Painlevé analysis for three integrable shallow water waves equations with time-dependent coefficients

PurposeThe purpose of this paper is to introduce a variety of new completely integrable Calogero–Bogoyavlenskii–Schiff (CBS) equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for each of the developed models.Design/methodology/approachThe newly developed models with time-dependent coefficients have been handled by using the simplified Hirota’s method. Moreover, multiple complex soliton solutions are derived by using complex Hirota’s criteria.FindingsThe developed models exhibit complete integrability, for specific determined functions, by investigating the compatibility conditions for each model.Research limitations/implicationsThe paper presents an efficient algorithm for handling integrable equations with analytic time-dependent coefficients.Practical implicationsThe work presents new integrable equations with a variety of time-dependent coefficients. The author showed that integrable equations with time-dependent coefficients give real and complex soliton solutions.Social implicationsThis study presents useful algorithms for finding and studying integrable equations with time-dependent coefficients.Originality/valueThe paper gives new integrable CBS equations which appear in propagation of waves and provide a variety of multiple real and complex soliton solutions.

In this paper, we develop two new fourth-order integrable equations represented by nonlinear PDEs of second-order derivative in time t. The new equations model both right- and left-going waves in a like manner to the Boussinesq equation. We will employ the Painleve analysis to formally show the complete integrability of each equation. The simplified Hirota’s method is used to derive multiple soliton solutions for this equation. We introduce a complex form of the simplified Hirota’s method to develop multiple complex soliton solutions. More exact traveling wave solutions for each equation will be derived as well.

(2+1)-Dimensional mKdV (N) equations by the mKdV recursion operator: Multiple soliton and multiple singular soliton solutions

Multiple soliton solutions for the Bogoyavlenskii’s generalized breaking soliton equations and its extension form

Some new exact solutions of the generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation (KPE) are explored in this study. Firstly, the resonant multiple soltion solutions (RMSs) are discussed via employing the linear superposition principle and weight algorithm. Then, by introducing pairs of the conjugate parameters to the RMSs, the complexiton solutions including the non-singular complexiton and singular complexiton solutions are extracted. In addition, the complex multiple kink soliton solutions are also probed by employing the bilinear approach. Finally, we investigate the rational wave solutions via the test function method and symbolic computation. By choosing the appropriate parameters, the graph descriptions of the derived solutions are presented to show the dynamical properties. The outcomes of this work are desirous to bring some new perspective to the study of the complexiton, complex solutions and rational wave solutions to the other PDEs.

Multiple soliton and lump solutions to a variety of novel integrable multi-dimensional Boussinesq-type equations

Numerial Methods for Volterra Integral Equations with Singular Kernels

In this paper, we construct an extended Bogoyavlenskii‐Kadomtsev‐Petviashvili (eBKP) equation. We use the Painlevé analysis to justify the integrability of the eBKP equation. We also examine the integrability of a related extended CBS equation. We derive multiple soliton solutions via using the Hirota's direct method. In addition, a variety of exact solutions, that include singular kink solutions, periodic solutions, and exponential solutions, are derived as well.

Because of the parabolic instability of the Shilov systems to change their coefficients, the definition parabolicity of Shilov for systems with time-dependent $t$ coefficients, unlike the definition parabolicity of Petrovsky, is formulated by imposing conditions on the matricant of corresponding dual by Fourier system. For parabolic systems by Petrovsky with time-dependent coefficients, these conditions are the property of a matricant, which follows directly from the definition of parabolicity. In connection with this, the question of the wealth of the class Shilov systems with time-dependent coefficients is important.A new class of linear parabolic systems with partial derivatives to the first order by the time $t$ with time-dependent coefficients is considered in this work. It covers the class by Petrovsky systems with time-dependent younger coefficients. A main part of differential expression of each such system is parabolic (by Shilov) expression with constant coefficients. The fundamental solution of the Cauchy problem for systems of this class is constructed by the Fourier transform method. Also proved their parabolicity by Shilov. Only the structure of the system and the conditions on the eigenvalues of the matrix symbol were used. First of all, this class characterizes the wealth by Shilov class of systems with time-dependents coefficients.Also it is given a general method for investigating a fundamental solution of the Cauchy problem for Shilov parabolic systems with positive genus, which is the development of the well-known method of Y.I. Zhitomirskii.

