Hamiltonian Structures And Conservation Laws For The Discrete Soliton Systems And Soliton Solutions Of The MKdV-SineGordon Equation

The paper proposes a systematic method to discussing the Hamiltonian structures for the discrete soliton systems. The isospertral evolution equation hierarehy that arise as compatibility conditions of a fairly general diserete Lax pair is considered. By using the so-called implicit represenlation of the isospoctral flows, the existence of the recursion operator L that is a strong and heroditary symmetry of the flows is demonstrated. It is then proved that every equation in the hierarchy possesses the Hamiltonian structure if L has a skew-symmetric factorization and the first equation ((?) = K⁽⁰⁾) in the hierarchy sarisfies some simple condition(K⁽⁰⁾ = 0f⁽⁰⁾ and f⁽⁰⁾ is a gradient function). The related properties, such as the impletie-symplectic factorization of L, the complete integrability in the Liouville sense and multi-llamiltonian structures for the equations, are obtained. Four examples, Toda lattice, Blaszak-Marciniak lattice, Ablowitz-Ladik lattice and a new lattice system, are given to show how to obtain the Hamiltonian systems from Lax pairs. This method is easily to be extended to be used for the continuous cases. We note that the conditions in light of which L is a strong and hereditary symmetry are so simple and natural that n large class of soliton systems (continuous and discrete) satisfy them. The related proof of us is much more precise then the one of I'okas and Anderson.The paper also proposes a systematic approach to constructing an infinite number of conservation laws for discrete soliton systems from the related Lax pair directly. we also illustrate by the above three well known examples how to construct conservation laws for the lattice hierarchies which is derived from various spectral problems, for example, ψ_φ = λ_φ, ψ =Apart from that, the soliton solutions to the mKdV-SineCordon equation are considered. The bilin-ear equations and bilinear Backlund transformation for the mKdV-SineGordon equation are obtained. and an ordinary form of Backlund transformation is given in passing. N-soliton solutions is constructed in the terms of Wronskian and is verified by direct substitution to satisfy the bilinear equation and the associated Backlud transformation respectively. The multi-soliton solutions and novel multi-soliton solutions for the mKdV-SineCordon equation are also obtained by using Hirota's direct method. We list the novel solutions as well for several other equations such as the KdV equation, mKdV equation and KP equation.