CyberSec Research

Using a general result of bidifferential calculus and recent results of other authors, a vectorial binary Darboux transformation is derived for the first member of the "negative" part of the potential Kaup-Newell hierarchy, which is a system of two coupled Fokas-Lenells equations. Miura transformations are found from the latter to the first member of the negative part of the AKNS hierarchy and also to its "pseudodual". The reduction to the Fokas-Lenells equation is implemented and exact solutions with a plane wave seed generated.

According to the classification of integrable complex Monge-Ampere equations by Doubrov and Ferapontov, the modified heavenly equation is a typical (3+1)-dimensional dispersionless and canonical integrable equation.In this paper we use the eigenfunctions of the Doubrov-Ferapontov modified heavenly equation to obtain a related hierarchy. Next we construct the Lax-Sato equations with Hamiltonian vector fields and Zakharov-Shabat type equations which are equivalent to the hierarchy. The nonlinear Riemann-Hilbert problem is also applied to study the solution of Doubrov-Ferapontov modified heavenly equation.

Weconsider Burgers equation on metric graphs for simplest topologies such as star, loops, and tree graphs. Exact traveling wave solutions are obtained for the vertex boundary conditions providing mass conservation and continuity of the solution at the nodes. Constraints for the nonlinearity coefficients ensuring integrability of the Burgers equation are derived. Numerical treatment of the soliton dynamics and their transmission through the graph vertex is presented.

We present three equivalence classes of rational non-invertible multidimensional compatible maps. These maps turns out to be idempotent and by construction they admit birational partial inverses (companion maps) which are Yang-Baxter maps. The maps in question can be reinterpreted as systems of difference equations defined on the edges of the $\mathbb{Z}^2$ graph. Finally, we associate these compatible systems of difference equations with integrable difference equations defined on the triangular lattice $Q(A2)$.

We propose a generalization of the BPS Skyrme model for simple compact Lie groups $G$ that leads to Hermitian symmetric spaces. In such a theory, the Skyrme field takes its values in $G$, while the remaining fields correspond to the entries of a symmetric, positive, and invertible $\dim G \times \dim G$-dimensional matrix $h$. We also use the holomorphic map ansatz between $S^2 \rightarrow G/H \times U(1)$ to study the self-dual sector of the theory, which generalizes the holomorphic ansatz between $S^2 \rightarrow CP^N$. This ansatz is constructed using the fact that stable harmonic maps of the two $S^2$ spheres for compact Hermitian symmetric spaces are holomorphic or anti-holomorphic. Apart from some special cases, the self-duality equations do not fix the matrix $h$ entirely in terms of the Skyrme field, which is completely free, as it happens in the original self-dual Skyrme model for $G=SU(2)$. In general, the freedom of the $h$ fields tend to grow with the dimension of $G$. The holomorphic ansatz enable us to construct an infinite number of exact self-dual Skyrmions for each integer value of the topological charge and for each value of $N \geq 1$, in case of the $CP^N$, and for each values of $p,\,q\geq 1$ in case of $SU(p+q)/SU(p)\otimes SU(q)\otimes U(1)$.

Integrable differential identities, together with ensemble-specific initial conditions, provide an effective approach for the characterisation of relevant observables and state functions in random matrix theory. We develop the approach for the unitary and orthogonal ensembles. In particular, we focus on a reduction where the probability measure is induced by a Hamiltonian expressed as a formal series of even interactions. We show that the order parameters for the unitary ensemble, that is associated with the Volterra lattice, solve the modified KP equation. The analogous reduction for the orthogonal ensemble, associated with the Pfaff lattice, leads to a new integrable chain. A key step for the calculation of order parameters solution for the orthogonal ensemble is the evaluation of the initial condition by using a map from orthogonal to skew-orthogonal polynomials. The thermodynamic limit leads to an integrable system (a chain for the orthogonal ensemble) of hydrodynamic type. Intriguingly, we find that the solution to the initial value problem for both the discrete system and its continuum limit are given by the very same semi-discrete dynamical chain.

We present novel Hamiltonian descriptions of some three-dimensional systems including two well-known systems describing the three-wave-interaction problem and some well-known chaotic systems, namely, the Chen, L\"u, and Qi systems. We show that all of these systems can be described in a Hamiltonian framework in which the Poisson matrix $\mathcal{J}$ is supplemented by a resistance matrix $\mathcal{R}$. While such resistive-Hamiltonian systems are manifestly non-conservative, we construct higher-degree Poisson matrices via the Jordan product as $\mathcal{N} = \mathcal{J} \mathcal{R} + \mathcal{R} \mathcal{J}$, thereby leading to new bi-Hamiltonian systems. Finally, we discuss conformal Hamiltonian dynamics on Poisson manifolds and demonstrate that by appropriately choosing the underlying parameters, the reduced three-wave-interaction model as well as the Chen and L\"u systems can be described in this manner where the concomitant non-conservative part of the dynamics is described with the aid of the Euler vector field.

