Yasuhiro Ohta(Kobe University, Japan)
The periodic phase soliton (PPS) is a recently discovered wave for the ultradiscrete hungry Lotka-Volterra (uhLV) equation. It is a traveling wave whose shape keeps changing periodically. Each soliton has its internal freedom describing the periodic structure. We construct discrete coupled soliton equations which admit the PPS solutions, by applying a reduction to the DKP hierarchy. The solutions are expressed in terms of two-component Pfaffians. Through the ultradiscrete limit, we also construct the ultradiscrete coupled system and its PPS solutions from which the PPS solutions of uhLV equation are recovered. This is a joint work with H. Nagai and R. Hirota
Gino Biondini (SUNY Buffalo)
The dynamics of self-focusing media with non-zero boundary conditions has attracted renewed attention in recent years in part thanks to its possible connection to the formation of rogue waves. This talk will discuss recent progress in this area. The most common tool to study self-focusing media is the nonlinear Schrödinger (NLS) equation. In the first part of the talk I will review the inverse scattering transform (IST) for the focusing NLS equation with non-zero background. Then I will discuss how the IST can be successfully used to study the nonlinear stage of modulational instability (MI). Recall that MI (a.k.a. Benjamin-Feir instability in deep water waves) is the instability of a constant background to long-wavelength perturbations, and is a ubiquitous nonlinear phenomenon. Even though MI was discovered in the 1960's, a characterization of the nonlinear stage of MI (i.e., the behavior of solutions once the perturbations have become comparable with the background) was missing until recently. I will show how one can identify the signature of MI within the IST. Contrary to a recent conjecture, the main mechanism for the instability is not the formation of solitons, but rather the exponential growth of the reflection coefficient along a certain portion of the continuous spectrum, which provides the precise nonlinear analogue of the unstable Fourier modes. Then I will show how one can characterize the nonlinear stage of MI by using the IST to compute the long-time asymptotics of solutions of the NLS equation with localized perturbations of the constant background. At large times, the space-time plane divides into three regions: a far left field and far right field and a central region. In the left far field and right far field the solution equals the background to leading order up to a phase. In the central wedge, the solution consists of a coherent oscillation structure comprised of a slow modulation of the periodic traveling wave solutions of the focusing NLS equation. I will also show that this kind of behavior is not limited to the NLS equation, but is instead shared by many different nonlinear models that exhibit MI (including several PDEs, nonlocal systems and differential-difference equations). Finally, I will briefly discuss solutions arising in more general scenarios, including soliton interactions and interactions between solitons and oscillatory wedge.
Kenichi Maruno (Waseda University, Japan)
We discuss soliton interactions of the KP and DKP (coupled KP) equations and show the network diagrams corresponding to line soliton solutions of the KP and DKP equations. We show that the network diagrams of the DKP line soliton solutions is a generalization of the network diagrams of the KP line solitons. We show that the DKP tau-function becomes the KP tau-function when the generalized network diagram is reduced to the KP network diagram. This is a joint work with Shinya Kido (Waseda), Yuta Tanaka (Waseda), Yasuyuki Watanabe (Waseda) and Saburo Kakei (Rikkyo).
Peter Miller (Univ. of Michigan, USA)
This talk will survey two recent studies of solutions of the Painlevé-III equation. First, we describe work with T. Bothner and Y. Sheng on the asymptotic behavior of rational solutions of the Painlevé-III equation in the limit where the degree is large. Then we turn to a class of rational solutions of a different equation, namely the focusing nonlinear Schrödinger equation, which are believed to model rogue waves. In work with D. Bilman and L. Ling we studied the fundamental rogue wave solutions in the limit of large order, and found a new transcendental solution of the focusing nonlinear Schrödinger equation that we call the rogue wave of infinite order. This solution turns out to also satisfy ordinary differential equations in the Painlevé-III hierarchy.
Babara Prinari (University of Colorado-Colorado Springs)
Soliton solutions of the focusing Ablowitz-Ladik (AL) equation with nonzero boundary conditions at infinity are derived within the framework of the inverse scattering transform (IST). After reviewing the relevant aspects of the direct and inverse problems, explicit soliton solutions will be discussed which are the discrete analog of the Tajiri-Watanabe and Kutznetsov-Ma solutions to the focusing NLS equation on a finite background. Then, by performing suitable limits of the above solutions, discrete analog of the celebrated Akhmediev and Peregrine solutions will also be presented. These solutions, which had been recently derived by direct methods, are obtained for the first time within the framework of the IST, thus providing a spectral characterization of the solutions and a description of the singular limit process.
Vladimir Dragovic (UT Dallas)
A new method to construct algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve represented as a ramified double covering of CP^1, a meromorphic differential is constructed with the following property: the common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of a Painleve VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve VI equations. A generalization of this differential to hyperelliptic curves is also constructed. The corresponding solutions of the rank two Schlesinger systems associated with elliptic and hyperelliptic curves are constructed in terms of this differential. The initial data for construction of the meromorphic differential include a point in the Jacobian of the curve, under the assumption that this point has nonvariable coordinates with respect to the lattice of the Jacobian while the branch points vary. The research has been partially supported by the NSF grant 1444147. This is joint work with Vasilisa Shramchenko.
Junchao Chen (Lishui University, China)
we will report Gram determinant solutions of local and nonlocal integrable discrete nonlinear Schrödinger (IDNLS)
equations via a pair reduction. A generalized IDNLS equation is firstly introduced which possesses
the single Casorati determinant solution. Two kinds of Gram determinant solutions are presented from Casorati
determinant ones due to the gauge freedom. The different pair constraint conditions for wave numbers are imposed
and then solutions of local and nonlocal IDNLS equations are derived in terms of Gram determinant.
