Moving boundary problems for an extended Dym equation. Reciprocal connection

Stefan-type moving boundary problems are investigated for an extended Dym equation originally introduced in work of Camassa and Holm. Reduction is made to an associated class of moving boundary value problems for the canonical integrable Dym equation and exact solution obtained in terms of Yablonski–Vorob’ev polynomials via a Painlevé II reduction.     

Chaos in Perturbed Planar Non-Hamiltonian Integrable Systems with Slowly-Varying Angle Parameters

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＭｅｌｎｉｋｏｖｍｅｔｈｏｄｆｏｒｄｅｔｅｃｔｉｎｇｃｈａｏｓ[1]ｈａｓｂｅｅｎｅｘｔｅｎｄｅｄｔｏｈｉｇｈ＿ｄｉｍｅｎｓｉｏｎａｌｓｙｓｔｅｍｓｗｉｔｈｓｌｏｗｌｙ＿ｖａｒｙｉｎｇａｎｇｌｅｐａｒａｍｅｔｅｒｓ ,ｂｕｔｔｈｅｃｏｒｒｅｓｐｏｎｄｉｎｇｕｎｐｅｒｔｕｒｂｅｄｉｎｔｅｇｒａｂｌｅｓｙｓｔｅｍｓａｒｅｒｅｑｕｉｒｅｄｔｏｂｅＨａｍｉｌｔｏｎｉａｎ[2 ].Ｆｏｒｐｅｒｔｕｒｂｅｄｐｌａｎａｒｎｏｎ＿Ｈａｍｉｌｔｏｎｉａｎｉｎｔｅｇｒａｂｌｅｓｙｓｔｅｍｓ,ｔｈ…     

Exact traveling wave solutions for an integrable nonlinear evolution equation given by M. Wadati

By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.     

First passage failure of quasi-partial integrable generalized Hamiltonian systems

The first passage failure of quasi-partial integrable generalized Hamiltonian systems is studied by using the stochastic averaging method. First, the stochastic averaging method for quasi-partial integrable generalized Hamiltonian systems is introduced briefly. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are derived from the averaged Itô equations. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained from solving these equations together with suitable initial condition and boundary conditions, respectively. Finally, one example is given to illustrate the proposed procedure in detail and the solutions are confirmed by using the results from Monte Carlo simulation of the original system.     

Darboux transformation for an integrable generalization of the nonlinear Schr?dinger equation

A Darboux transformation for an integrable generalization of the coupled nonlinear Schr?dinger equation is derived with the help of the gauge transformation between the Lax pair. As a reduction, a Darboux transformation for an integrable generalization of the nonlinear Schr?dinger equation is obtained, from which some new solutions for the integrable generalization of the nonlinear Schr?dinger equation are given.     

Horizontal Redistribution of Two Fluid Phases in a Porous Medium: Experimental Investigations

Classical models for flow and transport processes in porous media employ the so-called extended Darcy’s Law. Originally, it was proposed empirically for one-dimensional isothermal flow of an incompressible fluid in a rigid, homogeneous, and isotropic porous medium. Nowadays, the extended Darcy’s Law is used for highly complex situations like non-isothermal, multi-phase and multi-component flow and transport, without introducing any additional driving forces. In this work, an alternative approach by Hassanizadeh and Gray identifying additional driving forces were tested in an experimental setup for horizontal redistribution of two fluid phases with an initial saturation discontinuity. Analytical and numerical solutions based on traditional models predict that the saturation discontinuity will persist, but a uniform saturation distribution will be established in each subdomain after an infinite amount of time. The pressure field, however, is predicted to be continuous throughout the domain at all times and is expected to become uniform when there is no flow. In our experiments, we also find that the saturation discontinuity persists. But, gradients in both saturation and pressure remain in both subdomains even when the flow of fluids stops. This indicates that the identified additional driving forces present in the truly extended Darcy’s Law are potentially significant.     

