Exact Rational Solutions Research Articles

Rational solitons and rogue waves for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

In this paper, we investigate rational solitons and rogue wave (RW) solutions of the nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation by the generalized Darboux transformation method. The exact rational solutions are presented in terms of determinants whose entries are given by the initial eigenfunction and continuous-wave solution. Their dynamics investigated reveals the interesting patterns such as rational solitons, singular and nonsingular RWs depending on the selection of the free parameters. The compression effects of the RWs with respect to the strength of the higher-order nonlinear effects parameter γ are discussed and graphically illustrated.

Solutions of the matrix equation p(X)=A, with polynomial function p(λ) over field extensions of [formula omitted

Let H be a field with Q⊆H⊆C, and let p(λ) be a polynomial in H[λ], and let A∈Hn×n be nonderogatory. In this paper we consider the problem of finding a solution X∈Hn×n to p(X)=A. A necessary condition for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A=p(X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well.One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form.

Different dynamics of invariant solutions to a generalized (3+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation arising in shallow water-waves

Using the Lie symmetry technique, this paper studies a (3+1)-dimensional generalized Camassa-Holm-Kadomtsev-Petviashvili (GCHKP) equation that can possess various dynamical structures of solitary waves. The GCHKP equation is a special case of the internal shallow-water wave equation, which has recently gained popularity in ocean physics and hydrodynamics engineering. By taking advantage of the Lie symmetry technique, we derive Lie infinitesimals, geometric vector fields, commutator table, adjoint table, and an optimal system for the considered equation. Based on the one-dimensional optimal system, various attractive Lie symmetry reductions are carried out, and the GCHKP equation is reduced to a number of nonlinear ordinary differential equations (NODEs). Consequently, innumerable explicit exact solutions of a (3+1)-D GCHKP equation are obtained employing symmetry reductions via a one-dimensional optimal system of Lie subalgebra. With the assistance of symbolic computation, several 3D and 2D graphics exhibited their dynamic structures, and graphical illustrations were physically interpreted. For the first time, we furnish exact invariant solutions in terms of a variety of functions such as exact rational solutions, trigonometric and hyperbolic function solutions, V-shaped solitons, dark and bright solitons, and solitary waves, singular solitons, breather waves, and parabolic waves. In addition, the obtained results have significantly supplemented the exact invariant solutions of the GCHKP equation in the literature, allowing us to gain a better understanding of the dynamics of complex physical systems. Finally, the proposed method will significantly boost obtaining the exact explicit solutions of the nonlinear evolution equations.

Analytic solutions for a modified fractional three wave interaction equations with conformable derivative by unified method

The present work implements the unified method to a class of fractional partial differential systems corresponding to modified fractional three wave interaction equations (FTWIEs). The fractional derivative of the three waves envelopes is considered under the conformable sense. The conformable FTWIEs is derived in the (3 + 1) dimensions under the conformable time-fractional derivative of order α∈(0,1] based on a modification of the Lax-pair created by Zakharov-Manakov which includes three-dimensional velocity vector dotted with the usual del operator. Then, the unified method for creating solutions for nonlinear evolution FTWIEs is applied. Subsequently, a systematic algorithm is used to obtain an infinite set of exact rational solutions to the novel constructed system. By randomly selecting the special values for the parameters, three-dimensional graphs are also given for different patterns. The obtained solutions might play an essential role in many other nonlinear evolution equations that occur in the fields of engineering and mathematical physics.

Rogue wave solutions of the vector Lakshmanan–Porsezian–Daniel equation

We use a nonrecursive Darboux transformation method to obtain a special hierarchy of rogue wave solutions of the vector Lakshmanan–Porsezian–Daniel equation, which can govern the propagation of ultrashort optical pulses in a long-haul telecommunication fiber. In terms of the exact rational solutions, we demonstrate several interesting rogue wave dynamics such as rogue wave doublets, quartets and sextets. The modulation instability responsible for the excitation of rogue waves from an unstable continuous background in such a complex nonlinear system is also discussed.

Bright and dark rogue internal waves: The Gardner equation approach.

We have found "bright and dark" solutions of the Gardner equation which can model internal rogue waves in three-layer fluids. We provide the first four "bright" and "dark" exact rational solutions to the Gardner equation. These are the lowest-order solutions of the corresponding hierarchies of rogue-wave solutions of this equation. They have been obtained from the rogue-wave solutions of a modified Korteweg-de Vries equation by using a Lorentz-type transformation. The maximal (and minimal) amplitudes and the background levels of these solutions for arbitrary order are deduced, based on the lowest-order examples. These solutions can be useful for explanations of extremely large amplitude internal waves in the ocean, as well as for abnormally large-amplitude waves in other areas of nonlinear physics, such as optics and dusty plasmas.

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg-de Vries equation.

The formation of rogue waves in shallow water is presented in this Rapid Communication by providing the three lowest-order exact rational solutions to the Korteweg-de Vries (KdV) equation. They have been obtained from the modified KdV equation by using the complex Miura transformation. It is found that the amplitude amplification factor of such waves formed in shallow water is much larger than the amplitude amplification factor of those occurring in deep water. These solutions clearly demonstrate a potential hazard for coastal areas. They can also provide a solid mathematical basis for the existence of abnormally large-amplitude waves in other branches of nonlinear physics such as optics, unidirectional crystal growth, and in quantum mechanics.

New optical solitons of Tzitzeíca type evolution equations using extended trial approach

The properties of Tzitzeica equations in nonlinear optics have been the discussion of many recent studies. In this article, a new and productive implementation of trial equation method is chosen to handle this class of nonlinear evolution equations. As a result, the new and more exact periodic, singular, hyperbolic and rational solutions of Tzitzeica, Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations are formally obtained. The extended trial equation method along with the symbolic computation gives a versatile technique to deal such kinds of nonlinear evolution equations in nonlinear optics.

