Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations

We consider (3+1)-dimensional second-order evolutionary PDEs where the unknown $u$ enters only in the form of the 2nd-order partial derivatives. For such equations which possess a Lagrangian, we show that all of them have a symplectic Monge--Amp\`ere form and determine their Lagrangians. We develop a calculus for transforming the symmetry condition to a "skew-factorized" form from which we immediately extract Lax pairs and recursion relations for symmetries, thus showing that all such equations are integrable in the traditional sense. We convert these equations together with their Lagrangians to a two-component form and obtain recursion operators in a $2\times 2$ matrix form. We transform our equations from Lagrangian to Hamiltonian form by using the Dirac's theory of constraints. Composing recursion operators with the Hamiltonian operators we obtain the second Hamiltonian form of our systems, thus showing that they are bi-Hamiltonian systems integrable in the sense of Magri. By this approach, we obtain five new bi-Hamiltonian multi-parameter systems in (3+1) dimensions.Comment: 38 pages, LaTeX2e. Section 6 is modified and errors fixed. Section 9 added. This paper studies 4-dimensional PDEs, generalizing the paper arXiv:1712.01549 on 3-dimensional PDEs, but the method, though being essentially developed, is similar. This causes some text overlap with arXiv:1712.0154

Evolutionary Hirota Type (2+1)-Dimensional Equations: Lax Pairs, Recursion Operators and Bi-Hamiltonian Structures

We show that evolutionary Hirota type Euler-Lagrange equations in (2+1) dimensions have a symplectic Monge-Amp\`ere form. We consider integrable equations of this type in the sense that they admit infinitely many hydrodynamic reductions and determine Lax pairs for them. For two seven-parameter families of integrable equations converted to two-component form we have constructed Lagrangians, recursion operators and bi-Hamiltonian representations. We have also presented a six-parameter family of tri-Hamiltonian systems

Method of group foliation, hodograph transformation and non-invariant solutions of the Boyer-Finley equation

We present the method of group foliation for constructing non-invariant solutions of partial differential equations on an important example of the Boyer-Finley equation from the theory of gravitational instantons. We show that the commutativity constraint for a pair of invariant differential operators leads to a set of its non-invariant solutions. In the second part of the paper we demonstrate how the hodograph transformation of the ultra-hyperbolic version of Boyer-Finley equation in an obvious way leads to its non-invariant solution obtained recently by Manas and Alonso. Due to extra symmetries, this solution is conditionally invariant, unlike non-invariant solutions obtained previously. We make the hodograph transformation of the group foliation structure and derive three invariant relations valid for the hodograph solution, additional to resolving equations, in an attempt to obtain the orbit of this solution.Comment: to appear in the special issue of Theor. Math. Phys. for the Proceedings of NEEDS2002; Keywords: Heavenly equation, group foliation, non-invariant solutions, hodograph transformatio

Debt Deception: How Debt Buyers Abuse the Legal System to Prey on Lower-Income New Yorkers

In this report, we examine lawsuits filed by debt buyers and their profound impact on low- and moderate-income New Yorkers, lower-income communities, and communities of color. We begin, in Part I, with background on the debt buying industry, including an analysis of the debt buyer business model and collection methods. Part II focuses on debt buyer lawsuits, particularly the systemic problems at the root of these lawsuits. In Part III, we highlight specific findings from a study of debt buyer lawsuits in New York City. We draw results from two data sets: (1) a 365-case sample of lawsuits brought by the 26 debt buyers who filed the greatest number of cases in New York City between January 2006 and July 2008 ("Court Sample"); and (2) a 451-case sample of callers to NEDAP's legal hotline who were sued by a creditor or debt buyer in 2008 ("Client Sample"). Finally, in Part IV, we recommend policy and legislative reforms to address the problems documented in this repor

Recursions of Symmetry Orbits and Reduction without Reduction

We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Amp\`ere equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat K\"ahler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation

Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Pleba\'nski

We present first heavenly equation of Pleba\'nski in a two-component evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integrals of motion corresponding to each variational (Noether) symmetry. We derive two linearly independent recursion operators for symmetries of this system related by a discrete symmetry of both the two-component system and its symmetry condition. Acting by these operators on the first Hamiltonian operator $J_0$ we obtain second and third Hamiltonian operators. However, we were not able to find Hamiltonian densities corresponding to the latter two operators. Therefore, we construct two recursion operators, which are either even or odd, respectively, under the above-mentioned discrete symmetry. Acting with them on $J_0$, we generate another two Hamiltonian operators $J_+$ and $J_-$ and find the corresponding Hamiltonian densities, thus obtaining second and third Hamiltonian representations for the first heavenly equation in a two-component form. Using P. Olver's theory of the functional multi-vectors, we check that the linear combination of $J_0$, $J_+$ and $J_-$ with arbitrary constant coefficients satisfies Jacobi identities. Since their skew symmetry is obvious, these three operators are compatible Hamiltonian operators and hence we obtain a tri-Hamiltonian representation of the first heavenly equation. Our well-founded conjecture applied here is that P. Olver's method works fine for nonlocal operators and our proof of the Jacobi identities and bi-Hamiltonian structures crucially depends on the validity of this conjecture.Comment: Some text overlap with our paper arXiv:1510.03666 is caused by our use here of basically the same method for discovering the Hamiltonian and bi-Hamiltonian structures of the equation, but the equation considered here and the results are totally different from arXiv:1510.0366