Construction and separability of nonlinear soliton integrable couplings
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 . We define an algebra structure by the following multiplicationwhere are the basis vectors. Let , thenwhereWe find that the value of the coefficient is given by plus terms depending on lower order elements, with . 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 with m derivations . Let us construct its coupled counterpart , that is an algebra of functionswhere , 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 can be defined by
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 to be n arbitrary different solutions of the basic equation , the solution of coupled system (4) is given byMoreover, any solution of the coupled system is of the form (11).
After plugging (11) to (5),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 of dimension :where . The matrices constitute generating elements of commutative and associative sub-algebra of triangular matrices. Let us call the matrix algebra spanned by as a pseudo-scalar algebra of matrices, (n).
The algebra (n) is a matrix representation of the algebra of coupled scalars defined by the multiplication (1), which follows immediately by showing thatfor .
Decoupling procedure
Consider the transformation of the pseudo-scalar algebra in the form of a similarity relationwhere
Then, this transformation is apparently an isomorphism of matrix algebras. One finds that
In particular, for and for given by (12)
The above transformation naturally extends to any tensor product of some
References (13)
- et al.
Integrable theory of the perturbation equations
Chaos Solitons Fract.
(1996) - et al.
Constructing nonlinear discrete integrable Hamiltonian couplings
Comput. Math. Appl.
(2010) Nonlinear continuous integrable Hamiltonian couplings
Appl. Math. Comput.
(2011)A real nonlinear integrable couplings of continuous soliton hierarchy and its Hamiltonian structure
Phys. Lett. A
(2011)- et al.
Separable systems of coordinates for triangular Newton equations
Stud. Appl. Math.
(2007) - et al.
Integrable coupled KdV systems
J. Math. Phys.
(1998)
Cited by (6)
Multiple soliton solutions for an integrable couplings of the Boussinesq equation
2013, Ocean EngineeringCitation Excerpt :Boussinesq found the first analytical solution of the soliton solution in the sech form. The theory of nonlinear integrable couplings of ordinary soliton systems was presented in works of Ma and Fuchssteiner (1996), Ma and Zhu (2010) and further studied in the works of Fa-Jun and Li (2009), Zhang and Tam (2010) and Błaszak et al. (2012) and others. The perturbation method was developed by Ma and Fuchssteiner (1996) for establishing integrable couplings.
Bi-Integrable Couplings Associated with so (3 , R)
2019, Bulletin of the Malaysian Mathematical Sciences SocietyA block matrix loop algebra and bi-integrable couplings of the Dirac equations
2013, East Asian Journal on Applied MathematicsMultiple soliton solutions for two integrable couplings of the modified Korteweg-de Vries equation
2013, Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information ScienceIntegrable couplings of the Burgers equation and the Sharma-Tasso-Olver equation: Multiple kink solutions
2013, Romanian Reports in PhysicsMultiple soliton solutions for the integrable couplings of the KdV and the KP equations
2013, Central European Journal of Physics