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Register a new userComplete point-symmetry group, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system
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We carry out extended symmetry analysis of the (1+2)-dimensional Boiti-Leon-Pempinelli system, which enhances and generalizes many results existing in the literature. The complete point-symmetry group of this system is computed using an original megaideal-based version of the algebraic method. A number of meticulously selected differential constraints allow us to construct families of exact solutions of this system, which are significantly larger than all known ones. After classifying one- and two-dimensional subalgebras of the entire (infinite-dimensional) maximal Lie invariance algebra of this system, we study only its essential Lie reductions, which give solutions beyond the above solution families. Among reductions of the Boiti-Leon-Pempinelli system via differential constraints or Lie symmetries, we identify a number of famous partial and ordinary differential equations. We also show how all the constructed solution families can significantly be extended using Laplace and Darboux transformations.
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The Theory of (2+1) Systems based on 2D Schrodinger Operator was started by S.Manakov, B.Dubrovin, I.Krichever and S.Novikov in 1976. The Analog of Lax Pairs introduced by Manakov, has a form $L_t=[L,H]-fL$ (The $L,H,f$-triples) where $L=partial_xpartial_y+Gpartial_y+S$ and $H,f$-some linear PDEs. Their Algebro-Geometric Solutions and therefore the full higher order hierarchies were constructed by B.Dubrovin, I.Krichever and S.Novikov. The Theory of 2D Inverse Spectral Problems for the Elliptic Operator $L$ with $x,y$ replaced by $z,bar{z}$, was started by B.Dubrovin, I.Krichever and S.Novikov: The Inverse Spectral Problem Data are taken from the complex Fermi-Curve consisting of all Bloch-Floquet Eigenfunctions $Lpsi=const$. Many interesting systems were found later. However, specific properties of the very first system, offered by Manakov for the verification of new method only, were not studied more than 10 years until B.Konopelchenko found in 1988 analogs of Backund Transformations for it. He pointed out on the Burgers-Type Reduction. Indeed, the present authors quite recently found very interesting extensions, reductions and applications of that system both in the theory of nonlinear evolution systems (The Self-Adjoint and 2D Burgers Hierarhies were invented, and corresponding reductions of Inverse Problem Data found) and in the Spectral Theory of Important Physical Operators (The Purely Magnetic 2D Pauli Operators). We call this system GKMMN by the names of authors who studied it.
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