A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions

Searching for higher-dimensional integrable models is one of the most significant and challenging issues in nonlinear mathematical physics. This paper aims to extend the classic lower-dimensional integrable models to arbitrary spatial dimension. We investigate the celebrated Kadomtsev–Petviashvili (KP) equation and propose its (n+1)-dimensional integrable extension. Based on the singularity manifold analysis and binary Bell polynomial method, it is found that the (n+1)-dimensional generalized KP equation has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, Bäcklund transformation as well as infinite conservation laws, and thus the (n+1)-dimensional generalized KP equation is proven to be completely integrable. Moreover, various types of localized solutions can be constructed starting from the N-soliton solutions. The abundant interactions including overtaking solitons, head-on solitons, one-order lump, two-order lump, breather, breather-soliton mixed solutions are analyzed by some graphs.

     

Smooth soliton and kink solutions for a new integrable soliton equation

Nonlinear Dynamics - We establish a relationship between a new integrable soliton equation and Gardner’s equation by a transformation. Then, we use this transformation and solutions of...     

Coherent structure of Alice–Bob modified Korteweg de-Vries equation

To describe two-place events, Alice–Bob systems have been established by means of the shifted parity and delayed time reversal in the preprint arXiv:1603.03975v2 [nlin.SI], (2016). In this paper, we mainly study exact solutions of the integrable Alice–Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darboux transformation for the AB-mKdV equation is constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton, and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.     

Integration of linear and some model non-linear equations of motion of incompressible fluids

We suggest a new exact method that allows one to construct solutions to a wide class of linear and some model non-linear hydrodynamic-type systems. The method is based on splitting a system into a few simpler equations; two different representations of solutions (non-symmetric and symmetric) are given. We derive formulas that connect solutions to linear three-dimensional stationary and non-stationary systems (corresponding to different models of incompressible fluids in the absence of mass forces) with solutions to two independent equations, one of which being the Laplace equation and the other following from the equation of motion for any velocity component at zero pressure. To illustrate the potentials of the method, we consider the Stokes equations, describing slow flows of viscous incompressible fluids, as well as linearized equations corresponding to Maxwell's and some other viscoelastic models. We also suggest and analyze a differential-difference fluid model with a constant relaxation time. We give examples of integrable non-linear hydrodynamic-type systems. The results obtained can be suitable for the integration of linear hydrodynamic equations and for testing numerical methods designed to solve non-linear equations of continuum mechanics.     

Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r(1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system and a partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi partially integrable Hamiltonian system. In the present paper, the averaged Itô and Fokker-Planck-Kolmogorov (FPK) equations for quasi partially integrable Hamiltonian systems in both cases of non-resonance and resonance are derived. It is shown that the number of averaged Itô equations and the dimension of the averaged FPK equation of a quasi partially integrable Hamiltonian system is equal to the number of independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. The technique to obtain the exact stationary solution of the averaged FPK equation is presented. The largest Lyapunov exponent of the averaged system is formulated, based on which the stochastic stability and bifurcation of original quasi partially integrable Hamiltonian systems can be determined. Examples are given to illustrate the applications of the proposed stochastic averaging method for quasi partially integrable Hamiltonian systems in response prediction and stability decision and the results are verified by using digital simulation.     

Painlevé analysis for a new integrable equation combining the modified Calogero–Bogoyavlenskii–Schiff (MCBS) equation with its negative-order form

A new integrable equation is constructed by combining the recursion operator of the modified Calogero–Bogoyavlenskii–Schiff equation and its inverse recursion operator. The Painlevé is performed to demonstrate the complete integrability of the newly developed equation. Multiple-soliton solutions are depicted as manifestation of the integrability. We further show that this equation enjoys a variety of soliton solutions that include kinks, peakon, cuspon.     

First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r(1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system. A partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi-partially integrable Hamiltonian system. In the present paper, the procedures for studying the first-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems are proposed. First, the stochastic averaging method for quasi-partially integrable Hamiltonian systems is briefly reviewed. Then, based on the averaged Itô equations, a backward Kolmogorov equation governing the conditional reliability function, a set of generalized Pontryagin equations governing the conditional moments of first-passage time and their boundary and initial conditions are established. After that, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and of maximization of mean first-passage time are formulated. The relationship between the backward Kolmogorov equation and the dynamical programming equation for reliability maximization, and that between the Pontryagin equation and the dynamical programming equation for maximization of mean first-passage time are discussed. Finally, an example is worked out to illustrate the proposed procedures and the effectiveness of feedback control in reducing first-passage failure.     

