## 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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### References:

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