Narrow Results

Using the bilinear neural network method (BNNM) and the symbolic computation system Mathematica, this paper explains how to find an exact solution for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. In terms of activation function and weight coefficient, BNNM is a more appealing option for users than traditional symbolic computation methods. It is possible to develop a wide range of solutions and expand the classes of exact solutions by modifying the activation function. The activation function’s versatility allows it to generate a wide range of solutions with several theoretical and practical uses. The analytical solution is obtained by using a double layer type, while the rogue wave solution and mixed solutions are obtained by using a single layer type. The evolution of these waves is then illustrated using various 3D graphs, 2D graphs, and density plots.

This study discusses the $\mu$ and Lie symmetries, $\mu$-conservation laws, analytical solutions, chaotic phenomena, and sensitivity analysis of the geophysical Korteweg–de Vries equation (GKdVE). The GKdVE describes the propagation of long waves in geophysical systems like oceans, taking into account the influence of the Coriolis force due to Earth’s rotation. We aim to understand the behavior of waves better in geophysical settings and their potential applications across fields like oceanography, meteorology, and climate science. By using the similarity variables, the GKdVE is transformed into a reduced ordinary differential equation (RODE). We employ the ($g^{\prime }$)-expansion procedure in one of the RODEs to obtain soliton solutions. Thanks to the ($g^{\prime }$)-expansion procedure, we discover six wave solutions. Through the implementation of the variational problem strategy, we derive both the Lagrangian and the $\mu$-conservation law ($\mu$-CL). Additionally, we revisit the planar dynamical system associated with the equation of interest, conducting a sensitive inspection to assess its sensitivity. Moreover, the introduction of a perturbed term reveals chaotic and quasi-periodic behaviors across a range of parameter values. Furthermore, we provide visual demonstrations of these properties through figures depicting the exact solutions.

In this paper, the generalized nonlinear Schrödinger equation (GNLS) is studied. The bifurcation of solitary waves of the equation is discussed first, by using the bifurcation theory of planar dynamical systems. Then, the respective numbers of solitary waves are derived under different conditions on the equation parameters. Exact solutions of smooth solitary waves are obtained in the explicit form of a(ξ)e^{i(ψ(ξ)-ωt)}, ξ = x - vt by qualitatively seeking the homoclinic and heteroclinic orbits for a class of Liénard equations. Finally, nonsmooth solitary wave solutions of the GNLS are investigated.

Using analytic methods from the dynamical systems theory, some new nonlinear wave equations are investigated, which have exact explicit parametric representations of breaking loop-solutions under some fixed parameter conditions. It is shown that these parametric representations are associated with some families of open level-curves of traveling wave systems corresponding to such nonlinear wave equations, each of which lies in an area bounded by a singular straight line and the stable and the unstable manifolds of a saddle point of such a system.

This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

This paper considers a three-dimensional linear nonautonomous systems. It shows that for every integer frequency parameter value, this system has a distinct type of knotted periodic solutions, which lie in a bounded region of R³. Exact explicit parametric representations of the knotted periodic solutions are given. By using these parametric representations, two series of three-dimensional flows are constructed, such that these three-dimensional autonomous systems have knotted periodic orbits in the three-dimensional phase space.

For a class of nonlinear diffusion–convection–reaction equations, corresponding to two families of heteroclinic orbits connecting two nodes of the traveling wave system, the existence of uncountably infinite many global monotonic and nonmonotonic wavefront solutions is discussed. By using the method of planar dynamical systems, the dynamical behavior of the corresponding traveling wave system is studied. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink wave solutions are given.

In this paper, we apply the method of dynamical systems to the traveling wave solutions of the Novikov equation. Through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system and exact cuspon wave solution, as well as a family of breaking wave solutions (compactons). We find that the corresponding traveling system of Novikov equation has no one-peakon solution.

In this paper, we consider variform exact peakon solutions for four nonlinear wave equations. We show that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various explicit exact one-peakon solutions, which are different from the one-peakon solution pe^-α|x-ct|. In fact, when a traveling system has a singular straight line and a curve triangle surrounding a periodic annulus of a center under some parameter conditions, there exists peaked solitary wave solution (peakon).

In this paper, we consider a model which is a modulated equation in a discrete nonlinear electrical transmission line. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we derive all explicit exact parametric representations of solutions (including smooth solitary wave solutions, smooth periodic wave solutions, peakons, compactons, periodic cusp wave solutions, etc.) under different parameter conditions.

