Simplest equation method to look for exact solutions of nonlinear differential equations

DOI10.1016/j.chaos.2004.09.109zbMath1069.35018arXivnlin/0406007OpenAlexW1971169529WikidataQ115359428 ScholiaQ115359428MaRDI QIDQ2483676

Nikolay A. Kudryashov

Publication date: 25 July 2005

Published in: Chaos, Solitons and Fractals (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/nlin/0406007

zbMATH Keywords

solitary waves Kuramoto-Sivashinsky equation periodic waves

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) Solutions to PDEs in closed form (35C05)

Related Items

Simple equation method for nonlinear partial differential equations and its applications, A nonlinear model of the dynamics of radial dislocations in microtubules, A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation, Exact solutions of the KdV equation with dual-power law nonlinearity, On wave structures described by the generalized Kuramoto-Sivashinsky equation, Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives, Bright and dark soliton solutions for the perturbed nonlinear schrödinger's equation with Kerr law and non-Kerr law nonlinearity, Traveling waves and statistical distributions connected to systems of interacting populations, Exact closed-form solutions of a fully nonlinear asymptotic two-fluid model, Analytical properties and exact solutions of the Lotka-Volterra competition system, Exact solutions of one pattern formation model, Symbolic computation of conservation laws and exact solutions of a coupled variable-coefficient modified Korteweg-de Vries system, Soliton solutions of Davey-Stewartson equation by trial equation method and ansatz approach, Travelling wave solutions and soliton solutions for the nonlinear transmission line using the generalized Riccati equation mapping method, Travelling wave solutions of a two-dimensional generalized Sawada-Kotera equation, Painlevé analysis and exact solutions for the Belousov-Zhabotinskii reaction-diffusion system, Analytic solutions of nonlinear partial differential equations by the power index method, Exact solutions and conservation laws for a generalized double combined sinh-cosh-Gordon equation, New exact traveling wave solutions of the (4+1)-dimensional Fokas equation, New generalized hyperbolic functions to find new exact solutions of the nonlinear partial differential equations, Travelling wave solutions of nonlinear evolution equations using the simplest equation method, Exact solutions and conservation laws of a \((2 + 1)\)-dimensional nonlinear KP-BBM equation, A new variable-coefficient Riccati subequation method for solving nonlinear lattice equations, Nonlinear waves in fluid flow through a viscoelastic tube, A note on the \(G'/G\)-expansion method, Exact solutions of the Kudryashov-Sinelshchikov equation, Higher Painlevé transcendents as special solutions of some nonlinear integrable hierarchies, A simple method and its applications to nonlinear partial differential equations, Properties of nonlinear waves in an actively-dissipative dispersive medium, From Laurent series to exact meromorphic solutions: The Kawahara equation, Special polynomials associated with the Burgers hierarchy, Exact and traveling-wave solutions for convection-diffusion-reaction equation with power-law nonlinearity, Elliptic solutions for a family of fifth order nonlinear evolution equations, Exact solutions of the Swift-Hohenberg equation with dispersion, Lie symmetries of nonlinear boundary value problems, The simplest equation method to study perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity, Redundant exact solutions of nonlinear differential equations, Exact traveling discrete kink-soliton solutions for the discrete nonlinear electrical transmission lines, Exact solutions of the generalized Bretherton equation, On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation, Nonlinear waves in media with fifth order dispersion, Exp-function method for solving nonlinear evolution equations with higher order nonlinearity, One method for finding exact solutions of nonlinear differential equations, A simple approach for reducing the order of equations with higher order nonlinearity, Nonlinear differential equations of the second, third and fourth order with exact solutions, Symmetry reductions, exact solutions and conservation laws of a new coupled KdV system, Elliptic traveling waves of the Olver equation, Exact explicit solutions and conservation laws for a coupled Zakharov-Kuznetsov system, Solitary and periodic solutions of the generalized Kuramoto-Sivashinsky equation, Popular ansatz methods and solitary wave solutions of the Kuramoto-Sivashinsky equation, Comment on: ``A novel approach for solving the Fisher equation using exp-function method [Phys. Lett. A 372 (2008) 3836], Symmetries, traveling wave solutions, and conservation laws of a \((3+1)\)-dimensional Boussinesq equation, Prandtl's boundary layer equation for two-dimensional flow: exact solutions via the simplest equation method, On the solutions and conservation laws of the coupled Drinfeld-Sokolov-Satsuma-Hirota system, Painlevé analysis and exact solutions for the modified Korteweg-de Vries equation with polynomial source, New variable separation solutions to the \((2 + 1)\)-dimensional Burgers