The study of the integrability by separation of variables of the Hamilton-Jacobi equations is a classical problem in Mechanics, dating back to the foundational works of Liouville, Jacobi, Stäckel, Levi-Civita and others. Recently this kind integrability received a big attention due to its applications to the theory of integrable partial differential equations of Korteweg de Vries type and to the theory of quantum integrable systems.The most relevant development to the solution of the separation of variables due to Stäckel was to give the general solution in orthogonal coordinates.In the present paper by applying the Nambu bracket and its new properties we developed the Stäckel ideas and state and solve two inverse problems in ordinary differential equations. The first one is related with the problem of the construction of the Stäckel potential from a given special family of curves, and the second one is related with the problem of the construction of the Stäckel matrix from a given Riemann-Stäckel metric and consequently we construct the Stäckel mechanical system.The obtained results are illustrated solving some relevant applications. In particular we solve the second inverse Stäckel problem for the Kepler-Coulomb system, the Stark system in arbitrary dimension and for the first time we introduce and study the Stark-Coloumb system. For these Hamiltonian systems we prove that they are integrable by separation of variables. Moreover we solve the second Stäckel problem in the pseudo-Riemann space of constant curvature and prove that the Stäckel potential in this case is a complete homogeneous symmetric polynomial.
Considered herein is the Zakharov-Ito system modeling shallow water waves over a flat bottom with constant vorticity. We show the existence of the solitary waves in a certain regime of wave speeds. In addition, the orbital stability of such single solitary wave is established via the Grillakis-Shatah-Strauss framework in the energy space X=H1(R)×L2(R). Moreover, we demonstrate that a train of solitary waves, which are sufficiently decoupled, is also orbitally stable in X. The stability proof relies essentially on the modulation argument, the almost monotonicity of functionals, and localized coercivity estimates.
We consider a time fractional chemotaxis model in which the species density and the chemical concentration evolve according to Caputo-Hadamard derivatives of distinct orders. Under suitable hypotheses on the diffusion coefficients and initial data, we prove existence and uniqueness of mild solutions in Lebesgue spaces. We then prove a blow-up alternative and present numerical evidence for finite-time blow-up of the solution. To capture blow-up singularities and quantify the influence of key parameters, we develop an adaptive moving mesh method tailored to the system’s specific dynamics. Numerical experiments are consistent with the analytical blow-up alternative and provide detailed insights into the mechanisms of finite time blow-up in this fractional setting.
In this paper we discuss the traveling waves of the resonant Schrödinger equation with weak non-locality and dual-power law nonlinearity. We focus on the 6 important types of bounded traveling waves such as solitary wave, kink wave, periodic wave, compacton, peakon and periodic peakon and use a tuple of 6 non-negative integers, called wave tuple, to present the numbers of those waves. We give conditions of parameters for various wave tuples and obtain totally 15 different wave tuples for the equation. This provides 4 coexisting modes and shows that the largest coexisting number is 5.
This paper asks how Solomonoff-style program-length weighting can be represented by positive semidefinite kernels and Gaussian-process covariance operators. The answer developed here is the Solomonoff Feature Mixture (SFM), a kernel templatek(x,y)=∫Ωuω(x)uω(y)dπ(ω),π(ω)∝2−ℓ(ω),where uω may be uncentered, prior-centered, or data-centered as defined explicitly below. This includes ideal Solomonoff kernels, D2KE-based KC-kernel surrogates, Occam mixtures, and centered covariance kernels as special cases.The SFM construction induces a Solomonoff Kernel Covariance Operator (SKCO) and, when used as a covariance, classical Gaussian-process objects: Solomonoff Gaussian Processes, Solomonoff Gaussian Hilbert Spaces, and Solomonoff Gaussian Fields. The main result is structural rather than empirical or rate-optimal: under explicit non-degeneracy assumptions, the SKCO spectrum is controlled by program-length statistics, while truncation, landmark, and compressor-based approximations give computable surrogates. The paper also makes explicit the limits of this bridge: Gaussian processes do not reproduce Solomonoff induction at the level of discrete hypothesis probabilities or universal dominance; they preserve a second-order, operator-level form of algorithmic bias.
This paper investigates high-order lump patterns in a novel differential-difference KP equation, derived through the introduction of a new class of trigonometric-type bilinear Hirota operators. Rational solutions are obtained by applying two differential operators to the elements of Gram-type determinants, and are succinctly expressed in terms of Schur polynomials, establishing a direct connection between the lump patterns and Schur functions. Using concepts from integer partitions, we systematically construct these high-order patterns. Furthermore, asymptotic analysis in the large-parameter regime reveals that the distribution of lump centers is analytically governed by the root structures of special polynomials, including the Yablonskii-Vorob’ev, Umemura, Wronskian-Hermite, and Okamoto polynomials.
We present a systematic construction of enhanced Darboux transformations for the Harry Dym equation. The general steps for constructing these transformations are described in detail. For the resulting oscillatory N-cuspon solutions (continuous and with unbounded derivatives), N-soliton solutions (smooth to arbitrary order), and their interactions, compact explicit expressions applicable for analysis and numerical computation are derived by using the Neville theta functions. The effectiveness of our approach is demonstrated through numerical experiments, with an implementation available at https://github.com/lirm-math/osc-dym.git. Using this implementation, oscillatory 10-cuspon and 10-soliton solutions on a 3000 × 3000 grid are obtained within 5 min. Representative plots are also presented.
