Solitary Wave Type Solutions Research Articles

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Abundant variant wave patterns by coupled Boussinesq–Whitham–Broer–Kaup equations

This paper focus on searching for abundant wave patterns to time-fractional coupled Boussinesq–Whitham–Broer–Kaup equations, which plays a crucial role in obtaining the novel insights for propagational dynamics of shallow water. Through the innovative application of complete discrimination system for polynomial method (CDSPM) and conformable fractional transformation, the whole types of solutions for the fractional equations presented in the existing literatures are obtained. In particular, we obtain the solitary wave type solutions and Jacobian elliptic functions type solutions, and the later are initially founded. The existence and applicability of the obtained solutions are proved by concrete examples. Finally, the nature of wave propagation and the fractional characteristics are intuitively revealed by graphical simulations.

Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes

This survey presents results on the stability of elevation solitary waves in axisymmetric elastic membrane tubes filled with a fluid. The elastic tube material is characterized by an elastic potential (elastic energy) that depends non-linearly on the principal deformations and describes the compliant elastic media. Our survey uses a simple model of an inviscid incompressible fluid, which nevertheless makes it possible to trace the main regularities of the dynamics of solitary waves. One of these regularities is the spectral stability (linear stability in form) of these waves. The basic equations of the ‘axisymmetric tube – ideal fluid’ system are formulated, and the equations for the fluid are averaged over the cross-section of the tube, that is, a quasi-one-dimensional flow with waves whose length significantly exceeds the radius of the tube is considered. The spectral stability with respect to axisymmetric perturbations is studied by constructing the Evans function for the system of basic equations linearized around a solitary wave type solution. The Evans function depends only on the spectral parameter , is analytic in the right-hand complex half-plane , and its zeros in coincide with unstable eigenvalues. The problems treated include stability of steady solitary waves in the absence of a fluid inside the tube (the case of constant internal pressure), together with the case of local inhomogeneity (thinning) of the tube wall, the presence of a steady fluid filling the tube (the case of zero mean flow) or a moving fluid (the case of non-zero mean flow), and also the problem of stability of travelling solitary waves propagating along the tube with non-zero speed. Bibliography: 83 titles.

A (2+1)-dimensional shallow water equation and its explicit lump solutions

We introduce a new (2+1)-dimensional equation by modifying the potential form of the Calogero–Bogoyavlenskii–Schiff (CBS) equation. By applying the Hirota bilinear method, we construct explicit lump solutions to this new equation and establish necessary and sufficient conditions that guarantee that the solutions are analytic and rationally localized in all directions in space. We also depict the evolution of the profiles of some selected lump solutions with three-dimensional and contour plots. It is immediately observed that the lump solutions generated are solitary wave type solutions as is the case with the KP equation.

Solitary wave solutions with non-monotone shapes for the modified Kawahara equation

The propagation of stationary solitary waves of the non-linear modified Kawahara equation is considered. The asymptotic boundary conditions admit the trivial solution along with the solitary wave type solution, which is a bifurcation problem. The bifurcation is treated by reformulating the problem into a problem for identification of an unknown coefficient from over-posed boundary data, in which the trivial solution is excluded. Making use of the method of variational imbedding, the inverse problem for the coefficient identification is reformulated as a higher-order boundary value problem. This approach to solving the fifth-order modified Kawahara equation is allowing identification of non-trivial solutions. The obtained boundary-value problem is solved by means of an iterative difference scheme, which is thoroughly validated. New traveling wave solutions with monotone and non-monotone shapes are obtained for different values of the problem parameters.

Localized traveling wave solution for a logarithmic nonlinear Schrödinger equation

We investigate localized traveling wave solutions for a Schrödinger equation with two logarithmic nonlinear terms under no external potential. It is shown that it can have solitary wave type solutions whose envelope profile depends on the two types of nonlinearity. Remarkably, the profile has cutoffs in the coordinate of propagation. We argue also some fundamental properties that discriminate it from power law type nonlinear Schrödinger equations.

