Rational solutions of the KdV equation with damping

Calogero, F.; Olshanetsky, M.; Perelomov, A. M.

doi:10.1007/bf02725599

Cited by 6 publications

(1 citation statement)

References 3 publications

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“…In particular, shedding light on the existence of rational solutions comprising multiple simple poles seems to be a naturally due development. Indeed, singular solutions to remarkable integrable equations and their pole dynamics have proved to be intriguingly linked to the dynamics of particles in many-body systems (see, for instance, the seminal papers [36][37][38]) as well as to rogue waves (see [39][40][41]), and are still subject of active investigations about the rational solutions in the KP hierarchy and Painlevé equations [42][43][44][45]. In addressing this issue, guidelines can be taken from discussions concerning other integrable equations, as put forward for instance in [37].…”

Section: N-pole Dynamics Of Rational Solutionsmentioning

confidence: 99%

On solutions to a novel non-evolutionary integrable 1 + 1 PDE

Giglio,

Landolfi,

Martina

2023

J. Phys. A: Math. Theor.

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We investigate real solutions of a C-integrable non-evolutionary partial differential equation in the form of a scalar conservation law where the flux density depends both on the density and on its first derivatives with respect to the local variables. By performing a similarity reduction dictated by one of its local symmetry generators, a nonlinear ordinary differential equation arises that is connected to the Painlevé III equation. Exact solutions are secured and described provided a constraint holds among the coefficients of the original equation. In the most general case, we pinpoint the generation of additional singularities by numerical integration. Then, we discuss the evolution of given initial profiles. Finally, we mention aspects concerning rational solutions with a finite number of poles.

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