Construction and separability of nonlinear soliton integrable couplings

doi:10.1016/j.amc.2012.08.028

Applied Mathematics and Computation

Volume 219, Issue 4, 1 November 2012, Pages 1866-1873

https://doi.org/10.1016/j.amc.2012.08.028 Get rights and content

Abstract

The paper is motivated by recent works of several authors, initiated by articles of Ma and Zhu [W. X. Ma, Z. N. Zhu, Constructing nonlinear discrete integrable Hamiltonian couplings, Comput. Math. Appl. 60 (2010) 2601] and Ma [W. X. Ma, Nonlinear continuous integrable Hamiltonian couplings, Appl. Math. Comput. 217 (2011) 7238], where new class of soliton systems, being nonlinear integrable couplings, was introduced. Here, we present a general construction of such class of systems and we develop the decoupling procedure, separating them into copies of underlying original equations.

Introduction

For a given nonlinear integrable dynamical system there usually exists many different integrable extensions which equations of motion take a triangular form. They are usually called triangular systems. For Liouville integrable nonlinear ODE’s an interesting class of triangular systems is given in [1]. For soliton systems we know more examples. Triangular extensions of the KdV system was considered in [2]. In [3], the so-called ‘dark equations’, linear extensions of soliton systems which are also of the triangular form were constructed. Another class of linear extensions the so-called linear integrable couplings of soliton systems was introduced in [4] and then developed in [5] and many other papers. Recently, the theory of nonlinear integrable couplings of ordinary soliton systems was presented in [6], [7], [8] and further developed in [9], [10], [11], [12]. Particulary noteworthy are the constructions of integrable couplings based on the non-semisimple Lie algebras, see for instance [6].

In the present paper, we introduce very natural triangular nonlinear couplings of integrable systems, including in particular those in [7], [8], [9], [10], [11], which are also integrable. The construction is made on the level of evolution equations by a modification of the algebra of dynamical fields. We also propose the decoupling procedure for the considered class of integrable couplings in the form of an appropriate change of variables.

In Section 2 we introduce the algebra of coupled scalars, which is the underlying algebra for the nonlinear integrable couplings defined in Section 3. We derive soliton integrable couplings of field and lattice type. In Section 4, the general form of solution of the coupled systems is obtained. An example of soliton solutions of the nonlinearly coupled KdV system is presented. In Section 5 we derive a matrix representation of the algebra of coupled scalars and in consequence the matrix Lax representations for nonlinear couplings constructed in previous sections. Finally, in Section 6, we prove that any member of the constructed family of coupled systems separates into copies of the original soliton systems. We also show the source of the decoupling procedure.

Section snippets

The algebra of coupled scalars

Consider n-dimensional vector space over field of real numbers $R$ . We define an algebra structure by the following multiplication $e_{i} \cdot e_{j} ≔ e_{max (i, j)},$ where $e_{i}$ are the basis vectors. Let $a = \sum_{i = 1}^{n} a_{i} e_{i}$ , then $a \cdot b = (\begin{matrix} a_{1} \\ ⋮ \\ a_{n} \end{matrix}) \cdot (\begin{matrix} b_{1} \\ ⋮ \\ b_{n} \end{matrix}) = (\begin{matrix} c_{1} \\ ⋮ \\ c_{n} \end{matrix}) = c,$ where $c_{i} = a_{i} b_{i} + a_{i} (\sum_{k = 1}^{i - 1} b_{k}) + (\sum_{k = 1}^{i - 1} a_{k}) b_{i} .$ We find that the value of the coefficient $c_{i}$ is given by $a_{i} b_{i}$ plus terms depending on lower order elements, $a_{k}, b_{k}$ with $k < i$ . Therefore, we call this algebra as an algebra of coupled scalars. This algebra is unital, commutative and