Characterization and stability of localized bulging/necking in inflated membrane tubes

We consider localised bulging/necking in an inflated hyperelastic membrane tube with closed ends. We first show that the initiation pressure for the onset of localised bulging is simply the limiting pressure in uniform inflation when the axial force is held fixed. We then demonstrate analytically how, as inflation continues, the initial bulge grows continually in diameter until it reaches a critical size and then propagates in both directions. The bulging solution before propagation starts is of the solitary-wave type, whereas the propagating bulging solution is of the kink-wave type. The stability, with respect to axially symmetric perturbations, of both the solitary-wave type and the kink-wave type solutions is studied by computing the Evans function using the compound matrix method. It is found that when the inflation is pressure-controlled, the Evans function has a single non-negative real root and this root tends to zero only when the initiation pressure or the propagation pressure is approached. Thus, the kink-wave type solution is probably stable but the solitary-wave type solution is definitely unstable.

Evolution of lump solutions for the KP equation

The two (space)-dimensional generalisation of the Korteweg-de Vries (KdV) equation is the Kadomtsev-Petviashvili (KP) equation. This equation possesses two solitary wave type solutions. One is independent of the direction orthogonal to the direction of propagation and is the soliton solution of the KdV equation extended to two space dimensions. The other is a true two-dimensional solitary wave solution which decays to zero in all space directions. It is this second solitary wave solution which is considered in the present work. It is known that the KP equation admits an inverse scattering solution. However this solution only applies for initial conditions which decay at infinity faster than the reciprocal distance from the origin. To study the evolution of a lump-like initial condition, a group velocity argument is used to determine the direction of propagation of the linear dispersive radiation generated as the lump evolves. Using this information combined with conservation equations and a suitable trial function, approximate ODEs governing the evolution of the isolated pulse are derived. These pulse solutions have a similar form to the pulse solitary wave solution of the KP equation, but with varying parameters. It is found that the pulse solitary wave solutions of the KP equation are asymptotically stable, and that depending on the initial conditions, the pulse either decays to a pulse of lower amplitude (shedding mass) or narrows down (shedding mass) to a pulse of higher amplitude. The solutions of the approximate ODEs for the pulse evolution are compared with full numerical solutions of the KP equation and good agreement is found.

Nonlinear wave propagation in micropolar media—I. The general theory

In this work, the plane wave propagation in nonlinear micropolar solids is asymptotically investigated. Micropolar theory in the linear approximation predicts a dispersive optical (high-frequency) mode as well as a dispersive acoustical (low-frequency) mode for the harmonic waves in an unbounded medium. The acoustical mode has a weakly dispersive region when the wave number is small. If nonlinearity is also present in this weakly dispersive region and if both effects are small but finite, it may be expected that nonlinearity and dispersive effects can balance each other, and the wave propagation can be asymptotically governed by a nonlinear evolution equation which admits a solitary wave type solution. Using the reductive perturbation method to examine the plane wave propagation in a general nonlinear polar solid, it is found that far-field approximation of wave motion is governed by coupled Modified Korteweg-de Vries equations.

Nonlinear wave propagation in micropolar media—II. Special cases, solitary waves and Painlevé analysis

In this study, complex modified Korteweg-de Vries (CMKdV) equation describing far-field behaviour of a general micropolar solid obtained in [1] is first examined in the case of quadratic solid. Solitary wave type solutions of CMKdV equation are also presented and some physical interpretations are discussed. As a preliminary attempt to settle the question of complete integrability of the CMKdV equation, singular manifold (Painlevé) analysis is carried out.

On solitary waves for nonlinear schrödinger equations in higher dimensions

We investigate the existence of solitary waves of the nonlinear singular Schrodinger equation in arbitrary dimensions and prove that the phase φ of any solitary wave type solution u(x,t)= r(x,t).exp(iφ(x,t) is a linear function and can be determined by the velocity of the amplitude r . Thus the problem of finding solitary waves of the Schrodinger equation is equivalent to the investigation of solitary waves of the corresponding nonlinear scalar field equation. Further we prove some necessary conditions in the case χ = 0.

Stationary, oscillatory and solitary wave type solution of singular nonlinear schrödinger equations

AbstractIt is proven that for a certain class of singular nonlinear Schrödinger equations there exist stationary ground state and oscillatory solutions. Furthermore the existence of those solitary wave type solutions is considered which do not necessarely vanish at infinity. The results are applied to problems from plasma physics; to superfluid films, and to the Heisenberg ferromagnet spin chain.

The Schrödinger–Langevin equation: Special solutions and nonexistence of solitary waves

Two types of solutions of the Schrödinger–Langevin equation are investigated. It is proved that a special type of Gaussian solutions exist globally in time for the harmonic oscillator Hamiltonian. Furthermore, it is shown that the Schrödinger–Langevin equation can have no solitary wave type solutions in the damped free-particle case which lie in L2.