equation, Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications, On solutions of generalized modified Korteweg-de Vries equation of the fifth order with dissipation, Stochastic effects for the reaction-Duffing equation with Wick-type product, Classical symmetries, travelling wave solutions and conservation laws of a generalized Fornberg-Whitham equation, Two new reductions methods for polynomial differential equations and applications to nonlinear pdes, Exact solutions and conservation laws of a \({(3+1)}\)-dimensional B-type Kadomtsev-Petviashvili equation, Exact solutions of the generalized sinh-Gordon equation, On the solutions of the time-delayed Burgers equation, Conservation laws and exact solutions of a generalized Benjamin-Bona-Mahony-Burgers equation, Refinement of the Korteweg-de Vries equation from the Fermi-pasta-Ulam model, Numerical simulation of the process of autoregulation of the arterial blood flow, Nonlinear waves in liquids with gas bubbles with account of viscosity and heat transfer, Solution of a fractional transport equation by using the generalized quadratic form, On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis-Processi equation and \(b\)-equation, Explicit and exact solutions to cubic Duffing and double-well Duffing equations, On elliptic solutions of nonlinear ordinary differential equations, Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration, General exp\(_a\)-function method for nonlinear evolution equations, A note on the (\(G^{\prime }/G\))-expansion method again, Modified method of simplest equation and its application to nonlinear PDEs, An extended simplest equation method and its application to several forms of the fifth-order KdV equation, Comment on: New exact traveling wave solutions of the \((3 + 1)\)-dimensional Kadomtsev-Petviashvili (KP) equation, Be careful with the Exp-function method, On ``new travelling wave solutions of the KdV and the KdV-Burgers equations, On nonlinear dynamics of interacting populations: coupled kink waves in a system of two populations, Seven common errors in finding exact solutions of nonlinear differential equations, Exact solutions of the generalized \(K(m, m)\) equations, Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations, Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs, The method and topological soliton solution of the \(K(m, n)\) equation, Nonlinear evolution equation for describing waves in a viscoelastic tube, The exp-function approach to the Schwarzian Korteweg-de Vries equation, Comment on: ``Application of the \((\frac{G'}{G})\)-expansion method for the complex KdV equation [Huiqun Zhang, Commun. Nonlinear Sci. Numer. Simul. 15; 2010:1700-1704], A discrete generalization of the extended simplest equation method, Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity, Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics, Be careful with the Exp-function method - additional remarks, On nonlinear differential equation with exact solutions having various pole orders, Exact solutions of systems describing interacting populations with low critical densities, Note on exact solutions for a class of nonlinear diffusion equations, Traveling wave solutions of the generalized nonlinear evolution equations, A new rational auxiliary equation method and exact solutions of a generalized Zakharov system, On nonlinear population waves, A model of malignant gliomas through symmetry reductions, Dynamical features of the generalized Kuramoto-Sivashinsky equation, Dynamics of soliton waves of group-invariant solutions through optimal system of an extended KP-like equation in higher dimensions with applications in medical sciences and mathematical physics, On the solutions and conservation laws of a coupled Kadomtsev-Petviashvili equation, An analytical approach to obtain exact solutions of some space-time conformable fractional differential equations, Extended simplest equation method for nonlinear differential equations, On traveling waves in lattices: the case of Riccati lattices, Integrability of differential equations with fluid mechanics application: from Painlevé property to the method of simplest equation, Exact Solutions and Conservation Laws of the Joseph-Egri Equation with Power Law Nonlinearity, On the Exact Solutions of the Klein–Gordon–Zakharov Equations, Travelling Wave Solutions of a Generalized Variable-Coefficient Gardner Equation, Analytical and numerical studying of the perturbed Korteweg-de Vries equation, First integrals and general solution of the complex Ginzburg-Landau equation, Some more solutions of Caudrey-Dodd-Gibbon equation using optimal system of Lie symmetries, Painlevé integrability and new exact solutions of the \((4 + 1)\)-dimensional Fokas equation, Two component model of microtubules and continuum approximation, Analytical solitonic solutions of higher-order non-linear Ito equation, Breather-type and multi-wave solutions for \(( 2 + 1 )\)-dimensional nonlocal Gardner equation, Analytical properties of nonlinear dislocation equation, Exact solitary wave and periodic-peakon solutions of the complex Ginzburg-Landau equation: dynamical system approach, Lie symmetry analysis, Lie-Bäcklund symmetries, explicit solutions, and conservation laws of Drinfeld-Sokolov-Wilson system, Exact solutions and conservation laws of Zakharov-Kuznetsov