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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 Conservation laws 35B06 Symmetries, invariants, etc. (PDE) 35Q35 PDEs in connection with fluid mechanics
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##### References:
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