Rogue wave-type solutions of the mKdV equation and their relation to known NLSE rogue wave solutions

We present the first four exact rational solutions of the set of rational solutions of the modified Korteweg–de Vries equation. These solutions can be considered as rogue waves of the corresponding equation. Comparison with rogue wave solutions of the nonlinear Schrodinger equation shows a strong analogy between their characteristics, especially for amplitude-to-background ratio. The new solutions may be useful in the theory of rogue waves in shallow water and for light propagation in cubic nonlinear media involving only a few optical cycles.

The Exact Rational Solutions to a Shallow Water Wave-Like Equation by Generalized Bilinear Method

A Shallow Water Wave-like nonlinear differential equation is considered by using the generalized bilinear equation with the generalized bilinear derivatives D3,x and D3,t, which possesses the same bilinear form as the standard shallow water wave bilinear equation. By symbolic computation, four presented classes of rational solutions contain all rational solutions to the resulting Shallow Water Wave-like equation, which generated from a search for polynomial solutions to the corresponding generalized bilinear equation.

The Rational Solutions and Quasi-Periodic Wave Solutions as well as Interactions ofN-Soliton Solutions for 3 + 1 Dimensional Jimbo-Miwa Equation

The exact rational solutions, quasi-periodic wave solutions, andN-soliton solutions of 3 + 1 dimensional Jimbo-Miwa equation are acquired, respectively, by using the Hirota method, whereafter the rational solutions are also called algebraic solitary waves solutions and used to describe the squall lines phenomenon and explained possible formation mechanism of the rainstorm formation which occur in the atmosphere, so the study on the rational solutions of soliton equations has potential application value in the atmosphere field; the soliton fission and fusion are described based on the resonant solution which is a special form of theN-soliton solutions. At last, the interactions of the solitons are shown with the aid ofN-soliton solutions.

Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence.

We investigate dark rogue wave dynamics in normally dispersive birefringent optical fibers, based on the exact rational solutions of the coupled nonlinear Schrödinger equations. Analytical solutions are derived up to the second order via a nonrecursive Darboux transformation method. Vector dark "three-sister" rogue waves as well as their existence conditions are demonstrated. The robustness against small perturbations is numerically confirmed in spite of the onset of modulational instability, offering the possibility to observe such extreme events in normal optical fibers with random birefringence, or in other Manakov-type vector nonlinear media.

Rational solutions from Padé approximants for the generalized Hunter-Saxton equation

Exact rational solutions of the generalized Hunter-Saxton equation are obtained using Padé approximant approach for the traveling-wave and self-similarity reduction. A larger class of algebraic solutions are also obtained by extending a range of parameters within the solutions obtained by this approach.

Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions

This is the second part of our study of the solutions of a q -discrete second Painlevé equation ( q -P II ) of type ( A 2 + A 1 ) (1) via its iso-monodromy deformation problem. In part I, we showed how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions. In this paper, we study the case giving rise to an infinite sequence of q -hypergeometric-type solutions. We find a new determinantal representation of all such solutions and solve the iso-monodromy deformation problem in closed form.

Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.

Construction of new exact rational form non-travelling wave solutions to the (2 + 1)-dimensional generalized Broer–Kaup system

With the aid of symbolic computation Maple, several new families of rational form variable separation solutions with three arbitrary functions to the (2 + 1)-dimensional generalized Broer–Kaup system are derived by using an improved mapping approach and a variable separation approach. These solutions include rational solitary wave solutions, periodic wave solutions and rational wave solutions. The properties of the novel localized excitation are revealed by some figures.

New Travelling Wave solutions of the Korteweg De Vries Equation by - (G'/G') Expansion method.

Exact hyperbolic, trigonometric and rational function travelling wave solutions to the Korteweg De Vries (KDV) equation using the expansion method are presented in this paper. More travelling wave solutions to the KDV equation were obtained with Liu’s theorem. The solutions obtained were verified by putting them back into the equation with the aid of Mathematica. This shows that the expansion method is a powerful and effective tool for obtaining exact solutions to nonlinear partial differential equations in physics, mathematics and other applications.

Exact solutions to linear programming problems

The use of floating-point calculations limits the accuracy of solutions obtained by standard LP software. We present a simplex-based algorithm that returns exact rational solutions, taking advantage of the speed of floating-point calculations and attempting to minimize the operations performed in rational arithmetic. Extensive computational results are presented.

Integral Approximation of Rays and Verification of Feasibility

An algorithm is presented that produces an integer vector nearly parallel to a given vector. The algorithm can be used to discover exact rational solutions of homogeneous or inhomogeneous linear systems of equations, given a sufficiently accurate approximate solution. As an application, we show how to verify rigorously the feasibility of degenerate vertices of a linear program with integer coefficients, and how to recognize rigorously certain redundant linear constraints in a given system of linear equations and inequalities. This is a first step towards the handling of degeneracies and redundandies within rigorous global optimization codes.

Construction of New Exact Rational Solutions to the Veselov–Novikov Equation and New Exact Rational Potentials for the Two-Dimensional Schrödinger Stationary Equation by the \(\overline \partial \)-Dressing Method

With the help of the Zakharov–Manakov \(\overline \partial \)-dressing method, the scheme of obtaining exact rational solutions to the well-known two-dimensional Veselov–Novikov integrable nonlinear equation and exact rational potentials for the two-dimensional Schrodinger stationary equation corresponding to wave functions with multiple poles is developed. As an example, new exact rational nonsingular and singular solutions to the Veselov–Novikov equation and the corresponding exact rational potentials for the two-dimensional Schrodinger stationary equation with multiple second-order poles are obtained.