Breaking soliton equations and negative-order breaking soliton equations of typical and higher orders

We develop breaking soliton equations and negative-order breaking soliton equations of typical and higher orders. The recursion operator of the KdV equation is used to derive these models. We establish the distinct dispersion relation for each equation. We use the simplified Hirota’s method to obtain multiple soliton solutions for each developed breaking soliton equation. We also develop generalized dispersion relations for the typical breaking soliton equations and the generalized negative-order breaking soliton equations. The results provide useful information on the dynamics of the relevant nonlinear negative-order equations.     

Repeated application of the recursion operator for a new hierarchy of negative-order integrable KdV equations

ABSTRACT

In this work we use the repeated application of the recursion operator to establish a new hierarchy of negative-order integrable KdV equations of higher orders. The concept of the inverse recursion operator allows us to develop this new hierarchy. The complete integrability of each equation is guaranteed via the use of the recursion operator. We show that the dispersion relations of this hierarchy follow an infinite geometric series. Multiple soliton solutions for each equation of the hierarchy are obtained.     

A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV–nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions.     

Inverse problem of the variational calculus for higher KdV equations

It is shown that the usual Hamilton's variational principle supplemented by the methodology of the integer-programming problem can be used to construct expressions for the Lagrangian densities of higher KdV fields. This is demonstrated with special emphasis on the second and third members of the hierarchy. However, the method is general enough for applications to equations of any order. The expressions for Lagrangian densities are used to calculate results for Hamiltonian densities that characterize Zakharov-Faddeev-Gardner equation. Received 27 January 2002 / Received in final form 6 May 2002 Published online 24 September 2002     

Multiple soliton solutions for the integrable couplings of the KdV and the KP equations

In this work, we study the nonlinear integrable couplings of the KdV and the Kadomtsev-Petviashvili (KP) equations. The simplified Hirota’s method will be used for this study. We show that these couplings possess multiple soliton solutions the same as the multiple soliton solutions of the KdV and the KP equations, but differ only in the coefficients of the transformation used. This difference exhibits soliton solutions for some equations and anti-soliton solutions for others.     

KdV6: An integrable system

K²S²T [A. Karasu-Kalkani, A. Karasu, A. Sakovich, S. Sakovich, R. Turhan, nlin/0708.3247] recently derived a new 6th-order wave equation KdV6: , found a linear problem and an auto-Bäckclund transformation for it, and conjectured its integrability in the usual sense. We prove this conjecture by constructing an infinite commuting hierarchy KdV_n6 with a common infinite set of conserved densities. A general construction is presented applicable to any bi-Hamiltonian system (such as all standard Lax equations, continuous and discrete) providing a nonholonomic perturbation of it. This perturbation is conjectured to preserve integrability. That conjecture is verified in a few representative cases: the classical long-wave equations, the Toda lattice (both continuous and discrete), and the Euler top.     

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.     

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.     

Integrability of coupled KdV equations

The integrability of coupled KdV equations is examined. The simplified form of Hirota’s bilinear method is used to achieve this goal. Multiple-soliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation. The resonance phenomenon of each model will be examined.     

Collisionless Boltzmann equations and integrable moment equations

A collisionless Boltzmann equation, describing long waves in a dense gas of particles interacting via short-range forces, is shown to be equivalent to the Benney equations, which describe long waves in a perfect two-dimensional fluid with a free surface. These equations also describe, in a random phase approximation, the evolution, on long space and time scales, of multiply periodic solutions of the nonlinear Schrödinger equation. The derivative nonlinear Schrödinger equation is likewise shown to be related to an integrable system of moment equations.     

A remark on abstract operator completely integrable equations

We look at the isospectral deformation equation as compatibility conditions on abstract operators rationally depending on λ. Several concrete examples of abstract completely integrable operator equations in two and three dimensions are presented (non-linear Schrödinger, Korteweg-de Vries, Kadomtzev-Petviashvili, Benjamin-Ono).     

A family of integrable evolution equations of third order

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. Our construction of integrable equations and the corresponding differential substitutions involves geometry of a family of genus two curves and their Jacobians.     

Fluctuating solitons of the KdV equations

We introduce a new AKNS three-component system, which is convenient for finding periodic and/or almost periodic solutions to the hierarchy of the KdV equations. It conserves the spectral functions which determine the spectrum of the auxiliary Schrödinger equation containing the solutions of the Korteweg-de Vries equations as potentials. By means of the Darboux and Abraham-Moses transformations we derive new solutions of the KdV hierarchy, which can be grasped as solitons on the fluctuating background.Some parts of the paper were delivered in the talk at the III Potsdam-V Kiev international workshop on nonlinear processes in physics, Potsdam (USA), 1–11 August, 1991.     

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.     

Dirac structures of integrable evolution equations

An algebraic theory of Dirac structures is presented, enclosing finite-dimensional pre-symplectic and Poisson structures, as well as their infinite-dimensional analogs determined by local operators. The generalized Lenard scheme of integrability is considered together with examples of its action.     

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.     

Coupled KdV equations with multi-Hamiltonian structures

We present a large class of systems of N-equations which possess (N+1) purely differential, compatible Hamiltonian structures. Our generic equations are isospectral to [(Σ^N−1₀ε_iλ²ⁱ)∂²+Σ^N−1₀υ_iλ²ⁱ]ψ=λ^2Nψ, but our class also includes degenerate, nondispersive systems which are unrelated to this linear problem. Embedded in this class, for each N, there are N distinct coupled KdV systems. When N = 2 our class includes 3 known equations: dispersive water waves, Ito's equation and reduced Benney's equations. These equations are thus tri-Hamiltonian. We also present 2 examples of 3-component quadri-Hamiltonian systems, which generalise the above mentioned dispersive and nondispersive water waves.     

Second-order integrable Lagrangians and WDVV equations

Letters in Mathematical Physics - We investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $\begin{aligned} \int...     

AKNS方程族的一类扩展可积模型

首先构造了loop代数A～2的一个新的子代数,设计了一个等谱问题.应用屠格式求出了著名的AKNS方程族的一类扩展可积模型,即可积耦合.然后将这种求可积耦合的方法一般化,可用于一大类方程族,如KN族、GJ族、WKI族等谱系的可积耦合.提出的方法具有普遍应用价值.最后作为AKNS方程族的特例,求得了KdV方程和MKdV方程的可积耦合 关键词： 可积耦合 loop代数 AKNS方程族