We present an exact analytic solution for incompressible turbulent mixing described by 3D NS equations, with a passive scalar (concentration, temperature, or other scalar field) driven by the turbulent velocity field. Using our recent solution of decaying turbulence in terms of the Euler ensemble, we represent the correlation functions of a passive scalar as statistical averages over this ensemble. The statistical limit, corresponding to decaying turbulence, can be computed in quadrature. We find the decay spectrum and the scaling functions of the pair correlation and match them with physical and real experiments.

We present several known solutions to the two-dimensional Ising model. This review originated from the ``Ising 100'' seminar series held at Bo\u{g}azi\c{c}i University, Istanbul, in 2024.

Nonlinear stage of higher-order modulation instability (MI) phenomena in the frame of multicomponent nonlinear Schr\"odinger equations (NLSEs) are studied analytically and numerically. Our analysis shows that the $N$-component NLSEs can reduce to $N-m+1$ components, when $m(\leq N)$ wavenumbers of the plane wave are equal. As an example, we study systematically the case of three-component NLSEs which cannot reduce to the one- or two-component NLSEs. We demonstrate in both focusing and defocusing regimes, the excitation and existence diagram of a class of nondegenerate Akhmediev breathers formed by nonlinear superposition between several fundamental breathers with the same unstable frequency but corresponding to different eigenvalues. The role of such excitation in higher-order MI is revealed by considering the nonlinear evolution starting with a pair of unstable frequency sidebands. It is shown that the spectrum evolution expands over several higher harmonics and contains several spectral expansion-contraction cycles. In particular, abnormal unstable frequency jumping over the stable gaps between the instability bands are observed in both defocusing and focusing regimes. We outline the initial excitation diagram of abnormal frequency jumping in the frequency-wavenumber plane. We confirm the numerical results by exact solutions of multi-Akhmediev breathers of the multi-component NLSEs.

We investigate the Painlev\'{e} asymptotics for the Cauchy problem of the modified Camassa-Holm (mCH) equation with zero boundary conditions \begin{align*}\nonumber &m_t+\left((u^2-u_x^2)m\right)_x=0, \ (x,t)\in\mathbb{R}\times\mathbb{R}^+,\\ &u(x,0)=u_0(x), \lim_{x\to\pm\infty} u_0(x)=0, \end{align*} where $u_0(x)\in H^{4,2}(\mathbb{R})$. Recently, Yang and Fan (Adv. Math. 402, 108340 (2022)) reported the long-time asymptotic result for the mCH equation in the solitonic regions. The main purpose of our work is to study the long-time asymptotic behavior in two transition regions. The key to proving this result is to establish and analyze the Riemann-Hilbert problem on a new plane $(y;t)$ related to the Cauchy problem of the mCH equation. With the $\bar{\partial}$-generalization of the Deift-Zhou nonlinear steepest descent method and double scaling limit technique, in two transition regions defined by \begin{align}\nonumber \mathcal{P}_{I}:=\{(x,t):0\leqslant |\frac{x}{t}-2|t^{2/3}\leqslant C\},~~~~\mathcal{P}_{II}:=\{(x,t):0\leqslant |\frac{x}{t}+1/4|t^{2/3}\leqslant C\}, \end{align} where $C>0$ is a constant, we find that the leading order approximation to the solution of the mCH equation can be expressed in terms of the Painlev\'{e} II equation.

In this work, we prove that shifted nonlocal reductions of integrable $(2+1)$-dimensional $5$-component Maccari system are particular cases of shifted scale transformations. We present all shifted nonlocal reductions of this system and obtain new two-place and four-place integrable systems and equations. In addition to that we use the Hirota direct method and obtain one-soliton solution of the $5$-component Maccari system. By using the reduction formulas with the solution of the Maccari system we also derive soliton solutions of the shifted nonlocal reduced Maccari systems and equations. We give some particular examples of solutions with their graphs.