The concave or convex peaked and smooth soliton solutions of Camassa-Holm equation

ＩｎｔｒｏｄｕｃｔｉｏｎＣａｍａｓｓａ ,Ｈｏｌｍ[1]ｏｂｔａｉｎｅｄａｃｌａｓｓｏｆｎｅｗｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｓｈａｌｌｏｗｗａｔｅｒｅｑｕａｔｉｏｎ ,ｉ.ｅ.,Ｃａｍａｓｓａ＿Ｈｏｌｍｅｑｕａｔｉｏｎ2ｕｔ+ 2ｋｕｘ-12 ｕｘｘｔ+ 6ｕｕｘ =ｕｘｕｘｘｘ+ 12 ｕｕｘｘｘ. ( 1 )Ｆｏｒｅｖｅｒｙｋ,ｔｈｅＥｑ .( 1 )ｉｓａｃｌａｓｓｏｆｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｓｙｓｔｅｍ .Ｔｈｉｓｃｌａｓｓｏｆｅｑｕａｔｉｏｎｉｓａｃｌａｓｓｏｆｎｏｔｏｎｌｙｓｔｒａｎｇｅｂｕｔａｌｓｏ…     

Asymptotics of the Solutions of the Stochastic Lattice Wave Equation

We consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing. The multiplicative noise conserves energy, and in the unpinned case also conserves momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds for both square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.     

Geometrical Aspects in Hydrodynamics and Integrable Systems

The motion of fluid particles of an inviscid incompressible fluid on a bounded domain is formulated from a Lagrangian point of view. This is accomplished by observing that Euler's equation of motion is a geodesic equation on a group of volume-preserving diffeomorphisms with the metric defined by the kinetic energy. This formulation is based on Riemannian geometry and Lie group theory, first developed by Arnold (1966). Behaviors of the geodesics are characterized by Riemannian (sectional) curvatures, which are shown to be mostly negative (with some exceptions). This property is related to the mixing and ergodicity of the fluid motions. Free rotation of a rigid body fixed at a point gives a simplest example of the dynamical systems which are integrable and represented with such formulation. The same method is applied to the other integrable systems such as the vortex-filament equation or the KdV equation. In contrast to the hydrodynamic system, sectional curvatures are found to be mostly positive (with exceptions). Thus it is found that integrable systems are more stable in the behavior of geodesics than the hydrodynamic system governed by the Euler's equation of motion. Received 16 January 1997 and accepted 30 May 1997     

Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential

The nonlinear Schr?dinger equation with attractive quintic nonlinearity in periodic potential in 1D, modeling a dilute-gas Bose–Einstein condensate in a lattice potential, is considered and one family of exact stationary solutions is discussed. Some of these solutions have an analog neither in the linear Schr?dinger equation nor in the integrable nonlinear Schr?dinger equation. Their stability is examined analytically and numerically.     

Soliton molecules,rational positons and rogue waves for the extended complex modified KdV equation

Nonlinear Dynamics - In this paper, we consider the integrable extended complex modified Korteweg–de Vries equation. Based on Darboux transformation, we obtain soliton molecules, positon...     

Optimal control strategies for stochastically excited quasi partially integrable Hamiltonian systems

In this paper two different control strategies designed to alleviate the response of quasi partially integrable Hamiltonian systems subjected to stochastic excitation are proposed. First, by using the stochastic averaging method for quasi partially integrable Hamiltonian systems, an n-DOF controlled quasi partially integrable Hamiltonian system with stochastic excitation is converted into a set of partially averaged Itô stochastic differential equations. Then, the dynamical programming equation associated with the partially averaged Itô equations is formulated by applying the stochastic dynamical programming principle. In the first control strategy, the optimal control law is derived from the dynamical programming equation and the control constraints without solving the dynamical programming equation. In the second control strategy, the optimal control law is obtained by solving the dynamical programming equation. Finally, both the responses of controlled and uncontrolled systems are predicted through solving the Fokker-Plank-Kolmogorov equation associated with fully averaged Itô equations. An example is worked out to illustrate the application and effectiveness of the two proposed control strategies.     

Integrable (3+1)-dimensional Ito equation: variety of lump solutions and multiple-soliton solutions

Nonlinear Dynamics - In this work, we study an extended integrable (3+1)-dimensional Ito equation, where its complete integrability is justified via Painlevé analysis. The simplified...