Two-dimensional Solitary Waves Research Articles

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

Solitons and lumps in the cylindrical Kadomtsev-Petviashvili equation. II. Lumps and their interactions.

We study solitary waves in the cylindrical Kadomtsev-Petviashvili equation designated to media with positive dispersion (the cKP1 equation). By means of the Darboux-Matveev transform, we derive exact solutions that describe two-dimensional solitary waves (lumps), lump chains, and their interactions. One of the obtained solutions describes the modulation instability of outgoing ring solitons and their disintegration onto a number of lumps. We also derive solutions describing decaying lumps and lump chains of a complex spatial structure-ripplons. Then, we study normal and anomalous (resonant) interactions of lump chains with each other and with ring solitons. Results obtained agree with the numerical data presented in Part I of this study [Hu et al., Chaos (2024)].

Characterizing energy dissipation of shallow-water wave breaking in a storm surge

While understanding breaking waves is crucial for the development of parameterizations used in modeling ocean and climate, the complete description of wave breaking is not well understood. We present direct numerical simulations of two-dimensional solitary waves that break on a uniform beach in shallow water, including storm surge represented by an inshore region.

Degenerate lump interactions within the Kadomtsev–Petviashvili equation

We consider the anomalous scattering of lumps – fully localised two-dimensional solitary waves – within the framework of the Kadomtsev–Petviashvili equation. Such entities can exist in media with positive dispersion. As has been established, lumps of equal amplitudes can experience anomalously slow interactions. They can also form stationary bound states — multilumps. The features of anomalous interactions of lumps and multilumps have not been studied yet in detail. Therefore, this work aims to find a simple and fast method to derive bound states of lumps and study anomalous interactions of multilumps. The asymptotic behaviour of lumps is analysed analytically and numerically, and the results obtained are illustrated graphically. The approach introduced in this paper can be generalised to other (2+1)-dimensional integrable systems.

Lump Interactions with Plane Solitons

We analyze in detail the interactions of two-dimensional solitary waves called lumps and onedimensional line solitons within the framework of the Kadomtsev–Petviashvili equation describing wave processes in media with positive dispersion. We show that line solitons can emit or absorb lumps or periodic chains of lumps, as well as interact with each other by means of lumps. Within a certain time interval, lumps or lump chains can emerge between two line solitons and then disappear due to absorption by one of the solitons. This phenomenon resembles the appearance of rogue waves in the oceans. The results obtained are graphically illustrated and can be applicable to the description of the physical processes occurring in plasmas, fluids, solids, nonlinear optical media, etc.

Experimental observation of O-solitary waves in shallow water

Solitary waves have been well studied in various fields, including water waves, optical communications, plasmas, and biophysics. We present experimental results that a two-dimensional solitary wave, with its crest shaped like the letter “O,” occurs around an island terrain in shallow water. We name it the O-soliton (a cylindrical-type soliton) in this paper. This is the first experimental observation of the O-soliton including both outward and inward propagation in shallow water. O-solitons are observed with stable and periodic occurrences under typical coastal conditions. A water column at its center is higher than eight times the initial mean wave height measured by six wave gauges arranged in a regular hexagon. A one-dimension solitary wave incurs a phase shift when it collides with the O-soliton. However, no phase shift is observed in the O-soliton itself. Furthermore, when it collapses to its center point, it remains in this state for a short period. This is analogous to the pausing of the wave crest during a head-on collision of solitary waves or the reflection of a solitary wave at a vertical wall. These results also enrich the general understanding of solitary waves, whose repeated occurrence suggests that they may naturally occur in the ocean, and motivate analogous studies in other fields of physics.

Application of PIV Technique in Teaching Experiment of Solitary Wave Flow Field Measurement

Particle Image Velocimetry (PIV), as an advanced experimental technique of fluid mechanics, has been widely used in many fields such as experimental fluid mechanics and practical engineering applications. The solitary wave is a kind of advanced wave that has only one peak or trough and only appears in shallow water. This kind of wave is quasi-particle, and its theory is used to study the water depth of wave breaking, which has a guiding effect on the design of offshore structures. To let students understand PIV technology visually, and at the same time, learn the transmission and evolution process of solitary waves, this paper puts forward a PIV measurement platform for the fine flow field of solitary waves, which measures the velocity vector distribution, velocity scalar distribution and vorticity distribution of the flow field in the transmission, climbing deformation and crushing process of the two-dimensional solitary wave. Through this experiment, students have a more direct impression of the fluid phenomenon of transmission and deformation of solidarity waves and understand the principle and use of PIV technology. This experiment deepens students’ understanding of the knowledge, stimulates students’ interest in learning, cultivates their practical ability, improves their innovation ability, and lays a good foundation for future scientific research and exploration.

