Exact Periodic Solitary Solutions to the Shallow Water Wave Equation

Exact solutions to the shallow wave equation are studied based on the idea of the extended homoclinic test and bilinear method. Some explicit solutions, such as the one soliton solution, the doubly-periodic wave solution and the periodic solitary wave solutions, are obtained. In addition, the properties of the solutions are investigated.     

Note on wavefront dislocation in surface water waves

At singular points of a wave field, where the amplitude vanishes, the phase may become singular and wavefront dislocation may occur. In this Letter we investigate for wave fields in one spatial dimension the appearance of these essentially linear phenomena. We introduce the Chu-Mei quotient as it is known to appear in the ‘nonlinear dispersion relation’ for wave groups as a consequence of the nonlinear transformation of the complex amplitude to real phase-amplitude variables. We show that unboundedness of this quotient at a singular point, related to unboundedness of the local wavenumber and frequency, is a generic property and that it is necessary for the occurrence of phase singularity and wavefront dislocation, while these phenomena are generic too. We also show that the ‘soliton on finite background’, an explicit solution of the NLS equation and a model for modulational instability leading to extreme waves, possesses wavefront dislocations and unboundedness of the Chu-Mei quotient.     

Existence of Formal Conservation Laws of a Variable-Coefficient Korteweg--de Vries Equation from Fluid Dynamics and Plasma Physics via Symbolic Computation

Employing the method which can be used to demonstrate the infinite conservation laws for the standard Kortewegde Vries （KdV） equation, we prove that the variable-coeFficient KdV equation under the Painlevé test condition also possesses the formal conservation laws.     

Exact Analytic N-Soliton-Like Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg-de Vries Model from Plasmas and Fluid Dynamics

Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg-de Vries （vcKdV） model is investigated. The bilinear form and analytic N-soliton-like solution for such a model are derived by the Hirota method and Wronskian technique. Additionally, the bilinear auto-Bǎcklund transformation between （N-1）- soliton-like and N-soliton-like solutions is verified.     

Solitary waves in the granular chain

Solitary waves are lumps of energy. We consider the study of dynamical solitary waves, meaning cases where the energy lumps are moving, as opposed to topological solitary waves where the lumps may be static. Solitary waves have been studied in some form or the other for nearly 450 years. Subsequently, there have been many authoritative works on solitary waves. Nevertheless, some of the most recent studies reveal that these peculiar objects are far more complex than what we might have given them credit for. In this review, we introduce the physics of solitary waves in alignments of elastic beads, such as glass beads or stainless steel beads. We show that any impulse propagates as a new kind of highly interactive solitary wave through such an alignment and that the existence of these waves seems to present a need to re-examine the very definition of the concept of equilibrium. We further discuss the possibility of exploiting nonlinear properties of granular alignments to develop exciting technological applications.     

On the general form of the Benjamin-Bona-Mahony equation in fluid mechanics

In the study of many problems of mechanical and physical sciences, various versions of the Benjamin-Bona-Mahony (BBM) equation or the regularized-long-wave equation have been proposed. In this paper, we obtain the solitary-wave solutions to the general form of the BBM equation, which fully cover the variety of the BBM solitary waves previously reported. This work has been supported by the China Talent Fund, by the Cheung Kong Scholars Programme of China, by the Cheung-Kong-Scholar Research Concerted Fund of Beijing University of Aeronautics and Astronautics, and by the Doctoral Education Fund in Basic Sciences of Beijing University of Aeronautics and Astronautics.     

Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations

This Letter deals with a generalized Camassa-Holm equation and a nonlinear dispersive equation by making use of a mathematical technique based on using integral factors for solving differential equations. The peakons, solitary patterns and periodic solutions are expressed analytically under various circumstances. The conditions that cause the qualitative change in the physical structures of the solutions are highlighted.     

On exact N-loop soliton solution to nonlinear coupled dispersionless evolution equations

We investigate some nonlinear coupled dispersionless evolution equations (NLCDEE) modelling the dynamics of a current-fed string within an external magnetic field in 2D-space. Using a blend of transformations of independent variables, we derive from the previous equations a Schäfer-Wayne short pulse equation (SWSPE). By means of a transformation back to the original independent variables, we find the N-loop soliton solution to the coupled equations. We give some detail on the scattering behavior of two-loop solitons.     

On High-Frequency Soliton Solutions to a （2＋1）-Dimensional Nonlinear Partial Differential Evolution Equation

A （2＋1）-dimensional nonlinear partial differential evolution （NLPDE） equation is presented as a model equation for relaxing high-rate processes in active barothropic media. With the aid of symbolic computation and Hirota＇s method, some typical solitary wave solutions to this （2＋1）-dimensional NLPDE equation are unearthed. As a result, depending on the dissipative parameter, single and multivalued solutions are depicted.     

On the Conversion of High-Frequency Soliton Solutions to a （1＋1）-Dimensional Nonlinear Partial Differential Evolution Equation

From the dynamical equation of barotropic relaxing media beneath pressure perturbations, and using the reductive perturbative analysis, we investigate the soliton structure of a （1＋1）-dimensional nonlinear partial differential evolution （NLPDE） equation δy（δη ＋ uδy ＋ （u^2/2）δy）u ＋ auy ＋ u = 0, describing high-frequency regime of perturbations. Thus, by means of Hirota＇s bilinearization method, three typical solutions depending strongly upon a characteristic dissipation parameter are unearthed.     