Multiple soliton solutions are fundamental excitations. There are many kinds of equivalent representations for multiple soliton solutions such as the Hirota forms, Wronskian and/or double Wronskian expressions and Phaffian representations. Recently, in the studies of multi-place nonlocal systems, we find that there are a type of novel but equivalent simple and elegant forms to describe multiple soliton solutions for various integrable systems. In this paper, we mainly review novel types of expressions of multiple soliton solutions for some kinds of nonlinear integrable systems. Meanwhile, some completely new expressions for the Sawada-Kortera equations, the asymmetric Nizhnik-Novikov-Veselov system, the modified KdV equation, the sine-Gordon equation, the Ablowitz-Kaup-Newell-Segue system and the completely discrete <i>H</i><sub>1</sub> equation are firstly given in this paper. New expressions usually possess explicit full reversal symmetries including parity, time reversal, soliton initial position reversal and charge conjugate reversal. These kinds of explicitly symmetric forms are very useful and convenient in the studies on the nonlinear physical problems such as the multi-place nonlocal systems and the resonant structures.

A microcrystalline material refers to a crystallized substance or rock comprised of tiny crystals that can only be observed under a microscope. The strain wave equation is a fourth-order nonlinear partial differential equation encountered in the examination of non-dissipative strain wave propagation within microstructured solids. In this paper, the transmission of waves in microcrystalline materials is dictated by the non-dissipative case of strain wave equation’s structure, accounting for multiple dimensions within microcrystalline structures. The simplest equation method is employed to extract multi-soliton solutions, while the modified Sardar subequation method is applied to identify additional soliton solutions, including bright, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Furthermore, the dynamical system bifurcation theory approach is utilized to investigate the phase diagrams of the governing equation. Further elaboration on the physical dynamical representation of the presented solutions is provided through profile illustrations. A comparison with the existing literature is also provided, highlighting the efficacy of our work. The significance of the acquired outcomes lies in their capacity to portray a wide array of intricate and diverse phenomena observed in both mathematical and physical systems.

The bright, dark and kink soliton solutions of the coupled Zakharov-Kuznetsov (ZK) equation are obtained by using the solitary wave ansatz method, variational approximation (VA), variational iteration method (VIM) and Adomian's decomposition method (ADM). The bright and dark soliton solutions are multiple soliton solutions at time t = 0 which reduce into stationary soliton through single soliton for sufficiently large time t . The approximate solutions by the VA, VIM and ADM are compared with the exact solution obtained by the ansatz method. The VIM gives better approximate solution of the coupled ZK equation than the VA. The absolute error and convergence analysis of the approximate solutions by the VIM and ADM show that the approximate solutions converge to the exact solution. The modulation instability is used to discuss the stability of the steady state solution of the coupled ZK equation and it demonstrates that the nonlinear term in the equation decides the modulational stability of the soliton solution.

Diverse fields of modern science, including mathematical modeling and physics, plasma physics, atmospheric sciences, marine sciences, hydrodynamics, nonlinear mechanics, and other complex nonlinear physical phenomena, are expressed through Nonlinear Partial Differential Equations (NLPDEs). Exact solutions play an important role in understanding the behavior of solitary wave solutions and the dynamical properties of significant outcomes for higher-dimensional NLPDEs. As an integrable extension of the nonlinear Schrödinger equation, the nonlocal Kundu-Eckhaus model comprehends higher-order nonlinearities and nonlocal effects. The stability properties of dark, bright, periodic multiple, and singular soliton solutions to the nonlocal Kundu-Eckhaus (KE) equation within the Parity-Time (PT) symmetry model are studied here, whereas past studies have neglected the effects of nonlocal interactions and PT symmetry. is a well-established method, particularly for depicting wave dynamics in nonlinear optical and quantum systems. We attain exact traveling wave solutions evolved in hyperbolic and trigonometric forms, representing several classes of solitons by applying this method. The findings elucidate that the obtained soliton solutions are stable under small perturbations, highlighting their robustness and the persistence of localized waveforms in nonlocal circumstances. We plotted 3D, 2D, and contour graphs for simulating our findings using MATLAB. This research sheds light on the mysterious understanding of nonlocal nonlinear wave behaviors and offers analytical methodologies for future scrutiny in fields such as optical physics and complex nonlinear systems.

Two extended forms of a (3 + 1)-dimensional generalized B-type Kadomtsev–Petviashvili (BKP) equation are examined. We use the simplified Hereman–Nuseir form to establish one and two soliton solutions for each extended equation. We develop specific constraints that guarantee the existence of multiple soliton solutions for each equation.

The well‐known shallow wave equation can be reduced to the Hirota equation with the aid of corresponding transformation. We discuss its explicit solutions, including dark soliton solution, multiple soliton solution, multiple singular solution, and periodic solutions.