Extended Discrete KP Hierarchy and its Reductions from a Geometric Viewpoint

We interpret the recently suggested extended discrete KP (Toda lattice) hierarchy from a geometrical point of view. We show that the latter corresponds to the union of invariant submanifolds S ₀ ⁿ of the system which is a chain of infinitely many copies of Darboux–KP hierarchy, while the intersections yields a number of reduction s to l-field lattices.     

Recursion Operators for Dispersionless KP Hierarchy

Based on the corresponding theorem between dispersionless KP(dKP)hierarchy and -dependent KP ( KP) hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding KP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.     

Soliton Solutions of Coupled KdV System from Hirota＇s Bilinear Direct Method

With Hirota＇s bilinear direct method, we study the special coupled KdV system to obtain its new soliton solutions. Then we further discuss soliton evolution, corresponding structures, and interesting interactive phenomena in detail with plot. As a result, we find that after the interaction, the solitons make elastic collision and there are no exchanges of their physical quantities including energy, velocity and shape except the phase shift.     

Bilinear Forms on Novikov Algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. On the other hand, there can be geometry and Lagrangian mechanics on homogenous spaces related to Novikov algebras. The nondegenerate symmetric bilinear forms on Novikov algebras can be regarded as the pseudometrics, and some additional identities for these forms correspond to some conserved quantities. In particular, there is an important kind of conserved nondegenerate symmetric bilinear forms that correspond to the pseudo-Riemannian connections such that parallel translation preserves the bilinear form on the tangent spaces. Moreover, the fact that the left multiplication operators form a Lie algebra for a Novikov algebra is compatible with such a form. However, we show in this note that there are no such forms on most Novikov algebras in low dimensions.     

Bilinear Identities and Hirota’s Bilinear Forms for the (γn, σk)-KP Hierarchy

In this paper, we discuss how to construct the bilinear identities for the wave functions of the (γ_n, σ_k)-KP hierarchy and its Hirota’s bilinear forms. First, based on the corresponding squared eigenfunction symmetry of the KP hierarchy, we prove that the wave functions of the (γ_n, σ_k)-KP hierarchy are equal to the bilinear identities given in Sec.3 by introducing N auxiliary parameters z_i, i = 1, 2,?…?, N. Next, we derived the bilinear equations for the tau-function of the (γ_n, σ_k)-KP hierarchy. Then, we obtain the bilinear equations for the taufunction of the mixed type of KP equation with self-consistent sources (KPESCS), which includes both the first and the second type of KPESCS as special cases by setting n = 2 and k = 3. Finally, using the relation between the Hirota bilinear derivatives and the usual partial derivatives, we show the procedure of translating the Hirota’s bilinear equations into the mixed type of KPESCS.     

Wronskian Form Solutions for a Variable Coefficient Kadomtsev–Petviashvili Equation

Starting from a simple transformation, and with the aid of symbolic computation, we establish the relation- ship between the solution of a generalized variable coefficient Kadomtsev-Petviashvili （vKP） equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover, we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.     

Bilinear identities for the constrained modified KP hierarchy

In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ, the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ.     

Two new integrable Kadomtsev–Petviashvili equations with time-dependent coefficients: multiple real and complex soliton solutions

ABSTRACT

In this work, we develop two new integrable Kadomtsev–Petviashvili (KP) equations with time-dependent coefficients. The integrability property of each equation is explicitly demonstrated exhibiting the Painlevé test to confirm its integrability. Moreover, each equation admits multiple real and multiple complex soliton solutions. We introduce complex forms of the simplified Hirota's method to derive multiple complex soliton solutions. These two model equations are likely to be of applicative relevance, because it may be considered an application of a large class of nonlinear KP equations.     

Darboux Coverings and Rational Reductions of the KP Hierarchy

We use the method of Darboux coverings to discuss the invariant submanifolds of the KP equations presented as conservation laws in the space of monic Laurent series in the spectral parameter (the space of the Hamiltonian densities). We identify a special class of these submanifolds with the rational invariant submanifolds entering matrix models of two-dimensional gravity recently characterized by Dickey and Krichever. Four examples of the general procedure are provided.     

Characteristics of the solitary waves and rogue waves with interaction phenomena in a (2 + 1)-dimensional Breaking Soliton equation

Under inquisition in this paper is a $(2+1)$-dimensional Breaking Soliton equation, which can describe various nonlinear scenarios in fluid dynamics. Using the Bell polynomials, some proficient auxiliary functions are offered to apparently construct its bilinear form and corresponding soliton solutions which are different from the previous literatures. Moreover, a direct method is used to construct its rogue wave and solitary wave solutions using particular auxiliary function with the assist of bilinear formalism. Finally, the interactions between solitary waves and rogue waves are offered with a complete derivation. These results enhance the variety of the dynamics of higher dimensional nonlinear wave fields related to mathematical physics and engineering.     

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.     

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.     

A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations with logarithmic transformations is presented. In the algorithm, the general assumption of Hirota bilinear form is successfully reduced based on the property of uniformity in rank. Furthermore, we discard the integral operation in the traditional algorithm. The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety soliton equations.     

Coupling Integrable Couplings of an Equation Hierarchy

Based on a kind of Lie algebra G proposed by Zhang, one isospectral problem is designed. Under the framework of zero curvature equation, a new kind of integrable coupling of an equation hierarchy is generated using the methods proposed by Ma and Gao. With the help of variational identity, we get the Hamiltonian structure of the hierarchy.     

A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations with logarithmic transformations is presented. In the algorithm, the general assumption of Hirota bilinear form is successfully reduced based on the property of uniformity in rank. Furthermore, we discard the integral operation in the traditional algorithm. The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety soliton equations.     

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

Pairs of n×n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r×r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case r = 1, this reproduces well-known results of Wilson and others from the 1990’s relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators.      

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.     

Integrable Coupling of KN Hierarchy and Its Hamiltonian Structure

The Hamiltonian structure of the integrable couplings obtained by our method has not been solved. In this paper, the Hamiltonian structure of the KN hierarchy is obtained by making use of the quadratic-form identity.     

New DLW Hierarchy of an Integrable Coupling and Its Hamiltonian Structure

A type of higher-dimensional loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic identity.