Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: past and present

Nikolay K. Vitanov

Home/Archive/arXiv

arXiv · 1906.08053

Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: past and present

Nikolay K. Vitanov

View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a short review of the evolution of the methodology of the Method of simplest equation for obtaining exact particular solutions of nonlinear partial differential equations (NPDEs) and the recent extension of a version of this methodology called Modified method of simplest equation. This extension makes the methodology capable to lead to solutions of nonlinear partial differential equations that are more complicated than a single solitary wave.

Explore related subjects

nlin.SI

Keep this discovery

BibTeX RIS

Nikolay K. Vitanov. 2019-06-18. Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: past and present. https://arxiv.org/abs/1906.08053

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rogue-like waves from collision of mKdV solitons

Interactions of two solitary waves with an up and a down orientation in the modified Korteweg-de Vries equation are shown to produce rogue-like waves. For waves that asymptotically vanish, the maximum ratio between the height of the interaction profile and the height of the tallest incoming wave is 2.41 when the waves have approximately equal speeds, and this ratio decreases to 2 when the speed ratio is 1.5. For waves that approach a non-zero constant at infinity, the same ratio reaches a maximum of 2.65 when the speed ratio of the waves is 6.32.

nlin.SI↗

arXiv2026

On B\"acklund transformations preserving the Darboux integrability of hyperbolic equations

This paper deals with two kinds of B\"acklund transformations for scalar hyperbolic partial differential equations. We prove that both these types of transformations map solutions of a Darboux integrable equation into solutions of, generally speaking, another but also Darboux integrable equation. The latter fact can be used to roughly check the completeness of a list of Darboux integrable equations. To illustrate this, we apply the above transformations to several equations from a well-known list of Darboux integrable equations and, as a result, obtain a Darboux integrable equation which is absent in this list, but is already known at present. As a generalization of the last equation, we construct a family of Darboux integrable equations that is parametrized by three arbitrary functions, each of which depends on two arguments. This family is probably new.

nlin.SI↗

arXiv2026

Complex singularities for Burgers' equation with piecewise-continuous initial conditions

There is a body of research devoted to understanding how complex singularities of solutions of nonlinear partial differential equations (pdes) spontaneously emerge at $t=0^+$ and propagate for $t>0$, and how their behaviour affects the solution on the real axis. Despite the importance of the small-time limit in these studies, there is still a lack of understanding of how complex singularities are born at $t=0^+$, including for initial conditions that are not analytic functions of the spatial variable. In this paper, we use Burgers' equation as a prototype nonlinear pde and study the complex-plane singularities for initial conditions that are piecewise smooth. Using matched asymptotic expansions, we show how infinitely many singularities emerge from points of discontinuity in a pattern that can be described using branches of the Lambert-$W$ function. For various initial conditions, we observe how these singularities rearrange themselves to align with the appropriate exactly-described long-time behaviour, including sigmoid-shaped travelling waves, constant-area (triangular wave) similarity solutions and $N$-wave solutions. In terms of Burgers' equation, our small-time asymptotic analysis of the singularity propagation for piecewise-continuous initial conditions illustrates the types of generic behaviours that arise for inner regions when diffusion dominates advection. More generally, this work is a step towards understanding complex-plane behaviour of solutions of nonlinear partial differential equations with non-analytic initial conditions.

nlin.SI↗