Generalizations of the short pulse equation

Hone, Andrew N. W.; Novikov, V. S.; Wang, Jing Ping

doi:10.1007/s11005-017-1022-3

Cited by 24 publications

(28 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The developed methodology could be useful in many applied contexts, allowing one to bypass similar contradictions in the studies related, for example, to Kadomtsev‐Petviashvili (KP) and short‐pulse equations, as well as many similar equations and generalizations (see, for example, Refs. and and references therein).…”

Section: Discussionmentioning

confidence: 99%

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

Хуснутдинова

Tranter

2019

Stud Appl Math

View full text Add to dashboard Cite

In this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values.

show abstract

Section: Discussionmentioning

confidence: 99%

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

Хуснутдинова

Tranter

2019

Stud Appl Math

View full text Add to dashboard Cite

show abstract

“…We consider that p is constant and therefore equation (21) is satisfied and from (22)- (25) we obtain the following evolution equations [39,41]. Therefore, equations (29) reduce to the following super EB equations…”

Section: Osp(1 2) Wki Equationsmentioning

confidence: 99%

“…Generalizations of the SP equation were studied in [23]- [25], but in order to incorporate effects of polarization and anisotropy two-component (u, v) integrable generalizations of the SP equation were proposed [26]- [30]. These systems reduce to the SP equation (1) if u = 0 or v = 0 while others systems if u = v. The integrability of these systems were obtained mainly from a ZC representation or bilinear formalism and in [31,32] from a Hamiltonian point of view.…”

Section: Introductionmentioning

confidence: 99%

Super extensions of the short pulse equation

Brunelli

2018

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

show abstract

“…See, for instance, a review article [5] as for the soliton and periodic solutions and their properties. Quite recently, Hone et al [6] classified the integrable PDEs of second order with quadratic and cubic nonlinear terms. They found seven integrable PDEs including the SP equation.…”

Section: Introductionmentioning

confidence: 99%

Parametric solutions of the generalized short pulse equations

Matsuno

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

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.

show abstract

Generalizations of the short pulse equation

Abstract: We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.

Cited by 24 publications

References 33 publications

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

Super extensions of the short pulse equation

Parametric solutions of the generalized short pulse equations

Contact Info

Product

Resources

About