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Register a new userCommon Hirota Form B{a}cklund Transformation for the Unified Soliton System
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We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) $cong$ GL(2,$mathbb R$) $cong$ M{o}bius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the $N$-soliton solutions through the KdV type B{a}cklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the B{a}cklund transformations of the standard form into the same common Hirota form and the same common B{a}cklund transformation except the equation which has the time-derivative term. The difference is only the time-dependence and the main structure of the $N$-soliton solutions has same common form for KdV/mKdV/sinh-Gordon systems. Then the $N$-soliton solutions for the sinh-Gordon equation is obtained just by the replacement from KdV/mKdV $N$-soliton solutions. We also give general addition formulae coming from the KdV type B{a}cklund transformation which plays not only an important role to construct the trigonometric/hyperbolic $N$-soliton solutions but also an essential role to construct the elliptic $N$-soliton solutions. In contrast to the KdV type B{a}cklund transformation, the well-known mKdV/sinh-Gordon type B{a}cklund transformation gives the non-cyclic symmetric $N$-soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations.
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We study group theoretical structures of the mKdV equation. The Schwarzian type mKdV equation has the global M{o}bius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special local M{o}bius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV B{a}cklund transformation can generate the mKdV cyclic symmetric $N$-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.
We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with constant scalar curvature becomes an integrable differential equation. We show that such Einstein equation includes KdV/mKdV/sine-Gordon equations.
In this work, the $overline{partial}$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(mathbb{R})}$. The long-time asymptotic behavior of the solution $q(x,t)$ is derived in any fixed space-time cone $C(x_{1},x_{2},v_{1},v_{2})=left{(x,t)in mathbb{R}timesmathbb{R}: x=x_{0}+vt ~text{with}~ x_{0}in[x_{1},x_{2}]right}$. We show that solution resolution conjecture of the Hirota equation is characterized by the leading order term $mathcal {O}(t^{-1/2})$ in the continuous spectrum, $mathcal {N}(mathcal {I})$ soliton solutions in the discrete spectrum and error order $mathcal {O}(t^{-3/4})$ from the $overline{partial}$ equation.
The soliton solutions of the Camassa-Holm equation are derived by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by other methods.
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. According to a proposal by T. Ratiu, discretizations of the Hirota-Kimura type can be considered for numerous integrable systems of classical mechanics. Due to a remarkable and not well understood mechanism, such discretizations seem to inherit the integrability for all algebraically completely integrable systems. We introduce an experimental method for a rigorous study of integrability of such discretizations. Application of this method to the Hirota-Kimura type discretization of the Clebsch system leads to the discovery of four functionally independent integrals of motion of this discrete time system, which turn out to be much more complicated than the integrals of the continuous time system. Further, we prove that every orbit of the discrete time Clebsch system lies in an intersection of four quadrics in the six-dimensional phase space. Analogous results hold for the Hirota-Kimura type discretizations for all commuting flows of the Clebsch system, as well as for the $so(4)$ Euler top.
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