Constant vorticity equatorial flows beneath surface solitary waves with centripetal forces

In this paper, we consider equatorial flows with Coriolis and centripetal forces, with constant vorticity beneath a two-dimensional solitary wave and above a flat bed in the -plane approximation equations. We prove that the flow must be steady, two-dimensional, decayed and the vorticity may have only one nonzero component, which points in the horizontal direction orthogonal to the direction of wave propagation. Moreover, based on the vorticity equation and analytical techniques, we prove that the free surface of the flow exhibiting constant horizontal vorticity can only be two-dimensional.

Nonexistence of Subcritical Solitary Waves

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to “half-solitary” waves (e.g. bores) which decay in only one direction.

Experimental Study of the Formation of Three-Dimensional Waves from a Two-Dimensional Solitary Wave on Vertically Falling Liquid Films

The decay of a two-dimensional solitary wave into three-dimensional waves in film flow along a vertical plate has been studied using a high-speed laser-induced fluorescence technique that allows high spatial and temporal resolution measurements of the thickness of wave liquid films. It has been shown that the three-dimensional wave packet resulting from the decay of a two-dimensional solitary wave has a strong perturbing effect on the falling film and leads to the formation of rivulets on its surface.

Two-dimensional solitary wave solution to the quadratic-cubic nonlinear Schrödinger equation

The Madelung fluid transformation is applied to find the link between the modified Korteweg-de Vries and the quadratic-cubic nonlinear Schrödinger equation. The two-dimensional solitary wave solution of the quadratic-cubic nonlinear Schrödinger equation will be determined by the Petviashvili method. This solution will be used for the initial condition for the time evolution to study the stability analysis. The spectral method is applied for the time evolution.

Solitary waves in strongly non-local media with a harmonic potential

An exact analytical solution in strongly non-local media with a harmonic potential has been studied. Two-dimensional Bessel solitary wave clusters have been obtained by a self-similar method. The intensity distributions of optical beam with different parameters have been discussed in detail. It is found that the solitary waves have a symmetric necklace distribution and the number of facular points is double the value of the quantum number m. The modulation of the external potential field to the width of light beam is also discussed.

Two-dimensional fractional algebraic internal solitary waves model and its solution

With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. In this work, based on the basic system of equations for internal solitary waves, we have derived two-dimensional Benjamin–Ono equation. Then, the integer-order two-dimensional Benjamin–Ono equation is transformed into the time-fractional Benjamin–Ono equation. To study the properties of the algebraic internal solitary waves, we discuss the conservation laws of the new model. Also, using the Hirota bilinear method, the derived new model is solved. Finally, we explore the characteristics of motion of the algebraic internal solitary waves with the help of the multi-soliton solutions.

On the Existence and Instability of Solitary Water Waves with a Finite Dipole

This paper considers the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is vacuum, and that the waves are acted upon by gravity with surface tension effects on the air--water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted $\delta$-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of $0$. Our main result is that this family is conditionally orbitally unstable. This is proved using a modification of the Grillakis--Shatah--Strauss method recently introduced by Varholm, Wahl\'en, and Walsh.

Stability of gravity-capillary solitary waves on shallow water based on the fifth-order Kadomtsev-Petviashvili equation

Longitudinal and transverse instabilities of gravity-capillary solitary waves on shallow water are investigated based on the numerical analysis of the fifth-order Kadomtsev-Petviashvili (KP) equation, which describes the wave phenomena on shallow water where the relevant Bond number is less than and close to 1/3. Two-dimensional (2D) depression gravity-capillary solitary waves are stable to longitudinal perturbations. 2D elevation gravity-capillary solitary waves are unstable to longitudinal perturbations and finally evolve into 2D depression gravity-capillary solitary waves. Three-dimensional (3D) finite-amplitude depression gravity-capillary solitary waves are stable to longitudinal perturbations. 3D finite-amplitude elevation gravity-capillary solitary waves are unstable to longitudinal perturbations and finally evolve into an oscillatory state between two different 3D finite-amplitude depression gravity-capillary solitary waves. 3D small-amplitude depression and elevation gravity-capillary solitary waves are unstable to dilation-type longitudinal perturbations and eventually evolve into an oscillatory state between two different 3D finite-amplitude depression gravity-capillary solitary waves. 3D small-amplitude depression and elevation gravity-capillary solitary waves are unstable to contraction-type longitudinal perturbations and eventually become dispersed out toward still water surface. Finally, 2D depression and elevation gravity-capillary solitary waves are unstable to transverse perturbations and eventually evolve into 3D finite-amplitude depression gravity-capillary solitary waves. Therefore, the only stable gravity-capillary solitary waves on shallow water are 3D finite-amplitude depression gravity-capillary solitary waves. In particular, based on the linear stability analysis, a theoretical proof is presented for the long-wave transverse instability of 2D depression and elevation gravity-capillary solitary waves on shallow water.