This paper investigates the classification of solutions satisfying the polynomial energy growth condition near both the origin and infinity to the ${\mathrm SU}(n+1)$ Toda system on the punctured complex plane $\mathbb{C}^*$. The ${\mathrm SU}(n+1)$ Toda system is a class of nonlinear elliptic partial differential equations of second order with significant implications in integrable systems, quantum field theory, and differential geometry. Building on the work of A. Eremenko (J. Math. Phys. Anal. Geom., Volume 3 p.39-46), Jingyu Mu's thesis, and others, we obtain the classification of such solutions by leveraging techniques from the Nevanlinna theory. In particular, we prove that the unitary curve corresponding to a solution with polynomial energy growth to the ${\mathrm SU}(n+1)$ Toda system on $\mathbb{C}^*$ gives a set of fundamental solutions to a linear homogeneous ODE of $(n+1)^{th}$ order, and each coefficient of the ODE can be written as a sum of a polynomial in $z$ and another one in $\frac{1}{z}$.

This paper investigates the properties of the sequence of coefficients $(\b_n)_{n\geq0}$ in the recurrence relation satisfied by the sequence of monic symmetric polynomials, orthogonal with respect to the symmetric sextic Freud weight $$\omega(x; \tau, t) = \exp(-x^6 + \tau x^4 + t x^2), \qquad x \in \mathbb{R}, $$ with real parameters $\tau$ and $t$. We derive a fourth-order nonlinear discrete equation satisfied by $\beta_n$, which is shown to be a special case of {the second} member of the discrete Painlev\'{e} I hierarchy. Further, we analyse differential and differential-difference equations satisfied by the recurrence coefficients. The emphasis is to offer a comprehensive study of the intricate evolution in the behaviour of these recurrence coefficients as the pair of parameters $(\tau,t)$ change. A comprehensive numerical and computational analysis is carried out for critical parameter ranges, and graphical plots are presented to illustrate the behaviour of the recurrence coefficients as well as the complexity of the associated Volterra lattice hierarchy. The corresponding symmetric sextic Freud polynomials are shown to satisfy a second-order differential equation with rational coefficients. The moments of the weight are examined in detail, including their integral representations, differential equations, and recursive structure. Closed-form expressions for moments are obtained in several special cases, and asymptotic expansions for the recurrence coefficients are provided. The results highlight rich algebraic and analytic structures underlying the symmetric sextic Freud weight and its connections to integrable systems.

Recent studies by Copetti, C\'ordova and Komatsu have revealed that when non-invertible symmetries are spontaneously broken, the conventional crossing relation of the S-matrix is modified by the effects of the corresponding topological quantum field theory (TQFT). In this paper, we extend these considerations to $(1+1)$-dimensional quantum field theories (QFTs) with boundaries. In the presence of a boundary, one can define not only the bulk S-matrix but also the boundary S-matrix, which is subject to a consistency condition known as the boundary crossing relation. We show that when the boundary is weakly-symmetric under the non-invertible symmetry, the conventional boundary crossing relation also receives a modification due to the TQFT effects. As a concrete example of the boundary scattering, we analyze kink scattering in the gapped theory obtained from the $\Phi_{(1,3)}$-deformation of a minimal model. We explicitly construct the boundary S-matrix that satisfies the Ward-Takahashi identities associated with non-invertible symmetries.

Schur functions satisfy the relative Pl\"ucker relations which describe the projective embedding of the flag varieties and the Hirota bilinear equations for the modified KP hierarchies. These relative Pl\"ucker relations are generalized to the skew Schur functions.

We primarily study concave-downward and convex-upward types of elliptic dark soliton solutions for the Hirota equation, exhibiting a concave-downward shape on both upper and lower envelope surfaces and showing a convex-upward shape on the lower envelope surface, respectively. By analyzing the supremum and infimum of solutions, we provide the existence conditions for these two types of elliptic dark solitons. Additionally, we study two-elliptic dark soliton solutions combining both types with the same velocity and investigate the elastic collisions between these two types of solutions with different velocities.

The partial integrability of the Kuramoto model is often thought to be restricted to identically connected oscillators or groups thereof. Yet, the exact connectivity prerequisites for having constants of motion on more general graphs have remained elusive. Using spectral properties of the Koopman generator, we derive necessary and sufficient conditions for the existence of distinct constants of motion in the Kuramoto model with heterogeneous phase lags on any weighted, directed, signed graph. This reveals a broad class of network motifs that support conserved quantities. Furthermore, we identify Lie symmetries that generate new constants of motion. Our results provide a rigorous theoretical application of Koopman's framework to nonlinear dynamics on complex networks.