Deltons, peakons and other traveling-wave solutions of a Camassa-Holm hierarchy

In this letter, we study an integrable Camassa-Holm hierarchy whose high-frequency limit is the Camassa-Holm equation. Phase plane analysis is employed to investigate bounded traveling wave solutions. An important feature is that there exists a singular line on the phase plane. By considering the properties of the equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. Those paths in phase planes which represented bounded solutions are discussed one-by-one. Besides solitary, peaked and periodic waves, the equations are shown to admit a new type of traveling waves, which concentrate all their energy in one point, and we name them deltons as they can be expressed as some constant multiplied by a delta function. There also exists a type of traveling waves we name periodic deltons, which concentrate their energy in periodic points. The explicit expressions for them and all the other traveling waves are given.     

A hierarchy of Hamiltonian lattice equations associated with the relativistic Toda type system

By considering a discrete iso-spectral problem, a hierarchy of bi-Hamiltonian relativistic Toda type lattice equations are revisited. After introducing a semi-direct sum Lie algebras of four by four matrices, integrable coupling system associated with the relativistic Toda type lattice are derived. It is shown that the resulting lattice soliton hierarchy possesses Hamiltonian structures and infinitely many common commuting symmetries as well infinitely many conserved functions. The Liouville integrability of the resulting system is then demonstrated.     

Soliton solutions by Darboux transformation for a Hamiltonian lattice system

Starting from a new discrete iso-spectral problem, we derive a hierarchy of Hamiltonian lattice equations. A Darboux transformation is established for the lattice soliton hierarchy. As applications, the soliton solutions of resulted lattice hierarchy are given.     

Higher-order solution for dust acoustic solitary waves in an unmagnetized dusty plasma

The effect of higher-order nonlinearity on dust acoustic solitary waves is studied taking into account the dust-charge variation. The model of charge fluctuation, taken here, is of the formI _e+I _i=0,I _e andI _i being the electronic and ionic currents. The dust charge is determined self consistently from the current-balance equation. It is found that the higher-order correction modifies the amplitude and width of the dust acoustic solitary waves. The effect of dust-charge streaming is also discussed.     

On soliton structure of isospectral Vakhnenko equation: WKI-eigenvalue problem

In this Letter, an inverse scattering method is developed for the isospectral Vakhnenko equation, and the general N-solution is presented. Using this technique, a typical self-confined solitary wave hereafter named soliton, satisfying some vanishing boundary conditions is elicited. The detail on the scattering behavior of such structures including their phase shifts is outlined. This method is presented to be arguably more simple, tractable and straightforward than that recently investigated by Vakhnenko and Parkes [V.O. Vakhnenko, E.J. Parkes, Chaos Solitons Fractals 13 (2002) 1819] while solving the same equation. As a result, it is shown that when two single soliton solutions with ‘similar’ or ‘dissimilar’ amplitudes collide, there may be two types of features depending on the ratio of the two eigenvalues involved. It is then suggested an existence of some critical value for the ratio of the two eigenvalues at which the collision process changes its characteristic features.     

Multiple-mode wave solutions in Vakhnenko equation

A new type of wave solutions, called as multiple-mode waves, which can be expressed in the superposition forms of more than two types of single-mode waves of Vakhnenko equation have been investigated in this Letter. A new general method for obtaining the multiple-mode waves is proposed, based on which four cases of the possible forms of wave solutions with two-mode have been derived. The explicit expressions of the two-mode waves as well as the existence conditions have been presented, which may be the nonlinear combinations between periodic waves, solitons, compactons, etc., with different wave speeds, respectively. It is pointed out that more complicated multiple-mode waves with more than three single-mode waves can be derived accordingly, which can be used to reveal the evolution of interactions between different types of waves, especially between various solitons.     

Modification of the Clarkson-Kruskal Direct Method for a Coupled System

A new idea is put forward to modify the Clarkson-Kruskal （CK） direct method. Using the usual CK direct method to a coupled KdV system, two types of usual similarity reductions can be obtained. However, the application of the modified CK direct method leads to three types of new similarity reductions different from the usual ones.     

Quasi-Hamiltonian Structure Associated with an Integrable Coupling System

Starting from a spectrai problem, a corresponding soliton hierarchy is proposed, and we construct an integrable coupling system with five dependent variables for the hierarchy by using a class of semi-direct sums of Lie algebras. Moreover, it is shown that the coupling system possesses quasi-Hamiltionian structures, and that infinitely many conserved quantities are obtained.     

Evolutions of Wave Patterns in Whitham-Broer-Kaup Equation

Upon investigation of the parameter influence on the structure of WBK equation, transition boundaries are derived. All possible bounded waves as well as the existence conditions are obtained. The evolution of waves with variation of the parameters is discussed in detail, which reveals the bifurcation mechanism between different wave patterns.     

Two-Mode Wave Solutions to the Degasperis--Procesi Equation

By introducing a new type of solutions, called the multiple-mode wave solutions which can be expressed in nonlinear superposition of single-mode waves with different speeds, we investigate the two-mode wave solutions in Degasperis-Procesi equation and two cases are derived. The explicit expressions for the two-mode waves as well as the existence conditions are presented. It is shown that the two-mode waves may be the nonlinear combinations of many types of single-mode waves, such as periodic waves, solJtons, compactons, etc., and more complicated multiple-mode waves can be obtained if higher order or more single-mode waves are taken into consideration. It is pointed out that the two-mode wave solutions can be employed to display the typical mechanism of the interactions between different single-mode waves.