Novel composite function solutions of the modified KdV equation

A Wronskian form expansion method is proposed to construct novel composite function solutions to the modified Korteweg-de Vries (mKdV) equation. The method takes advantage of the forms and structures of Wronskian solutions to the mKdV equation, and Wronskian entries do not satisfy linear partial differential equations. The method can be automatically carried out in computer algebra (for example, Maple).     

The N-soliton solutions of the fifth-order KdV equation under Bargmann constraint

We derive the N-soliton solutions for the fifth-order KdV equation under Bargmann constraint through Hirota method and Wronskian technique, respectively. Some novel determinantal identities and properties are presented to finish the Wronskian verifications. The uniformity of these two kinds of N-soliton solutions is proved.     

Self-self-dual spaces of polynomials

A space of polynomials V of dimension 7 is called self-dual if the divided Wronskian of any 6-subspace is in V. A self-dual space V has a natural inner product. The divided Wronskian of any isotropic 3-subspace of V is a square of a polynomial. We call V self-self-dual if the square root of the divided Wronskian of any isotropic 3-subspace is again in V. We show that the self-self-dual spaces have a natural non-degenerate skew-symmetric 3-form defined in terms of Wronskians.We show that the self-self-dual spaces correspond to G₂-populations related to the Bethe Ansatz of the Gaudin model of type G₂ and prove that a G₂-population is isomorphic to the G₂ flag variety.     

Jeu de taquin and a monodromy problem for Wronskians of polynomials

The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n−d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. When the roots of the Wronskian are real, we show that the monodromy is combinatorially encoded by Schützenberger's jeu de taquin; hence we obtain new geometric interpretations and proofs of a number of results from jeu de taquin theory, including the Littlewood-Richardson rule.     

Gauge transformation, elastic and inelastic interactions for the Whitham-Broer-Kaup shallow-water model

Whitham-Broer-Kaup (WBK) model is a model for the dispersive long wave in shallow water. With symbolic computation, gauge transformation between the WBK model and a parameter Ablowitz-Kaup-Newell-Segur (AKNS) system is hereby constructed. By selecting seeds, we derive two sorts of multi-soliton solutions for the WBK model via a N-fold Darboux transformation (DT) of the parameter AKNS system, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Different from the bilinear way, the double Wronskian solutions can be obtained via the N-fold DT with a linear algebraic system and matrix differential equation solved. A novel inelastic interaction is graphically discussed, in which the soliton complexes are formed after the collision. Our results could be helpful for interpreting certain shallow-water-wave phenomena.     

Multi-component Wronskian solution to the Kadomtsev-Petviashvili equation

It is known that the Kadomtsev-Petviashvili (KP) equation can be decomposed into the first two members of the coupled Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy by the binary non-linearization of Lax pairs. In this paper, we construct the N-th iterated Darboux transformation (DT) for the second- and third-order m-coupled AKNS systems. By using together the N-th iterated DT and Cramer’s rule, we find that the KPII equation has the unreduced multi-component Wronskian solution and the KPI equation admits a reduced multi-component Wronskian solution. In particular, based on the unreduced and reduced two-component Wronskians, we obtain two families of fully-resonant line-soliton solutions which contain arbitrary numbers of asymptotic solitons as y → ?∞ to the KPII equation, and the ordinary N-soliton solution to the KPI equation. In addition, we find that the KPI line solitons propagating in parallel can exhibit the bound state at the moment of collision.     

The solution to the embedding problem of a (differential) Lie algebra into its Wronskian envelope

Any commutative algebra equipped with a derivation may be turned into a Lie algebra under the Wronskian bracket. This provides an entirely new sort of a universal envelope for a Lie algebra, the Wronskian envelope. The main result of this paper is the characterization of those Lie algebras which embed into their Wronskian envelope as Lie algebras of vector fields on a line. As a consequence we show that, in contrast to the classical situation, free Lie algebras almost never embed into their Wronskian envelope.     

On the Wronskian combinants of binary forms

Given a sequence A=(A₁,…,A_r) of binary d-ics, we construct a set of combinants C={C_q:0≤q≤r,q≠1}, to be called the Wronskian combinants of A. We show that the span of A can be recovered from C as the solution space of an SL(2)-invariant differential equation. The Wronskian combinants define a projective imbedding of the Grassmannian G(r,S_d), and, as a corollary, any other combinant of A is expressible as a compound transvectant in C.Our main result characterises those sequences of binary forms that can arise as Wronskian combinants; namely, they are the ones such that the associated differential equation has the maximal number of linearly independent polynomial solutions. Along the way we deduce some identities which relate Wronskians to transvectants. We also calculate compound transvectant formulae for C in the case r=3.     

Differential and integral relations involving fractional derivatives of Airy functions and applications

Various differential and integral relations are deduced that involve fractional derivatives of the Airy function Ai(x) and the Scorer function Gi(x). Several new Wronskian relations are obtained that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the Airy functions. New fractional derivative conservation laws are derived for equations of the Korteweg-de Vries type.     

Soliton solutions, Bäcklund transformation and Wronskian solutions for the extended (2+1)-dimensional Konopelchenko-Dubrovsky equations in fluid mechanics

This paper is to investigate the extended (2+1)-dimensional Konopelchenko-Dubrovsky equations, which can be applied to describing some phenomena in the stratified shear flow, the internal and shallow-water waves and plasmas. Bilinear-form equations are transformed from the original equations and N-soliton solutions are derived via symbolic computation. Bilinear-form Bäcklund transformation and single-soliton solution are obtained and illustrated. Wronskian solutions are constructed from the Bäcklund transformation and single-soliton solution.     

Triple Wronskian solutions of the coupled derivative nonlinear Schrödinger equations in optical fibers

A type of the coupled derivative nonlinear Schrödinger (CDNLS) equations are studied by means of symbolic computation, which can describe the wave propagation in birefringent optical fibers. Soliton solutions in the triple Wronskian form of the CDNLS equations are obtained. Elastic and inelastic collisions are both presented under some parametric conditions. In addition, generalized triple Wronskian solutions of a set of the coupled general derivative nonlinear Schrödinger (CGDNLS) equations are derived. Triple Wronskian identities are given to prove such solutions, which may also be used for other coupled nonlinear equations. Rational solutions of the CGDNLS equations are also obtained.     

The order of a zero of a Wronskian and the theory of linear dependence

Every root of the top Wronskian of a Wronskian matrix whose rank at the root is equal to the number of columns, is of integer order even if the highest derivatives exist only at the root. If the rank of a Wronskian matrix is constant and smaller than the number of rows, then the number of independent linear relations between the functions in the first row is equal to the number of functions minus the rank. These results were proved under additional assumptions by Bôcher, Curtiss, and Moszner. Their proofs are simplified.     

Associative homotopy Lie algebras and Wronskians

We analyze representations of Schlessinger-Stasheff associative homotopy Lie algebras by higher-order differential operators. W-transformations of chiral embeddings of a complex curve related with the Toda equations into Kähler manifolds are shown to be endowed with the homotopy Lie-algebra structures. Extensions of the Wronskian determinants preserving Schlessinger-Stasheff algebras are constructed for the case of n ≥ 1 independent variables.     

Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation

Wronskian and Grammian formulations are established for a (3 + 1)-dimensional generalized KP equation, based on the Plücker relation and the Jacobi identity for determinants. Generating functions for matrix entries satisfy a linear system of partial differential equations involving a free parameter. Examples of Wronskian and Grammian solutions are computed and a few particular solutions are plotted.     

非线性KdV-mKdV方程的新的复合函数解

利用孤立子方程KdV-mKdV的朗斯基解的形式和结构,我们提出了朗斯基形式展开法,运用这一方法获得了KdV-mKdV方程的丰富的新的复合函数解,并且朗斯基行列式中的元素不满足任何线性偏微分方程组.所得到的复合函数解是使用其它的方法得不到的.     

Symbolic computation, Bäcklund transformation and analytic N-soliton-like solution for a nonlinear Schrödinger equation with nonuniformity term from certain space/laboratory plasmas

With symbolic computation, a bilinear Bäcklund transformation is presented for a nonlinear Schrödinger equation with nonuniformity term from certain space/laboratory plasmas, and correspondingly the one-soliton-like solution is derived from the Bäcklund transformation. Simultaneously, the N-soliton-like solution in double Wronskian form is also given. Besides, the authors verify that the (N−1)- and N-soliton-like solutions satisfy the Bäcklund transformation. The results obtained in this paper might be valuable for the study of the nonuniform media.     

A relation between Gram and Wronsky determinants

The orders of the zeros at T₀ = 0 of the Wronskian of a system of functions defined in the interval [0,T], and of its Gramian, considered as a function of T, are calculated, and a relation between the corresponding derivatives is obtained. These results are then applied to obtain an estimate for the distance [in the sense of the norm of L²(0,T)] from one element of the system to the subspace spanned by the others.     

Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation

A set of sufficient conditions consisting of systems of linear partial differential equations is obtained which guarantees that the Wronskian determinant solves the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form. Upon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield rational solutions, solitons, negatons, positons and interaction solutions.     

On scattering on a matrix potential with symplectic structure

The scattering problem for the matrix Schrödinger operator with a non-Hermitian potential is considered. It is shown that there exists a set of unsymmetric potentials for which the Wronskian can be introduced. For a real k, an explicit expression for the Wronskian is derived. For a complex k, the asymptotic value of the Wronskian, as x± , is determined.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 133–139.     

Generalized double Wronskian solutions of the third-order isospectral AKNS equation

The generalized double Wronskian solutions of the third-order isospectral AKNS equation are obtained. Thus we found rational solutions, Matveev solutions, complexitons and interaction solutions. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction.