noether’s approach is then used to derive the conservation laws. Finally, the conservation laws are expressed in the variables \(u,v\) and they constitute the conservation laws for the third-order variant Boussinesq system. Infinitely many nonlocal conserved quantities are found for the variant Boussinesq system.

MSC:

35L65 Hyperbolic conservation laws
35B06 Symmetries, invariants, etc. in context of PDEs
35Q35 PDEs in connection with fluid mechanics
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