Nonlinear couplings of soliton systems

Consider a commutative and associative algebra, with respect to the ordinary dot multiplication, of smooth functions on $R^{m}$ with m derivations $\frac{\partial}{\partial x_{i}} : C^{\infty} (R^{m}) \to C^{\infty} (R^{m})$ . Let us construct its coupled counterpart $C_{d}^{\infty} (R^{m})$ , that is an algebra of functions $f (x) = f_{1} (x) e_{1} + \dots + f_{n} (x) e_{n} = (\begin{matrix} f_{1} (x) \\ ⋮ \\ f_{n} (x) \end{matrix}),$ where $x = (x_{1}, \dots, x_{m})$ , taking values in the algebra of coupled scalars. So, it is a commutative and associative algebra with respect to the multiplication (1) and the derivations in $C_{d}^{\infty} (R^{m})$ can be defined by $\frac{\partial}{\partial x_{i}} ≔ \frac{\partial}{\partial x_{i}} e_{1} i = 1, \dots, m .$

Solutions of coupling systems

The solutions of the coupled systems (4) are completely determined by solutions of the basic equations.

We will show now that, assuming $S^{1}, \dots, S^{n}$ to be n arbitrary different solutions of the basic equation $u_{t} = K [u]$ , the solution of coupled system (4) is given by $u_{1} = S^{1}, u_{k} = S^{k} - S^{k - 1}, k = 2, \dots, n .$ Moreover, any solution of the coupled system is of the form (11).

After plugging (11) to (5), $(S^{k} - S^{k - 1})_{t} - (K [S^{k}] - K [S^{k - 1}]) = 0,$ it is evident that (11) solves (4). On the other hand, if the solution of the basic equation is

Matrix representation of the algebra of coupled scalars

Consider n quadratic matrices $E_{k}$ of dimension $n \times n$ : ${(E_{k})}_{ij} = \{\begin{matrix} 1 & if j = k, i ⩽ k, \\ 1 & if j = i, i > k, \\ 0 & otherwise, \end{matrix}$ where $k = 1, \dots, n$ . The matrices $E_{i}$ constitute generating elements of commutative and associative sub-algebra of triangular matrices. Let us call the matrix algebra spanned by $E_{k}$ as a pseudo-scalar algebra of matrices, $ps$ (n).

The algebra $ps$ (n) is a matrix representation of the algebra of coupled scalars defined by the multiplication (1), which follows immediately by showing that $E_{i} E_{j} = E_{j} E_{i} = E_{max (i, j)}$ for $i, j = 1, \dots, n$ .

Decoupling procedure

Consider the transformation of the pseudo-scalar algebra $ps (n)$ in the form of a similarity relation $T (A) ≔ S^{- 1} A S,$ where $S_{ij} = \{\begin{matrix} 1 & for i ⩽ j \\ 0 & for i > j, \end{matrix} (S^{- 1})_{ij} = \{\begin{matrix} 1 & for j = i, \\ - 1 & for j = i + 1, \\ 0 & otherwise. \end{matrix}$

Then, this transformation is apparently an isomorphism of matrix algebras. One finds that $T (E_{k}) = \{\begin{matrix} 1 & for j = i, j ⩾ k, \\ 0 & otherwise. \end{matrix}$

In particular, for $n = 4$ $S = (\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix}), S^{- 1} = (\begin{matrix} 1 & - 1 & 0 & 0 \\ 0 & 1 & - 1 & 0 \\ 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 1 \end{matrix})$ and for $A$ given by (12) $T (A) : = (\begin{matrix} a_{1} & 0 & 0 & 0 \\ 0 & a_{1} + a_{2} & 0 & 0 \\ 0 & 0 & a_{1} + a_{2} + a_{3} & 0 \\ 0 & 0 & 0 & a_{1} + a_{2} + a_{3} + a_{4} \end{matrix}) .$

The above transformation naturally extends to any tensor product of some

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View full text

Construction and separability of nonlinear soliton integrable couplings

Abstract

Introduction

Section snippets

The algebra of coupled scalars

Nonlinear couplings of soliton systems

Solutions of coupling systems

Matrix representation of the algebra of coupled scalars

Decoupling procedure

Chaos Solitons Fract.

Comput. Math. Appl.

Appl. Math. Comput.

Phys. Lett. A

Separable systems of coordinates for triangular Newton equations

Stud. Appl. Math.

Integrable coupled KdV systems

J. Math. Phys.