This work is dedicated to addressing the challenge of constructing exact solutions for discrete equations that possesses Lax pairs and Hirota bilinear representations on oscillatory backgrounds, with a focus on the semi-discrete Korteweg-de Vries (sd-KdV) equation. A systematic method is developed for constructing exact, bounded and real-valued solutions of the sd-KdV equation. First, Darboux transformations are formulated by using Casoratians of the spectral functions. Second, by using the Hirota bilinear representations and the quasiperiodicity of the Neville theta functions, a seed solution of the sd-KdV equation is constructed. Third, the spectral functions of the spectral problems with the potential given by the first step are obtained from Bäcklund transformations. Fourth, the Casoratians of the spectral functions are converted to Hirota summations to the determine the parameters ensuring the reality and the boundedness of the new solutions. Consequently, oscillatory N-soliton solutions of the sd-KdV equation are obtained. Finally, some new solutions are plotted to illustrate these new solutions.
The basin of attraction is one of the most fundamental concepts in modern dynamical systems theory. However, its definition for spatially extended systems, particularly those capable of supporting a wide spectrum of partial synchronization states, remains challenging. In this paper, we analyze a neuronal network that is simple in a configurational sense, developing full and partial synchronization states from specially structured large-scale periodic initial conditions. This approach allows restructuring the basin of attraction within a lower-dimensional space and examine its structure comprehensively. Given the empirically established link between cortical waves, partial synchronization states, and information processing in cortical structures, this study may contribute to a fundamental understanding of functional zone dynamics from the perspective of dynamical systems theory. It may also reveal the coordination mechanisms mediated by effective connectivity.
Nonlinear, integrable Hamiltonian systems offer a promising framework for developing high-intensity particle accelerators. Fermilab’s Integrable Optics Test Accelerator provides a central example of this idea in four-dimensional transverse phase space. In this work, we study a direct extension of this framework to six-dimensional phase space, including longitudinal motion and acceleration, under a specified set of Hamiltonian approximations and electromagnetic field assumptions. We search for additional invariants that are at most quadratic in the canonical momenta, motivated by the Courant-Snyder invariants and by the quadratic invariants appearing in transverse integrable-optics constructions. Within this restricted setting, we find that the known transverse quadratic-in-momenta invariant structure does not persist under the six-dimensional extension considered here. We also construct invariants for a purely longitudinal reduction of the model in the extended autonomous Hamiltonian formulation. These results should be interpreted as a restricted obstruction to one natural extension of transverse integrable optics, rather than as a general nonexistence theorem for six-dimensional integrable accelerator Hamiltonians.
In this paper, using the reciprocal transformation and the associated Camassa-Holm equation, we introduce a new approach to construct two distinct types of Darboux transformations for the Camassa-Holm equation. Furthermore, we derive a general formula for the composition of N Bäcklund transformations for the Camassa-Holm equation, expressed in terms of determinants. Notably, the first type of Darboux transformation is shown to be equivalent to the composition of N Bäcklund transformations. Additionally, we develop a generalized binary Darboux transformation incorporating an arbitrary time-dependent function for the Camassa-Holm equation with self-consistent sources.
We construct explicit, closed-form algebro-geometric solutions of the Jaulent–Miodek equation in terms of genus-two hyperelliptic ℘-functions and present numerical visualizations of these solutions. To resolve the long-standing inversion problem for even-degree hyperelliptic curves, we introduce an explicit birational transformation from the original sextic spectral curve to an associated odd-degree quintic curve. This geometric bridge yields the exact hyperelliptic parametrization of the Viète variables associated with the divisors. By incorporating the nonlinearization of the Lax pair, the Jaulent–Miodek equation is decomposed into a pair of compatible finite-dimensional Hamiltonian systems and linearized on the Jacobian variety to produce the exact solutions. Finally, to investigate the localized dynamical evolution without numerical singularities, we compute the solution profiles using the classical fourth-order Runge–Kutta method formulated in terms of desingularized local parameters.
In this paper, we investigate the Cauchy problem for a nonlinear heat equation driven by the Grushin operator and involving both instantaneous reaction and nonlocal memory effects u(t) - Delta(g)u = k(1) integral(t) (0 ) ( t-s)(-gamma )|u(s)|p(1 )ds + k(2) |u(t)|p(2) (z,t) is an element of R N+k x (0,infinity ) with locally integrable initial data u(z, 0) = epsilon u(0)(z), where gamma is an element of (0, 1), k(1), k(2) >= 0,( p1, p2) > 1, and epsilon > 0 is a small parameter. The Grushin operator Delta(G) encodes anisotropic and degenerate diffusion, while the integral term represents a long-memory nonlinear feedback with a power-law kernel. We aim to derive upper estimates for the lifespan of solutions, interpreted as the maximal time of existence before blow-up, which in physical terms corresponds to the onset of a finite-time instability driven by nonlinear amplification. Our analysis highlights the interplay between the classical Fujita exponent and the effects of the nonlocal memory term. Using a combination of the standard test function method and the refined approach recently developed by Ikeda and Sobajima in [Nonlinear Anal. 182 (2019) 57-74], we show that when k(1 )= 0, the lifespan is governed by the classical Fujita exponent, while the presence of the memory term modifies the lifespan according to the scaling-predicted exponent. Our results extend classical Fujita-type phenomena to the degenerate Grushin setting with nonlocal memory effects, providing a detailed analysis of the lifespan estimates.
