可精确求解与可积系统

Inspired by the recent developments in the study of the thermodynamics of van der Waals fluids via the theory of nonlinear conservation laws and the description of phase transitions in terms of classical (dissipative) shock waves, we…

We give a simple form of the evolution equation and a scalar Lax pair for the $q$-Garnier system. Some reductions to the $q$-Painlev\'e equations and the autonomous case as a generalized QRT system are discussed. Using two kinds of Pad\'e…

The B\"acklund transformation (BT) for the Camassa-Holm (CH) equation is presented and discussed. Unlike the vast majority of BTs studied in the past, for CH the transformation acts on both the dependent and (one of) the independent…

Using exact formulae for the scattering data of the Benjamin-Ono equation valid for general rational potentials recently obtained by Miller and Wetzel (2015), we rigorously analyze the scattering data in the small-dispersion limit. In…

We study integrable Euler equations on the Lie algebra $\mathfrak{gl}(3,\mathbb{R})$ by interpreting them as evolutions on the space of hexagons inscribed in a real cubic curve.

In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric $\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out…

We consider two Lax systems for the homogeneous Painlev\'{e} II equation: one of size $2\times 2$ studied by Flaschka and Newell in the early 1980's, and one of size $4\times 4$ introduced by Delvaux-Kuijlaars-Zhang and Duits-Geudens in the…

In this work we classify the fifth-order integrable symmetrically coup led systems of weight 0 that possess seventh-order symmetry. we obtained 2 new integrable systems that related bi-Hamiltonian formulations are constructed too.

In this work we develop some fifth-order integrable coupled systems of weight $0$ and $1$ which possess seventh-order symmetry. We establish four new systems, where in some cases, related recursion operator and bi-Hamiltonian formulations…

Li\'enard-type equations are used for the description of various phenomena in physics and other fields of science. Here we find a new family of the Li\'enard-type equations which admits a non-standard autonomous Lagrangian. As a by-product…

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schr\"{o}dinger equation (AKNS$_r$)\,($r=1,2$). Using the dressing transformation (DT) method and the related tau functions we study the…

We develop a uniform semiclassical trace formula for the density of states of a three-dimensional isotropic harmonic oscillator (HO), perturbed by a term $\propto r^4$. This term breaks the U(3) symmetry of the HO, resulting in a spherical…

We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term…

Painlev\'{e} test of the coupled Gross-Pitaevskii equations has been carried out with the result that the coupled equations pass the P-test only if a special relation containing system parameters (masses, scattering lengths) is satisfied.…

The connection between the complex Sine and Sinh-Gordon equations on the complex plane associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We…

The general method of the cojmplex supersymmetrization (supercomplexifications) of the soliton equations with the odd (bi) hamiltoninan structure is established. New version of the supercomplexified Kadomtsev-Petvishvili hierarchy is given.…

We construct a family of integrable equations of the form $v_t=f(v,v_x,v_{xx},v_{xxx})$ such that $f$ is a transcendental function in $v,v_x,v_{xx}$. This family is related to the Krichever-Novikov equation by a differential substitution.…

In this paper, we analyze the long-time behavior of the solution of the initial value problem (IVP) for the short pulse (SP) equation. As the SP equation is a complete integrable system, which posses a Wadati-Konno-Ichikawa (WKI)-type Lax…

In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms…

Using the Chiellini condition for integrability we derive explicit solutions for a generalized system of Riccati equations $\ddot{x}+\alpha x^{2n+1}\dot{x}+x^{4n+3}=0$ by reduction to the first-order Abel equation assuming the parameter…