Symmetries for exact solutions to the nonlinear Schrödinger equation

Aktosun, Tuncay; Busse, Theresa; Demontis, Francesco; Mee, Cornelis van der

doi:10.1088/1751-8113/43/2/025202

Cited by 37 publications

(31 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Though reflection symmetries are not quite compatible with the structure of a superposition, we still have the following result (based on arguments in [143]).…”

mentioning

confidence: 84%

“…The following result has its origin in [143], where only the (focusing) continuous NLS equation was considered, however. We show that for a solution determined by matrix data via proposition 6.1, in general equivalent matrix data (S, U, V ) exist such that all eigenvalues s of S with Re(s) = 0 in the continuous NLS case, respectively |s| = 1 in the discrete cases, satisfy (i) Re(s) > 0 in the continuous NLS case, (ii) |s| < 1 in the semi-and fully discrete NLS case.…”

Section: The Use Of Reflection Symmetriesmentioning

confidence: 99%

“…There is a certain redundancy in the matrix data, since different matrix data can determine the same solution, but we will narrow this down to a considerable extent (section 4.2, see also [143]).…”

Section: )mentioning

confidence: 99%

“…Hence a reflection is a symmetry of the solutions given in proposition 4.1 for all the NLS systems. A symmetry transformation of this kind for continuous NLS solutions already appeared in [143].…”

Section: )mentioning

confidence: 99%

See 3 more Smart Citations

Solutions of matrix NLS systems and their discretizations: a unified treatment

Dimakis¹,

Müller‐Hoissen²

2010

Inverse Problems

View full text Add to dashboard Cite

Using a bidifferential graded algebra approach to "integrable" partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework, by specifying a representation of the bidifferential graded algebra and imposing a reduction. By application of a general result, corresponding families of exact solutions are obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The solutions are parametrised in terms of constant matrix data subject to a Sylvester equation (which previously appeared as a rank condition in the integrable systems literature). These data exhibit a certain redundancy, which we diminish to a large extent.More precisely, we first consider more general AKNS-type systems from which two different matrix NLS systems emerge via reductions. In the continuous case, the familiar Hermitian conjugation reduction leads to a continuous matrix (including vector) NLS equation, but it is well-known that this does not work as well in the discrete cases. On the other hand there is a complex conjugation reduction, which apparently has not been studied previously. It leads to square matrix NLS systems, but works in all three cases (continuous, semi-and fully-discrete). A large part of this work is devoted to an exploration of the corresponding solutions, in particular regularity and asymptotic behaviour of matrix soliton solutions.

show abstract

“…Though reflection symmetries are not quite compatible with the structure of a superposition, we still have the following result (based on arguments in [143]).…”

mentioning

confidence: 84%

Section: The Use Of Reflection Symmetriesmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

See 2 more Smart Citations

Solutions of matrix NLS systems and their discretizations: a unified treatment

Dimakis¹,

Müller‐Hoissen²

2010

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…The log function we use is the principal branch of the complex-valued logarithm function and it has its branch cut along the negative real axis while log(1) = 0. The tan −1 function we use is the single-valued branch related to the principal branch of the logarithm as 5) and its branch cut is (−i∞, −i] ∪ [i, +i∞). For any square matrix M not having eigenvalues on that branch cut, we define…”

Section: Introductionmentioning

confidence: 99%

Exact solutions to the sine-Gordon equation

Aktosun

Demontis

Mee

2010

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable x and the temporal variable t, and they are exponentially asymptotic to integer multiples of 2π as x → ±∞. The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation using a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation. C

show abstract

Novel integrable Hamiltonian hierarchies with six potentials

2024

Acta Math Sci

View full text Add to dashboard Cite

Symmetries for exact solutions to the nonlinear Schrödinger equation

Cited by 37 publications

References 13 publications

Solutions of matrix NLS systems and their discretizations: a unified treatment

Solutions of matrix NLS systems and their discretizations: a unified treatment

Exact solutions to the sine-Gordon equation

Novel integrable Hamiltonian hierarchies with six potentials

Contact Info

Product

Resources

About