Eleftherios Gkioulekas (University of Texas Rio Grande Valley)
In the two-layer quasi-geostrophic model, the friction between the flow at the bottom layer and the surface layer, placed beneath the bottom layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the bottom layer. The Ekman term appears in the governing equations asymmetrically, it is placed at the bottom layer, but does not appear at the top layer. A variation, proposed by Salmon, uses extrapolation to place the Ekman term between the bottom layer and the surface layer, or at the surface layer. We present theoretical results that show that in either the standard or the extrapolated configurations, the Ekman term dissipates energy at large scales, but does not dissipate potential enstrophy. It also creates an almost symmetric stable distribution of potential enstrophy between the two layers. The behavior of the Ekman term changes fundamentally at large wavenumbers. Under the standard formulation, the Ekman term will dissipate both energy and potential enstrophy unconditionally at large wavenumbers. However, under the extrapolated formulation, there exist small ”negative regions”, which are defined over a twodimensional phase space, capturing the distribution of energy per wavenumbee between baroclinic energy and barotropic energy, and the distribution of potential enstrophy per wavenumber between the top layer and the bottom layer, where the Ekman term may inject energy or potential enstrophy.
Sergey Grigorian (University of Texas Rio Grande Valley)
G2-structures on 7-dimensional manifolds play a very important role in both geometry and physics. One of the ways of better understanding the relationships between different types of G2-structures is to study their flows. A natural flow of G2-structures is an analog of the heat equation. However since the Laplacian is determined by the G2-structure itself, this becomes a nonlinear PDE. Here we analyze such a flow when the 7-manifold is a warped product of a 6-dimensional Calabi-Yau or nearly Khler manifold and either a circle or an interval, in which case the equations can be written out explicitly as a system of nonlinear PDEs with three dependent and two independent variables. We then look at the soliton equations for this system, where in some cases explicit non-trivial solutions are obtained.
Brandt Kronholm (University of Texas Rio Grande Valley)
In this presentation we will discuss several recent results on partitions restricted by both the number and sizes of parts. Our emphasis will be on the methods used to obtain these results. We will begin by offering two q-series proofs of the following fact: The number of partitions of n into at most four parts, denoted p(n, 4), is divisible by 3 at least 50% of the time. However, the second proof will make use of a novel approach revealing far more information about the result than the first proof.
These methods have been used to provide a new decomposition of partitions, which in turn allows us to define statistics called supercranks that combinatorially witness every instance of divisibility of p(n, 3) by any prime of the form 6j − 1.
If time allows, we will discuss recent results and conjectures on the coefficients of Gaussian polynomials which are known as partitions of n into at most m parts, no part larger than N and denoted by p(n, m, N).
Arturo Martinez (University of Texas Rio Grande Valley)
The goal of this presentation is to show that for a fixed m, all Gaussian polynomials [N+m m ] come in exactly 2lcm(m) m varieties, where lcm(m) represents the least common multiple of the numbers 1 through m. The set of partitions of n into at most m parts, p(n, m), can be decomposed into the sum of two collections; partitions with parts not larger than N, denoted p(n, m, N), and partitions with parts larger than N, denoted P(n, m, N). It is well known that the quasipolynomial for p(n, m) is periodic with period lcm(m). The period of P(n, m, N) is shorter, and strangely the quasipolynomial for p(n, m, N) appears to not be periodic at all. We will discuss these observations and other behavior of these functions.”
Karen Yagdjian (University of Texas Rio Grande Valley)
In this talk we will present the integral transform that allows us to construct solutions of the hyperbolic partial differential equation with variable coefficients via solutions of a simpler equation. This transform was suggested by the author in the case when the last equation is a wave equation. Then it was used to investigate several well-known equations such as generalized Tricomi equation, the KleinGordon equation of the quantum field theory in the de Sitter and Einstein-de Sitter space-times of the expanding universe. In particular it was shown that a field with the mass √ 2 is Huygensian. Moreover, the numbers √ 2, 0 are the only values of the mass such that equation obeys an incomplete Huygens’ Principle. Then, it was shown that in the de Sitter space-time the existence of two different scalar fields (with mass √ 2 and 0 ), which obey incomplete Huygens’ principle, is equivalent to the condition that the spatial dimension of the physical world is 3. In fact, Paul Ehrenfest in 1917 addressed the question: “Why has our space just three dimensions?”. In this talk a special attention will be also given to the global in time existence of self-interacting scalar field in the de Sitter universe and to the Higuchi bound of the quantum field theory and equations with the Higgs potential.
The talk is based on the contents of following publications:
[1] K.Y; A. Galstian: Comm. Math. Phys. 285 (2009), no. 1, 293–344. [2] K.Y.: Comm. Partial Differential Equations 37 (2012), no. 3, 447–478. [3] K.Y.: J. Math. Phys. 54 (2013), no. 9, 091503, 18 pp. [4] K.Y.: J. Differential Equations 259 (2015), no. 11, 59275981. [5] K.Y.: Math. Nachr. 288 (2015), no. 17-18, 2129–2152. [6] A. Galstian; K.Y.:: Global solutions for semilinear Klein-Gordon equations in FLRW spacetimes. Nonlinear Anal. 113 (2015), 339–356 [7] A. Galstian; K.Y.: Nonlinear Anal. Real World Appl. 34 (2017), 110–139.