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Register a new userHodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation
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It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock type singularities are presented.
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We present the fundamental solutions for the spin-1/2 fields propagating in the spacetimes with power type expansion/contraction and the fundamental solution of the Cauchy problem for the Dirac equation. The derivation of these fundamental solutions is based on formulas for the solutions to the generalized Euler-Poisson-Darboux equation, which are obtained by the integral transform approach.
The equations of Loewner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approaches. After recalling the derivation of Lowner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarhies (dKP, dBKP, dToda, and dDKP). The one-vaiable reductions are described by solutions of differe
We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noethers theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.
The paper begins with a review of the well known Novikovs equations and corresponding finite KdV hierarchies. For a positive integer $N$ we give an explicit description of the $N$-th Novikovs equation and its first integrals. Its finite KdV hierarchy consists of $N$ compatible integrable polynomial dynamical systems in $mathbb{C}^{2N}$. Then we discuss a non-commutative version of the $N$-th Novikovs equation defined on a finitely generated free associative algebra $mathfrak{B}_N$ with $2N$ generators. In $mathfrak{B}_N$, for $N=1,2,3,4$, we have found two-sided homogeneous ideals $mathfrak{Q}_Nsubsetmathfrak{B}_N$ (quantisation ideals) which are invariant with respect to the $N$-th Novikovs equation and such that the quotient algebra $mathfrak{C}_N = mathfrak{B}_Ndiagup mathfrak{Q}_N$ has a well defined Poincare-Birkhoff-Witt basis. It enables us to define the quantum $N$-th Novikovs equation on the $mathfrak{C}_N$. We have shown that the quantum $N$-th Novikovs equation and its finite hierarchy can be written in the standard Heisenberg form.
We study the spectral (linear) stability and orbital (nonlinear) stability of the elliptic solutions for the focusing modified Korteweg-de Vries (mKdV) equation with respect to subharmonic perturbations and construct the corresponding breather solutions to exhibit the unstable or stable dynamic behavior. The elliptic function solutions of mKdV equation and the fundamental solutions of Lax pair are exactly represented by using the theta function. Based on the `modified squared wavefunction (MSW) method, we construct all linear independent solutions of the linearized KdV equation, and then provide a necessary and sufficient condition of the spectral stability for the elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stable, the orbital stability of the elliptic function solutions with respect to subharmonic perturbations is established under a suitable Hilbert space. Using Darboux-Backlund transformation, we construct the breather solutions to exhibit the unstable or stable dynamic behavior. Through analyzing the asymptotical behavior, we find the breather solution under the $mathrm{cn}$-background is equivalent to the elliptic function solution adding a small perturbation as $ttopminfty$.
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