modified equal width equation with power law nonlinearity, Analytical properties of the perturbed FitzHugh-Nagumo model, Traveling wave solutions to Kawahara and related equations, On the instability of monotone traveling-wave solutions for a generalized Korteweg-de Vries-Burgers equation, A note on the \(\exp (-\varphi (z))\) expansion method, Fifth order evolution equation for long wave dissipative solitons, Nonlinear physical complex Hirota dynamical system: construction of chirp free optical dromions and numerical wave solutions, On the modified generalized multidimensional KP equation in plasma physics and fluid dynamics in \((3+1)\) dimensions, From the Fermi-Pasta-Ulam model to higher-order nonlinear evolution equations, Exact solutions of \(\phi^4\) equation using Lie symmetry approach along with the simplest equation and exp-function methods, Operator-based approach for the construction of analytical soliton solutions to nonlinear fractional-order differential equations, An extended Korteweg-de Vries equation: multi-soliton solutions and conservation laws, A (2+1)-dimensional breaking soliton equation: solutions and conservation laws, Painlevé analysis and exact solutions of the fourth-order equation for description of nonlinear waves, Reductions of PDEs to first order ODEs, symmetries and symbolic computation, Symmetries, solutions and conservation laws of a class of nonlinear dispersive wave equations, New exact solutions for some power law nonlinear diffusion equation, Solutions and conservation laws of a generalized second extended \((3+1)\)-dimensional Jimbo-Miwa equation, Exact solutions of damped improved Boussinesq equations by extended \((G'/G)\)-expansion method, Bifurcations in Kuramoto-Sivashinsky equations, Classical solution for the linear sigma model, On the analytical expression of the multicompacton and some exact compact solutions of a nonlinear diffusive Burgers' type equation, Traveling wave reduction of the modified KdV hierarchy: the Lax pair and the first integrals, Numerical treatment of non-linear fourth-order distributed fractional sub-diffusion equation with time-delay, Lie group analysis, exact solutions and new conservation laws for combined KdV-mKdV equation, The generalized Duffing oscillator, The extended multiple \(\left(G'/ G\right)\)-expansion method and its application to the Caudrey-Dodd-Gibbon equation, Exact solutions for \((4 + 1)\)-dimensional nonlinear Fokas equation using extended \(F\)-expansion method and its variant, On the solutions and conservation laws of a coupled KdV system, Nonlinearly self-adjoint, conservation laws and solutions for a forced BBM equation, Exact solutions of the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations, Exact solutions for power-law regularized long-wave and \(R(m, n)\) equations with time-dependent coefficients, Application of the method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg-de Vries equation and generalized Camassa-Holm equation, Lie symmetry analysis, traveling wave solutions, and conservation laws to the \((3+1)\)-dimensional generalized B-type Kadomtsev-Petviashvili equation, On the exact and numerical solution of the time-delayed Burgers equation, Quasi-exact solutions of the dissipative Kuramoto-Sivashinsky equation, Polynomials in logistic function and solitary waves of nonlinear differential equations, Lie group analysis and dynamical behavior for classical Boussinesq-Burgers system, Solutions of two nonlinear evolution equations using Lie symmetry and simplest equation methods, Existence of solitary solutions in a class of nonlinear differential equations with polynomial nonlinearity, Exact traveling wave solutions to the fractional coupled nonlinear Schrödinger equations, Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations, Reductions of PDEs to second order ODEs and symbolic computation, Direct and inverse relationships between Riccati systems coupled with multiplicative terms, Highly dispersive optical solitons of an equation with arbitrary refractive index, Exact solutions and integrability of the Duffing-van der Pol equation, On solitary wave solutions of a class of nonlinear partial differential equations based on the function \(1/ \cosh^n(\alpha x + \beta t)\), Polygons of differential equations for finding exact solutions, Analytical features of the SIR model and their applications to COVID-19, Solitary wave solutions of time-space nonlinear fractional Schrödinger's equation: two analytical approaches, Quasi-exact solutions of nonlinear differential equations, Travelling wave solutions to the Kuramoto-Sivashinsky equation, Simplest equation method for some time-fractional partial differential equations with conformable derivative, A new method to obtain either first- or second-order reductions for parametric polynomial ODEs, Conserved quantities and solutions of a \((2+1)\)-dimensional Hǎrǎgus-Courcelle-Il'ichev model, Exact solutions of the equation for surface waves in a convecting fluid, Solutions and conservation laws of a \((2+1)\)-dimensional Boussinesq equation, Painlevé analysis and exact solutions of the nonlinear diffusion equation with a polynomial source, Logistic function as solution of many nonlinear differential equations, Numerical investigation of nonlinear waves connected