Matrix generalisation of the modified Korteweg-de Vries equation

Khalilov, F A; Khruslov, E. Ya.

doi:10.1088/0266-5611/6/2/004

Cited by 29 publications

(21 citation statements)

References 5 publications

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“…u . This equation was rst described in [26], where the Lax pair formulation and the inverse scattering problem were studied. Equation (1.6) does not admit a Miura transformation.…”

Section: Introductionmentioning

confidence: 99%

Integrable Evolution Equations on Associative Algebras

Olver

Соколов

1998

Communications in Mathematical Physics

196

274

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This paper surveys the classication of integrable evolution equations whose eld variables take v alues in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to associative algebra-valued version of the Painlev e transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the bi-Hamiltonian structures for several examples are found.

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“…u . This equation was rst described in [26], where the Lax pair formulation and the inverse scattering problem were studied. Equation (1.6) does not admit a Miura transformation.…”

Section: Introductionmentioning

confidence: 99%

Integrable Evolution Equations on Associative Algebras

Olver

Соколов

1998

Communications in Mathematical Physics

196

274

View full text Add to dashboard Cite

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“…Finally, the solution given in Theorem 13 is mapped 5 to a solution of the alternative mKdV hierarchy,Ṽ t2n−1 =Ψ(Ṽ ) n−1Ṽ x , 1 ≤ n ≤ N , whereΨ(Ṽ ) is the recursion operator in (17). This leads to Corollary 14.…”

Section: Solving the Meta-mkdv Hierarchymentioning

confidence: 91%

A novel noncommutative KdV-type equation, its recursion operator, and solitons

Carillo

Schiavo

Porten

et al. 2018

Journal of Mathematical Physics

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A noncommutative KdV-type equation is introduced extending the Bäcklund chart in [4]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [22, Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived. 1991 Mathematics Subject Classification. 35Q53; 46L55; 37K35.

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“…was first described in [34], where the Lax pair formulation and the inverse scattering problem were studied. In contrast to the non-Abelian mKdV (2.5) studied in the previous section, (3.1) does not admit a Miura transformation [44].…”

Section: On the Non-abelian Mkdv Equationsmentioning

confidence: 99%

Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart

Carillo¹,

Schiavo²,

Schiebold³

2016

SIGMA

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Abstract. Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative Bäcklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The implied applications referring to finite-dimensional cases will be considered separately.

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Matrix generalisation of the modified Korteweg-de Vries equation

Abstract: The matrix-modified Korteweg-de Vries equation (MMKdV)

Cited by 29 publications

References 5 publications

Integrable Evolution Equations on Associative Algebras

Integrable Evolution Equations on Associative Algebras

A novel noncommutative KdV-type equation, its recursion operator, and solitons

Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart

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