In this paper, we consider a model which is the modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems and investigating the dynamical behavior, we obtain bifurcations of the phase portraits of the system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including smooth solitary wave and periodic wave solutions, periodic cusp wave solutions) under different parameter conditions.

In this paper, we consider a model which is a generalization of the nonlinear Schrödinger equation where the dispersive term was substituted by a nonlocal integral term with given kernel. The study on this model derives a planar dynamical system with two singular straight lines. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical system, we obtain all possible explicit exact parametric representations of solutions (including kink wave solutions, unbounded wave solutions, compactons, etc.) under different parameter conditions. The existence of bounded solutions of the planar dynamical system implies that there exist infinitely many breather solutions of this generalized nonlinear Schrödinger system.

In this paper, we consider a modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems to investigate the dynamical behavior for this system, we obtain bifurcations of phase portraits under different parameter conditions. Corresponding to some special level curves, we derive exact explicit parametric representations of solutions (including smooth solitary wave solutions, peakons, compactons, periodic cusp wave solutions) under different parameter conditions.

In this paper, we consider a model created by diffraction in periodic media. The study of the traveling wave solutions for this model derives a planar dynamical system with a singular straight line. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, periodic peakon solutions, compactons, etc.) under different parameter conditions.

Propagating modes in a class of nonic derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions $\varphi (\xi )$ and $\psi (\xi )$ in the solutions $A(x,t)=[\varphi (\xi )+i\psi (\xi )]exp(i(px-\mathrm{\Omega }t))$, $(\xi =x-ct)$ satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude $\mathrm{\Phi }={\varphi }^2+{\psi }^2$ can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions $A(x,t)$ and $\varphi (\xi ),\psi (\xi )$ can be exactly obtained. Thirty six exact explicit solutions of equation are derived.

In this paper, we consider the traveling wave solutions for a shallow water equation. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. On the basis of the theory of the singular traveling wave systems, we obtain the bifurcations of phase portraits and explicit exact parametric representations for solitary wave solutions and smooth periodic wave solutions, as well as periodic peakon solutions. We show the existence of compacton solutions of the equation under different parameter conditions.

In this paper, we consider the exact explicit solutions for the famous generalized Hénon–Heiles (H–H) system. Corresponding to the three integrable cases, on the basis of the investigation of the dynamical behavior and level curves of the planar dynamical systems, we find all possible explicit exact parametric representations of solutions in the invariant manifolds of equilibrium points in the four-dimensional phase space. These solutions contain quasi-periodic solutions, homoclinic solutions, periodic solutions as well as blow-up solutions. Therefore, we answer the question: what are the flows in the center manifolds and homoclinic manifolds of the generalized Hénon–Heiles (H–H) system. As an application of the above results, we consider the traveling wave solutions for the coupled $(n+1)$-dimensional Klein–Gordon–Schrödinger Equations with quadratic power nonlinearity.

In this paper, we study a model of generalized Dullin–Gottwald–Holm equation, depending on the power law nonlinearity, that derives a series of planar dynamical systems. The study of the traveling wave solutions for this model derives a planar Hamiltonian system. By investigating the dynamical behavior and bifurcation of solutions of the traveling wave system, we derive all possible exact explicit traveling wave solutions, under different parametric conditions. These results completely improve the study of traveling wave solutions to the mentioned model stated in [Biswas & Kara, 2010].

Raman soliton model in nanoscale optical waveguides, with metamaterials, having polynomial law nonlinearity is investigated by the method of dynamical systems. The functions $\varphi (\xi )$ in the solutions $q(x,t)=\varphi (\xi )\mathrm{\text{exp}}(i(-kx+\omega t)),$$(\xi =x-vt)$ satisfy a singular planar dynamical system having two singular straight lines. By using the bifurcation theory method of dynamical systems to the equations of $\varphi (\xi )$, under 23 different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons and peakons as well as compacton solutions for this planar dynamical system are obtained. 92 exact explicit solutions of system (6) are derived.

This paper considers a class of three-dimensional systems constructed by a rotating planar symmetric cubic vector field. To study its periodic orbits including homoclinic orbits, which may be knotted in space, we classify the types of periodic orbits and then calculate their exact parametric representations. Our study shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on three families of invariant tori. Numerical examples of $(m,n)$-torus knot periodic orbits have also been provided to illustrate our theoretical results.