One- and Two-dimensional Solitary Wave States in the Nonlinear Kramers Equation with Movement Direction as a Variable

We study self-propelled particles by direct numerical simulation of the nonlinear Kramers equation for self-propelled particles. In our previous paper, we studied self-propelled particles with velocity variables in one dimension. In this paper, we consider another model in which each particle exhibits directional motion. The movement direction is expressed with a variable $\phi$. We show that one-dimensional solitary wave states appear in direct numerical simulations of the nonlinear Kramers equation in one- and two-dimensional systems, which is a generalization of our previous result. Furthermore, we find two-dimensionally localized states in the case that each self-propelled particle exhibits rotational motion. The center of mass of the two-dimensionally localized state exhibits circular motion, which implies collective rotating motion. Finally, we consider a simple one-dimensional model equation to qualitatively understand the formation of the solitary wave state.

Two-dimensional solitary waves and periodic waves on coupled nonlinear electrical transmission lines

By using the dynamical system approach, the exact travelling wave solutions for a system of coupled nonlinear electrical transmission lines are studied. Based on this method, the bifurcations of phase portraits of a dynamical system are given. The two-dimensional solitary wave solutions and periodic wave solutions on coupled nonlinear transmission lines are obtained. With the aid of Maple, the numerical simulations are conducted for solitary wave solutions and periodic wave solutions to the model equation. The results presented in this paper improve upon previous studies.

Multidimensional nonlinear ion-acoustic waves in a plasma in view of relativistic effects

The structure and dynamics of ion-acoustic waves in an unmagnetized plasma, including the case of weakly relativistic collisional plasma (when it is necessary to take into account the high energy particle flows which are observed in the magnetospheric plasma), are studied analytically and numerically on the basis of a model of the Kadomtsev-Petviashvili (KP) equation. It is shown that, if the velocity of plasma particles approaches the speed of light, the relativistic effects start to strongly influence on the wave characteristics, such as its phase velocity, amplitude, and characteristic wavelength, with the propagation of the twodimensional solitary ion-acoustic wave. The results can be used in the study of nonlinear wave processes in the magnetosphere and in laser and astrophysical plasma.

Nonlinear magnetosonic solitary and shock waves in strongly coupled quantum electron–positron–ion plasmas

Two-dimensional nonlinear magnetosonic solitary and shock waves propagating perpendicular to the applied magnetic field are presented in quantum electron–positron–ion plasmas with strongly coupled classical ions and weakly coupled quantum electrons and positrons. The generalized viscoelastic hydrodynamic model is used for the ions and a quantum hydrodynamic model is introduced for the electrons and positrons. In the weakly nonlinear limit, a modified Kadomstev–Petviashvili (KP) equation with a damping term and a KP–Burgers equation have been derived in the kinetic regime and hydrodynamic regime, respectively. The analytical and numerical solutions of the modified KP and KP–Burgers equations are also presented and analysed with the typical parameters of a white dwarf star and pulsar magnetosphere, which show that the quantum plasma beta and the variation of positron number density have remarkable effects on the propagation of magnetosonic solitary and shock waves.

Selection of solitary waves in vertically falling liquid films

Two-dimensional solitary waves at the surface of a film flow down a vertical plane are considered. When the system is subjected to inlet white noise, solitary waves are formed after an inception region and interact with each other. Using open-domain simulations of reduced equation models, we investigate numerically their late time process dynamics. Close to the instability threshold, the waves synchronize themselves into bound states. For higher values of the Reynolds number, the separation distance between the waves increases and the synchronization process at work is weaker. Performing statistics, we show that the mean characteristics of the waves correspond to the minimal value of the mean film thickness along the traveling-wave branch of solutions. In this regime, synchronization occurs through the waves tails which is associated with a change of scaling of the waves features. A similar behavior is observed performing simulations in periodic domains: the selected waves maximize the mean flow rate.