Related papers: Evolution equations: Frobenius integrability, cons…
Travelling-wave solutions of the standard and compound form of Korteweg-de Vries-Burgers equations are found using factorizations of the corresponding reduced ordinary differential equations. The procedure leads to solutions of Bernoulli…
Symmetries and conservation laws are studied for a generalized Westervelt equation which is a nonlinear partial differential equation modelling the propagation of sound waves in a compressible medium. This nonlinear wave equation is widely…
We study various properties of solutions of an extended nonlinear Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric evolution problems -- including vortex filament dynamics -- and governs propagation of short pulses…
A set of traveling wave solution to convection-reaction-diffusion equation is studied by means of methods of local nonlinear analysis and numerical simulation. It is shown the existence of compactly supported solutions as well as solitary…
In this paper, we investigate the instability of one-dimensionally stable periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Zakharov-Kuznetsov…
We investigate the presence of soliton solutions in some classes of nonlinear partial differential equations, namely generalized Korteweg-de Vries-Burgers, Korteveg-de Vries-Huxley, and Korteveg-de Vries-Burgers-Huxley equations, which…
This manuscript embarks on an in-depth exploration of the modified Korteweg-de Vries (mKdV) equation, with a particular emphasis on unraveling the intricate structure of its infinite symmetries and their physical interpretations. Central to…
Generalizing results by Bryant and Griffiths [Duke Math. J., 1995, V.78, 531-676], we completely describe local conservation laws of second-order (1+1)-dimensional evolution equations up to contact equivalence. The possible dimensions of…
We present analytic self-similar or traveling wave solutions for a one-dimensional coupled system of continuity, compressible Euler and heat conduction equations. Different kind of equation of states are investigated. In certain forms of…
We introduce a model for nonlinear viscoelastic solids where traveling shear waves with compact support are possible. We obtain an exact compact solution. We also derive a new Burger's type evolution equation associated with the introduced…
We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp…
The Euler's equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymtotic expansion of the Euler's equations is taken (to a certain order of smallness of the scale parameters), relations to…
A potential representation for the subset of traveling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves a reduction of a third order partial differential equation to a first order ordinary…
The present paper deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves. Our method can be seen as an extension to spatially periodic waves…
We study solitary wave solutions of the fifth-order Korteweg - de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear…
We prove compactness and hence existence for solutions to a class of non linear transport equations. The corresponding models combine the features of linear transport equations and scalar conservation laws. We introduce a new method which…
The spatial version of the fourth-order Dysthe equations describe the evolution of weakly nonlinear narrowband wave trains in deep waters. For unidirectional waves, the hidden Hamiltonian structure and new invariants are unveiled by means…
We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation…
Obstacles to integrability in perturbed evolution equations are overcome by allowing higher-order terms in the expansion of the solution to depend explicitly on time and position. With a special expansion algorithm, obstacles vanish…
The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation…