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Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

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Book cover Nonlinear Dynamics of Structures, Systems and Devices

Abstract

Nonlinear non-abelian Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations and their links via Bäcklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg–de Vries equation. Matrix equations can be viewed as a specialisation of operator equations in the finite dimensional case when operators admit a matrix representation. Bäcklund transformations allow to reveal structural properties Carillo and Schiebold (J Math Phys 50:073510, 2009) enjoyed by non-commutative KdV-type equations, such as the existence of a recursion operator. Operator methods combined with Bäcklund transformations allow to construct explicit solution formulae Carillo and Schiebold (J Math Phys 52:053507, 2011). The latter are adapted to obtain solutions admitted by the 2 × 2 and 3 × 3 matrix mKdV equation. Some of these matrix solutions are visualised to show the solitonic behaviour they exhibit. A further key tool used to obtain the presented results is an ad hoc construction of computer algebra routines to implement non-commutative computations.

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Notes

  1. 1.

    It is generally assumed that M is space of functions u(x, t) which, ∀ fixed t, belong to the Schwartz space S of rapidly decreasing functions on , in the, i.e., , where , .

  2. 2.

    An overview on non-commutative equations is given in [22].

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Acknowledgements

Under the financial support of G.N.F.M.-I.N.d.A.M., I.N.F.N. and Università di Roma La Sapienza, Rome, Italy. C.Schiebold acknowledges Dip. S.B.A.I., Università di Roma La Sapienza, for the kind hospitality.

Author information

Authors and Affiliations

  1. Università di Roma “La Sapienza”, Dip. S.B.A.I., Rome, Italy

    Sandra Carillo

  2. I.N.F.N. – Sezione Roma 1, Gr. IV – M.M.N.L.P., Rome, Italy

    Sandra Carillo

  3. Università di Roma “La Sapienza”, Dip. D.I.S.G., Rome, Italy

    Mauro Lo Schiavo

  4. Department of Mathematics and Science Education, Mid Sweden University, Sundsvall, Sweden

    Cornelia Schiebold

  5. Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Kielce, Poland

    Cornelia Schiebold

Authors
  1. Sandra Carillo
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  2. Mauro Lo Schiavo
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  3. Cornelia Schiebold
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Corresponding author

Correspondence to Sandra Carillo .

Editor information

Editors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy

    Walter Lacarbonara

  2. Department of Mechanical Engineering, University of Maryland, College Park, MD, USA

    Balakumar Balachandran

  3. Department of Physics, Lanzhou University of Technology, Lanzhou, Gansu, China

    Jun Ma

  4. Department of Electrical Engineering, Polytechnic of Porto - School of Engineering (ISEP), Porto, Portugal

    J. A. Tenreiro Machado

  5. Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

    Gabor Stepan

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Carillo, S., Schiavo, M.L., Schiebold, C. (2020). Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation. In: Lacarbonara, W., Balachandran, B., Ma, J., Tenreiro Machado, J., Stepan, G. (eds) Nonlinear Dynamics of Structures, Systems and Devices. Springer, Cham. https://doi.org/10.1007/978-3-030-34713-0_8

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