Nonlocal KdV equations

Writing the Hirota-Satsuma (HS) system of equations in a symmetrical form we find its local and new nonlocal reductions. It turns out that all reductions of the HS system are Korteweg-de Vries (KdV), complex KdV, and new nonlocal KdV equations. We obtain one-soliton solutions of these KdV equations by using the method of Hirota bilinearization.     

Integrable extensions of N = 2 supersymmetric KdV hierarchy associated with the nonuniqueness of the roots of the Lax operator

We present new supersymmetric integrable extensions of the a = 4, N = 2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generates a new hierarchy of integrable equations, for which we investigate the Hamiltonian structure. In a special case our system describes the interaction of the KdV equation with the two MKdV equations.     

Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy

In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. From the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation.     

New analytical solutions for propagation of small but finite amplitude ion-acoustic waves in a dense plasma

Abstract

The theoretical and numerical studies have been investigated on the nonlinear propagation of electrostatic ion-acoustic waves (IAWs) in an un-magnetized Thomas–Fermi plasma system consisting of electron, positrons, and positive ions for both of ultra-relativistic and non-relativistic degenerate electrons. Korteweg-de Vries (K-dV) equation is derived from the model equations by using the well-known reductive perturbation method. This equation is solved by employing the generalized Riccati equation mapping method. The hyperbolic functions type solutions to the K-dV equation are only considered for describing the effect of plasma parameters on the propagation of electrostatic IAWs for both of ultra-relativistic and non-relativistic degenerate electrons. The obtained results may be helpful in proper understanding the features of small but finite amplitude localized IAWs in degenerate plasmas and provide the mathematical foundation in plasma physics.     

A simple-looking relative of the Novikov,Hirota-Satsuma and Sawada-Kotera equations

We study the simple-looking scalar integrable equation f_xxt 3( f_x f_t 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).     

Integrable coupling system of fractional soliton equation hierarchy

In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained.     

Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.     

Constraints and Soliton Solutions for KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equationcan be generated by its recursion operator.We generalize the result to a special form of Lax pair,from which a method to constrain the integrable system to alower-dimensional or fewer variable integrable system is proposed.A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depictedby a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.     

Multiple complex soliton solutions for the integrable KdV,fifth-order Lax,modified KdV,Burgers, and Sharma–Tasso–Olver equations

We show that completely integrable equations give multiple complex soliton solutions in addition to multiple real real soliton solutions. We exhibit complex categories of the simplified Hirota’s method to confirm these new findings. To demonstrate the power of the new complex forms, we test it on integrable KdV, fifth-order Lax, modified KdV, fifth-order modified KdV, Burgers, and Sharma–Tasso–Olver equations.     

Alice-Bob Peakon Systems

We study the Alice-Bob peakon system generated from an integrable peakon system using the strategy of the socalled Alice-Bob non-local KdV approach [Scientific Reports 7(2017) 869]. Nonlocal integrable peakon equations are obtained and shown to have peakon solutions.     

Partial differential equations possessing Frobenius integrable decompositions

Frobenius integrable decompositions are introduced for partial differential equations. A procedure is provided for determining a class of partial differential equations of polynomial type, which possess specified Frobenius integrable decompositions. Two concrete examples with logarithmic derivative Bäcklund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations with cubic nonlinearity. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations.     

The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics

By introducing a new general ansätz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarie?s modified Riemann-Liouville derivative. By means of this method, the space-time fractional Whitham-Broer-Kaup and generalized Hirota-Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations.     

A New Solution to the Hirota-Satsuma Coupled KdV Equations by the Dressing Method

The Hirota-Satsuma coupled KdV equations associated 2×2 matrix spectral problem is discussed by the dressing method, which is based on the factorization of integral operator on a line into a product of two Volterra integral operators. A new solution is obtained by choosing special kernel of integral operator.     

Global Solutions to the Ultra-Relativistic Euler Equations

We show that when entropy variations are included and special relativity is imposed, the thermodynamics of a perfect fluid leads to two distinct families of equations of state whose relativistic compressible Euler equations are of Nishida type. (In the non-relativistic case there is only one.) The first corresponds exactly to the Stefan-Boltzmann radiation law, and the other, emerges most naturally in the ultra-relativistic limit of a γ-law gas, the limit in which the temperature is very high or the rest mass very small. We clarify how these two relativistic equations of state emerge physically, and provide a unified analysis of entropy variations to prove global existence in one space dimension for the two distinct 3 × 3 relativistic Nishida-type systems. In particular, as far as we know, this provides the first large data global existence result for a relativistic perfect fluid constrained by the Stefan-Boltzmann radiation law.     

Pseudo-differential equations,and the Bethe ansatz for the classical Lie algebras

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed.     

W-symmetry and the rigid particle

We prove that W₃ is the gauge symmetry of the scale-invariant rigid particle, whose action is given by the integrated extrinsic curvature of its world line. This is achieved by showing that its equations of motion can be written in terms of the Boussinesq operator. The W₃ generators T and W then appear respectively as functions of the induced world line metric and the extrinsic curvature. We also show how the equations of motion for the standard relativistic particle arise from those of the rigid particle whenever it is consistent to impose the “zero-curvature gauge”, and how to rewrite them in terms of the KdV operator. The relation between particle models and integrable systems is further pursued in the case of the spinning particle, whose equations of motion are closely related to the SKdV operator. We also partially extend our analysis in the supersymmetric domain to the scale-invariant rigid particle by explicitly constructing a supercovariant version of its action.     

Relativistic particle in a three-dimensional box

We generalize the work of Alberto, Fiolhais and Gil and solve the problem of a Dirac particle confined in a 3-dimensional box. The non-relativistic and ultra-relativistic limits are considered and it is shown that the size of the box determines how relativistic the low-lying states are. The consequences for the density of states of a relativistic fermion gas are briefly discussed.     

A variety of negative-order integrable KdV equations of higher orders

We develop a variety of negative-order integrable KdV equations of higher orders. We use the inverse recursion operator to construct these new equations. The complete integrability of each established equation is investigated via the Painlevé test, where each equation shows distinct branch of resonances. We use the simplified form of the Hirota’s direct method to obtain multiple soliton solutions for the generalized negative-order KdV equation.     

形式变量分离法及一般Hirota-Satsuma方程新的精确解

将形式变量分离方法推广应用于一个不可积的一般Hirota-Satsuma方程,得到了一些新的孤波解和周期波解. 关键词：     

Interface Shape and Concentration Distribution in Crystallization from Solution under Microgravity

The usual formally variable separation approach is valid only for completely integrable models. In this paper, we extend the method to a nonintegrable generalized Hirota-Satsuma equations. Some new exact solitary wave solutions and periodic wave solutions of the equations are also obtained.