Application of the Gkm to Some Nonlinear Partial Equations

In this manuscript, the strain wave equation, which plays an important role in describing different types of wave propagation in microstructured solids and the (2+1) dimensional Bogoyavlensky Konopelchenko equation, is defined in fluid mechanics as the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis, were studied. The generalized Kudryashov method (GKM), which is one of the solution methods of partial differential equations, was applied to these equations for the first time. Thus, a series of solutions of these equations were obtained. These found solutions were compared with other solutions. It was seen that these solutions were not shown before and were presented for the first time in this study. The new solutions of these equations might have been useful in understanding the phenomena in which waves are governed by these equations. In addition, 2D and 3D graphs of these solutions were constructed by assigning certain values and ranges to them

Separation Transformation and a Class of Exact Solutions to the Higher-Dimensional Klein-Gordon-Zakharov Equation

The separation transformation method is extended to the n+1-dimensional Klein-Gordon-Zakharov equation describing the interaction of the Langmuir wave and the ion acoustic wave in plasma. We first reduce the n+1-dimensional Klein-Gordon-Zakharov equation to a set of partial differential equations and two nonlinear ordinary differential equations of the separation variables. Then the general solutions of the set of partial differential equations are given and the two nonlinear ordinary differential equations are solved by extended F-expansion method. Finally, some new exact solutions of the n+1-dimensional Klein-Gordon-Zakharov equation are proposed explicitly by combining the separation transformation with the exact solutions of the separation variables. It is shown that, for the case of n≥2, there is an arbitrary function in every exact solution, which may reveal more nontrivial nonlinear structures in the high-dimensional Klein-Gordon-Zakharov equation

Analytical solutions for nonlinear systems using Nucci's reduction approach and generalized projective Riccati equations

In this study, the Nucci's reduction approach and the method of generalized projective Riccati equations (GPREs) were utilized to derive novel analytical solutions for the (1+1)-dimensional classical Boussinesq equations, the generalized reaction Duffing model, and the nonlinear Pochhammer-Chree equation. The nonlinear systems mentioned earlier have been solved using analytical methods, which impose certain limitations on the interaction parameters and the coefficients of the guess solutions. However, in the case of the double sub-equation guess solution, analytic solutions were allowed. The soliton solutions that were obtained through this method display real positive values for the wave phase transformation, which is a novel result in the application of the generalized projective Riccati method. In previous applications of this method, the real positive properties of the solutions were not thoroughly investigated

Exact spatiotemporal traveling and solitary wave solutions for the generalized nonlinear Schrödinger equation

Напредак у нелинеарној оптици умногоме зависи од наше способности да нађемо нова решења разних диференцијалних једначина које се природно јављају у системима где светлост интерагује са нелинеарном средином. Иако су рекреирање ових система кроз експеримент и компјутерска симулација система два најчешћа и плодотворна приступа, крајњи циљ остаје да се нађу егзактна решења ових система. Циљ ове тезе је да комбинује раније технике налажења егзактних решења диференцијалних једначина и примени их на нелинеарну Шредингерову диференцијалну једначину (НШДЈ). Конкретно, настао је недавно пробој у применама одређених техника експанзије у налажењу одређених егзактних решења НШДЈ. Упркос ограничењу у комбиновању решења због нелинеарности система и чињенице да не могу општа решења да се нађу, сама чињеница да можемо идентификовати нека егзактна решења је од великог значаја за област, посебно код евалуирања какве су појаве могуће у таквим системима. Ова теза ће се фокусирати на примену технике Ф-експанзије користећи се Јакобијевим елиптичним функцијама (ЈЕФ) да би се решиле разне форме НШДЈ са нелинеарношћу трећег степена. НШДЈ са нелинеарношћу трећег степена је од фундаменталне важности за област нелинеарне оптике јер описује путовање светлости кроз материјал са Керовом нелинеарношћу. Одређеним модификацијама технике Ф-експанзије можемо наћи егзактна решења за широку класу система. Системи које ја презентујем у тези имају одређен скуп заједничких особина. Све једначине имају једну лонгитудиналну промењиву, или просторну или временску, због параксијалне апроксимације, и до три трансферзалне димензије, такође или просротне или временске понаособ. Ако су све трансферзалне вариабле просторне vii онда суму њихових других извода множим са коефицијентом дифракције β, а ако је нека од варијабли темпорална, онда говорим о коефицијенту дифракције/дисперзије. Та два коефицијента (дифракција и дисперзија) могу да се нормализују у један до на знак. У случају аномалне дисперзије коефицијенти имају исти знак, а у случају нормалне дисперзије супротан знак. Осим ова два коефицијента редукована у један, имамо такође и коефицијент χ који одређује јачину нелинеарности трећег степена, и коефицијент γ који одређује добитак (за позитивно γ) или губитак сигнала у нашем систему...The progress of the field of non-linear optics greatly depends on our ability to find solutions of various differential equations that naturally occur in the systems where light interacts with nonlinear media. Though re-creating the systems through experiment and performing computer simulations are the two most common and fruitful approaches, the ultimate goal remains to find exact solutions of these systems. The goal of this Thesis is to combine the work done in the field of finding exact solutions to certain classes of non-linear differential Schrödinger equations (NLSE). Most notably, there has been a breakthrough as of late in applying various expansion techniques in finding certain exact solutions to various NLSE. Despite the limitations of combining said solutions due to the non-linear nature of the solutions and the fact that not all solutions can be found using these techniques, the very fact that we can identify certain exact solutions is of tremendous importance to the field, especially when it comes to evaluating the kinds of functions and behavior that are possible within such systems. This Thesis will focus primarily on applying the F-expansion technique using the Jacobi elliptic functions (JEFs) to solve various forms of the NLSE with the cubic nonlinearity. The NLSE with a cubic nonlinearity is one of fundamental importance in the field of nonlinear optics because it describes the travelling of a light wave through a medium with a Kerr-like nonlinearity. Through certain modification of the technique we can find exact solutions in a very large class of systems. The systems I present in this Thesis will share a certain set of common properties. All of the equations I will tackle have a single longitudinal variable, either temporal or spatial, due to the application of the paraxial wave approximation, and up to three transverse dimensions, again both temporal and spatial. If all the transverse variables are spatial I x assign to the sum of their second derivatives a diffraction coefficient β whereas if one of them is temporal, I speak of the diffraction/dispersion coefficient. The two coefficients can be normalized into one, up to their sign. In the case of anomalous dispersion, the two coefficients have the same sign. In the case of normal dispersion, the two coefficients have the opposite signs. Apart from these terms which are present in the ordinary wave equation of linear optics, we also have the third order nonlinearity whose strength is determined by a parameter χ and we also have the term γ which describes the gain of loss of the signal inside our system..