Traveling Soliton Solutions Research Articles

An innovative method for solving the nonlinear fractional diffusion reaction equation with quadratic nonlinearity analysis

Abstract This study generates and investigates spreading solitons in the fractional DR quadratic equation (FDE) with fractional derivatives using the Extended Direct Algebraic Method (EDAM). In population growth, mathematical biology, and reaction-diffusion mechanisms, the FDE is crucial. Applying the series form solution to the NODE from the FDE conversion into a recommended EDAM yields many traveling soliton solutions. To characterize and explore soliton structure propagation, we draw shock and kink soliton solutions. Through reaction-diffusion mechanics and mathematical biology, we may explain complex processes in many academic subjects.

Particle trajectories in the KP-II equation

In the current work, we consider particle trajectories beneath traveling soliton solutions described by the Kadomtsev–Petiviashvili-II (KP-II) equation, which is a model for small-amplitude water waves in shallow water. The KP-II equation has a large number of exact solutions describing crossing line solitons. Here, we describe the particle drift induced by the single line soliton and various two- and three-soliton solutions on a two-dimensional horizontal domain.First, the derivation of the KP-II equation is used to exhibit expressions for the three components of the fluid velocity vector induced by a surface wave profile given by the KP-II equation. These velocity components can be used to specify a coupled system of ordinary differential equations (ODEs) which describe the motion of a fluid particle. Given an initial position inside the fluid for a specific particle, as well as continuously providing time-dependent values for the surface deflection, the ODE system can be solved numerically to find the particle path induced by the passage of a wave. A special numerical integration grid tracking the particle is used to handle integral terms occurring in the fluid velocity expressions.

Darboux transformation of a two‐component generalized Sasa–Satsuma equation and explicit solutions

Under investigation in this paper is a two‐component generalized Sasa–Satsuma equation associated with a 4 × 4 matrix spectral problem. We construct Darboux transformation for the two‐component generalized Sasa–Satsuma equation on basis of the gauge transformation between the Lax pairs. The corresponding N‐fold Darboux transformation is derived in terms of both iterative representation and compact determinant form. As applications of the Darboux transformation together with the limit technique, some interesting solutions including traveling soliton solutions, breather soliton solutions, and rogue wave solutions are explicitly obtained and graphically illustrated.

Exact Travelling Envelope Solitons and Kink-soliton Solutions for the Josephson Nonlinear Left-handed Transmission Line

Exact traveling soliton solutions for the Josephson nonlinear Left-handed transmission line (NL-JLHTL) based on the periodic structure of an array of Josephson junctions (JJs) are investigated. The nonlinearity of the Josephson left-handed transmission line (JLHTL) is due to the highly nonlinear nature of the JJs that provide the shunt inductances required to realize an LHTL. Applying the generalized Ricati methods, we analytically and successfully derive exact traveling kink solitons, bright and dark solitary wave solutions on this network. The left-handedness of the line is explicitly confirmed in numerical simulations with the existence of bright and dark soliton solutions in good agreement with analytical predictions.

A new class of traveling solitons for cubic fractional nonlinear Schrödinger equations

We consider the one-dimensional cubic fractional nonlinear Schrödinger equationi∂tu−(−Δ)σu + |u|2u=0,where and the operator is the fractional Laplacian of symbol . Despite the lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form0\\end{eqnarray} ?>u(t,x)=e−it(|k|2σ−ω2σ)Qω,k(x−2tσ|k|2σ−2k),k∈R, ω>0by a rather involved variational argument.

Exact traveling soliton solutions for the scalar Qiao equation

We investigate exact traveling wave solutions of the scalar Qiao equation proposed by Li and Qiao (2010 J. Math. Phys. 51 042703) using the generalized auxiliary equation method. This equation is known to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation for a fixed value of k. Symmetries of the scalar Qiao equation and its solutions are also considered. The obtained solutions include kink and antikink solitons, bright and dark solitons, singular solutions and exponential solutions. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics. This work could be also relevant for numerical studies of the scalar Qiao equation.

Exact solutions of the nonlinear differential—difference equations associated with the nonlinear electrical transmission line through a variable-coefficient discrete (G′/G)-expansion method

We investigated exact traveling soliton solutions for the nonlinear electrical transmission line. By applying a concise and straightforward method, the variable-coefficient discrete (G′/G)-expansion method, we solve the nonlinear differential—difference equations associated with the network. We obtain some exact traveling wave solutions which include hyperbolic function solution, trigonometric function solution, rational solutions with arbitrary function, bright as well as dark solutions.

Cusped and Smooth Solitons for the Generalized Camassa-Holm Equation on the Nonzero Constant Pedestal

We investigate the traveling solitary wave solutions of the generalized Camassa-Holm equationut - uxxt + 3u2ux=2uxuxx + uuxxxon the nonzero constant pedestallimξ→±∞⁡uξ=A. Our procedure shows that the generalized Camassa-Holm equation with nonzero constant boundary has cusped and smooth soliton solutions. Mathematical analysis and numerical simulations are provided for these traveling soliton solutions of the generalized Camassa-Holm equation. Some exact explicit solutions are obtained. We show some graphs to explain our these solutions.

Traveling solitons in the parametrically driven nonlinear Schrödinger equation.

We show that the (undamped) parametrically driven nonlinear Schrödinger equation has wide classes of traveling soliton solutions, some of which are stable. For small driving strengths stable nonpropagating and moving solitons co-exist while strongly forced solitons can only be stable when moving sufficiently fast.