to blood flow in an elastic tube with variable radius, Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations, Analytical solutions of the Lorenz system, Asymptotic and exact solutions of the Fitzhugh-Nagumo model, Travelling wave solutions for some time-delayed equations through factorizations, The logarithmic \((1+1)\)-dimensional KdV-like and \((2+1)\)-dimensional KP-like equations: Lie group analysis, conservation laws and double reductions, Symmetry analysis and exact solutions for a generalized Fisher equation in cylindrical coordinates, Painlevé analysis and a solution to the traveling wave reduction of the Radhakrishnan-Kundu-Lakshmanan equation, Elliptic solutions in the Hénon-Heiles model, Bifurcation analysis and solutions of a three-dimensional Kudryashov-Sinelshchikov equation in the bubbly liquid, Generalization of the simplest equation method for nonlinear non-autonomous differential equations, Lie symmetries and conservation laws for a generalized \((2+1)\)-dimensional nonlinear evolution equation, Comment on: ``The modified extended tanh-function method for solving Burgers-type equations, An efficient method for solving fractional Black-Scholes model with index and exponential decay kernels, Symmetry analysis and conservation laws of the Zoomeron equation, A study on the (2+1)-dimensional first extended Calogero-Bogoyavlenskii-Schiff equation, Exact solutions for a new nonlinear KdV-like wave equation using simplest equation method and its variants, New exact solutions for a higher-order wave equation of KdV type using the multiple simplest equation method, A new perspective for analytical and numerical soliton solutions of the Kaup-Kupershmidt and Ito equations, Painlevé analysis and exact solutions of the Korteweg-de Vries equation with a source, A note on solutions of the generalized Fisher equation, Some Solutions of Generalised Variable Coefficients KdV Equation by Classical Lie Approach, On the exact solutions, lie symmetry analysis, and conservation laws of Schamel–Korteweg–de Vries equation, Numerical solutions of differential equations having cubic nonlinearity using Boole collocation method, Symmetry analysis and some new exact solutions of some nonlinear KdV-like equations, Solitons and other solutions of perturbed nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index, Exact solutions and bifurcation for the resonant nonlinear Schrödinger equation with competing weakly nonlocal nonlinearity and fractional temporal evolution, Dynamical behavior of analytical soliton solutions, bifurcation analysis, and quasi-periodic solution to the \((2+1)\)-dimensional Konopelchenko-Dubrovsky (KD) system, Numerical study of the model described by the fourth order generalized nonlinear Schrödinger equation with cubic-quintic-septic-nonic nonlinearity, Kinks and bell-type solitons in microtubules, A study on Caudrey–Dodd–Gibbon–Sawada–Kotera partial differential equation, On solutions of the fifth-order dispersive equations with porous medium type non-linearity, Closed-form solutions and conserved quantities of a new integrable \((2 +1)\)-dimensional Boussinesq equation of nonlinear sciences, Propagation of dark-bright soliton and kink wave solutions of fluidized granular matter model arising in industrial applications, On the classical and nonclassical symmetries of a generalized Gardner equation, Properties of the generalized Chavy-Waddy-Kolokolnikov model for description of bacterial colonies, Nonlinear dynamics of microtubules and series expansion unknown function method, Optical soliton solutions to a \((2+1)\) dimensional Schrödinger equation using a couple of integration architectures, The polygonal method for constructing exact solutions to certain nonlinear differential equations describing water waves, Classical symmetries of the Klein-Gordon-Zakharov equations with time-dependent variable coefficients, Unnamed Item, An Overview of the Generalized Gardner Equation: Symmetry Groups and Conservation Laws, On Symmetries and Conservation Laws for a Generalized Fisher–Kolmogorov–Petrovsky–Piskunov Equation, Stationary solitary and kink solutions in the helicoidal Peyrard-Bishop model of DNA molecule, Traveling waves in nonlinear media with dispersion, dissipation, and reaction, Meromorphic solutions of nonlinear ordinary differential equations, Effect of cubic nonlinearity on soliton solutions of the Benjamin-Bona-Mahony equation, On the exact solitary wave solutions of a special class of Benjamin-Bona-Mahony equation, Unnamed Item, Exact analytical solutions for 3D-Gross-Pitaevskii equation with periodic potential by using the Kudryashov method, A novel method for new solutions of time fractional (1+2)-dimensional nonlinear Schrödinger equation involving dual-power law nonlinearity, Exact solutions and conservation laws for a generalized improved Boussinesq equation, Lie group theory, stability analysis with dispersion property, new soliton solutions and conserved quantities of 3D generalized nonlinear wave equation in liquid containing gas bubbles with applications in mechanics of fluids, biomedical sciences and cell biology, Painlevé analysis and exact solutions of a predator–prey system with diffusion, New wave behaviors of the system of equations for the ion sound and Langmuir Waves

Cites Work