Results 31 to 40 of about 6,078 (188)

On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems [PDF]

, 2015
Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems.
Santoprete, Manuele
core   +3 more sources

Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Complexity, 2019
An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws.
Qianqian Yang, Qiulan Zhao, Xinyue Li
doaj   +1 more source

Darboux-Integrability and Uniform Convergence

Real Analysis Exchange, 2004
A concept of Darboux-integrability for Banach space-valued functions defined on a basic space \((\Omega, {\mathcal D}, \mu)\) is introduced. In the finite-dimensional case for functions on \([a,b]\) this concept generalizes the Riemann integral, but in the infinite-dimensional case the Darboux-integral is more restrictive than the standard definition ...
openaire   +3 more sources

Integrability of Nonholonomic Heisenberg Type Systems [PDF]

, 2016
We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and classical $r ...
Grigoryev, Yury A., Sozonov, Alexey P., Tsiganov, Andrey V.   +2 more
core   +3 more sources

Effect of Third-Order Dispersion on the Solitonic Solutions of the Schrödinger Equations with Cubic Nonlinearity

Advances in Mathematical Physics, 2014
We derive the solitonic solution of the nonlinear Schrödinger equation with cubic nonlinearity, complex potentials, and time-dependent coefficients using the Darboux transformation.
H. Chachou Samet, M. Benarous, M. Asad-uz-zaman, U. Al Khawaja   +3 more
doaj   +1 more source

Finiteness of integrable $n$-dimensional homogeneous polynomial potentials

, 2007
We consider natural Hamiltonian systems of $n>1$ degrees of freedom with polynomial homogeneous potentials of degree $k$. We show that under a genericity assumption, for a fixed $k$, at most only a finite number of such systems is integrable.
Ablowitz, Ablowitz, Furta, Goriely, Grammaticos, Grammaticos, Guillot, Hietarinta, Hietarinta, Iwasaki, Kowalevski, Kowalevski, Kozlov, Lakshmanan, Lyapunov, Maciejewski, Maciejewski, Maciejewski, Maria Przybylska, Morales-Ruiz, Morales-Ruiz, Morales-Ruiz, Nakagawa, Nowicki, Painlevé, Ramani, Ramani, van der Put, Whittaker, Yoshida, Yoshida, Ziglin, Ziglin   +32 more
core   +2 more sources

On integrability aspects of the supersymmetric sine-Gordon equation

, 2017
In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zero-curvature representations which are equivalent to the supersymmetric sine-Gordon equation.
Bertrand, Sébastien
core   +1 more source

Inverse Problems in Darboux’ Theory of Integrability [PDF]

Acta Applicandae Mathematicae, 2012
The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result,
Christopher, Colin, Llibre, Jaume, Pantazi, Chara, Walcher, Sebastian   +3 more
openaire   +5 more sources

Analytical and Numerical Soliton Solutions of the Shynaray II‐A Equation Using the G′G,1G$$ \left(\frac{G^{\prime }}{G},\frac{1}{G}\right) $$‐Expansion Method and Regularization‐Based Neural Networks

Mathematical Methods in the Applied Sciences, EarlyView.
ABSTRACT Nonlinear differential equations play a fundamental role in modeling complex physical phenomena across solid‐state physics, hydrodynamics, plasma physics, nonlinear optics, and biological systems. This study focuses on the Shynaray II‐A equation, a relatively less‐explored parametric nonlinear partial differential equation that describes ...
Aamir Farooq, H. W. A. Riaz, Sadique Rehman, M. Mamun Miah, Wen Xiu Ma   +4 more
wiley   +1 more source

C∞‐Structures for Liénard Equations and New Exact Solutions to a Class of Klein–Gordon Equations

Mathematical Methods in the Applied Sciences, Volume 49, Issue 4, Page 2795-2822, 15 March 2026.
ABSTRACT Liénard equations are analyzed using the recent theory of 𝒞∞‐structures. For each Liénard equation, a 𝒞∞‐structure is determined by using a Lie point symmetry and a 𝒞∞‐symmetry. Based on this approach, a novel method for integrating these equations is proposed, which consists in solving sequentially two completely integrable Pfaffian equations.
Beltrán de la Flor, Adrián Ruiz, Concepción Muriel   +2 more
wiley   +1 more source