On Integrability of a Third-Order Complex Nonlinear Wave Equation

Sakovich, Sergei

doi:10.33581/1561-4085-2022-25-4-381-386

Cited by 3 publications

(3 citation statements)

References 25 publications

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“…In addition, by means of the reciprocal transformation, equations (1.5) and (1.6) were shown to be transformed to the Tzitzeica equation whereas equation (1.7) was found to be related to the sine-Gordon (sG) equation [6]. See also an analogous work demonstrating that equation (1.7) is transformed to the sG equation by means of the reciprocal transformation combined with a sequence of dependent variable transformations [13]. These reciprocal links between equations (1.5)-(1.7) and the integrable Tzitzeica and sG equations suggest the existence of exact solutions of the former equations.…”

Section: Introductionmentioning

confidence: 93%

“…The above equation has been considered in [13], where the parametric solutions are shown to be constructed in terms of solutions of the sG equation. The parametric solutions of equation (4.44) are obtained simply by applying the scalings y → −iy, τ → iτ , y 0 → −iy 0 in addition to the scalings mentioned above for u, x and t to (4.2).…”

Section: If One Carries Out the Scalingsmentioning

confidence: 99%

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Parametric solutions of the generalized short pulse equations

Matsuno

2020

J. Phys. A: Math. Theor.

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We consider three novel PDEs associated with the integrable generalizations of the short pulse equation classified recently by Hone et al (2018 Lett. Math. Phys. 108 927-947). In particular, we obtain a variety of exact solutions by means of a direct method analogous to that used for solving the short pulse equation. The main results reported here are the parametric representations of the multisoliton solutions. These solutions include cusp solitons, unbounded solutions with finite slope and breathers, as well as traveling periodic waves with cusp singularity. In addition, the smooth periodic traveling wave solutions are provided by employing phase plane analysis. Several new features of solutions are exhibited. As for non-periodic solutions, smooth breather solutions are of particular interest from the perspective of applications to real physical phenomena. The cycloid reduced from the periodic traveling wave with cusps is also worth remarking in connection with Gerstner's trochoidal solution in deep gravity waves. A number of works are left for future study, some of which will be addressed in concluding remarks.

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Section: Introductionmentioning

confidence: 93%

Section: If One Carries Out the Scalingsmentioning

confidence: 99%

Parametric solutions of the generalized short pulse equations

Matsuno

2020

J. Phys. A: Math. Theor.

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“…Dodd [7] gives the one-and two-soliton solutions of the KW equation by Hirota direct method [5] but Pekcan [8] shows that the KW equation (1) does not have three-soliton solution and thus it is not integrable in Hirota sense. Moreover, Sakovich [9] claims that the KW equation (1) is not Painlevé integrable and he believes that it cannot possess any good Lax representation [10].…”

Section: Introductionmentioning

confidence: 99%

Some new types of exact solutions for the Kac–Wakimoto equation associated with ${{\mathfrak{e}}}_{6}^{(1)}$

2020

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Recently, it is shown that the Kac-Wakimoto equation associated with ( ) e 6 1 is not integrable since it does not pass Painlevé test and does not have three-soliton solution, even it has one-and twosoliton solutions. Thus in this paper, we investigate some new types of exact solutions for this equation based on its bilinear form. As a result, the rational solutions, kink-type breather solution and degenerate three-solitary wave solutions of this equation are found. The properties and space structures of these exact solutions are analyzed by displaying their profiles in (x, y)-directions. Furthermore, the Lie symmetry analysis is done to present the one-parameter group of symmetries for the KW equation.

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A vectorial binary Darboux transformation for the first member of the negative part of the AKNS hierarchy

Mueller-Hoissen

2023

J. Phys. A: Math. Theor.

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Using bidifferential calculus, we derive a vectorial binary Darboux transformation for the first member  of the ``negative'' part of the AKNS hierarchy. A reduction leads to the first ``negative flow" of the  NLS hierarchy, which in turn is a reduction of a rather simple nonlinear complex PDE in two dimensions, with a leading  mixed third derivative. This PDE may be regarded as describing geometric dynamics of a complex scalar field  in one dimension, since it is invariant under coordinate transformations in one of the two independent variables.  We exploit the correspondingly reduced vectorial binary Darboux transformation to generate multi-soliton  solutions of the PDE, also with additional rational dependence on the independent variables, and  on a plane wave background. This includes rogue waves.

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On Integrability of a Third-Order Complex Nonlinear Wave Equation

Cited by 3 publications

References 25 publications

Parametric solutions of the generalized short pulse equations

Parametric solutions of the generalized short pulse equations

Some new types of exact solutions for the Kac–Wakimoto equation associated with ${{\mathfrak{e}}}_{6}^{(1)}$

A vectorial binary Darboux transformation for the first member of the negative part of the AKNS hierarchy

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