In this article dispersive optical soliton solutions that are governed by the time-fractional Schrödinger–Hirota equation have been presented. Some new soliton solutions of the time-fractional SH ...equations are obtained in this work. Using extended auxiliary equation method, optical soliton solutions are sought for the fractional SH equations with power law nonlinearity as well as Kerr law nonlinearity. Using fractional complex transform the time-fractional SH equation is converted into the nonlinear ordinary differential equation, and then, the resulting equation is solved using a novel analytical method viz. extended auxiliary equation method. As a result of this, some new optical soliton solutions have been successfully obtained. The obtained results show that the proposed method is a straightforward technique to find out new optical solutions of time-fractional SH equation.
In this paper, using the Lie group analysis method, the infinitesimal generators for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation are obtained. The new concept of nonlinear self-adjointness ...of differential equations is used for construction of nonlocal conservation laws. The conservation laws for the (2+1)-dimensional Bogoyavlensky–Konopelchenko equation are obtained by using the new conservation theorem method and the formal Lagrangian approach. Transforming this equation into a system of equations involving with two dependent variables, it has been shown that the resultant system of equations is quasi self-adjoint and finally the new nonlocal conservation laws are constructed by using the Lie symmetry operators.
In this paper, new exact solutions of fractional nonlinear acoustic wave equations have been devised. The travelling periodic wave solutions of fractional Burgers–Hopf equation and ...Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation have obtained by first integral method. Nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible. Taking nonlinear effects into account, the ultrasound beam analysis makes more accurate in these applications. The Burgers–Hopf equation is one of the extensively studied models in mathematical physics. In addition, the KZK parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3D diffraction sound beams in dissipative media. In the present analysis, these nonlinear equations have solved by first integral method. As a result, new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations. The obtained results are presented graphically to demonstrate the efficiency of this proposed method.
In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics, namely the time fractional KdV–Zakharov–Kuznetsov (KdV–ZK) and ...space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equations by using improved fractional sub equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie’s modified Riemann–Liouville sense.
This paper focuses on the fourth-order nonlinear potential Ito equation, which describes wave patterns in shallow waters. To reveal the integrable characteristics of the considered equation, such as ...bilinear BT and Lax pair, the Bell polynomials method is used. By employing this technique, the bilinear representation in the form of classical Hirota operators is derived. Moreover, with the help of bilinear form, the bilinear Bäcklund transformation and the Lax pair of the considered equation are obtained successfully. The three wave method in the form of test function is adopted to generate the analytical solutions of the considered equation. By applying this approach, ten analytical solutions are obtained successfully. For each of the obtained solutions, a 3D graph has been plotted by varying free parametric values. These graphs show the different kinds of wave behaviour, including kink-soliton, anti-kink soliton, periodic wave, dark-soliton, bright-soliton, and some complex kink and periodic-type wave solutions.
In this paper, the invariance properties of the time fractional (2+1)-dimensional Zakharov–Kuznetsov modified equal width (ZK-MEW) equation have been investigated using the Lie group analysis method. ...Lie point symmetries of the time fractional (2+1)-dimensional ZK-MEW equation have been derived by using the Lie group analysis method of fractional differential equations. Using the Lie symmetry analysis, the vector fields and the symmetry reduction of this equation are obtained. It is shown that the time fractional (2+1)-dimensional ZK-MEW equation can be transformed to an equation with Erdélyi–Kober fractional derivative. Finally using new conservation theorem with formal Lagrangian, the new conserved vectors are well constructed with a detailed derivation, which constitutes the conservation analysis for the time fractional (2+1)-dimensional ZK-MEW equation.
This article deals with the (3+1)-dimensional variable coefficient Date–Jimbo–Kashiwara–Miwa equation which describes the behavior of waves as they travel through nonlinear dispersive media. The ...integrability of this equation has been evaluated by mean of Painlevé property. An auto-Bäcklund transformation of the considered equation is being produced with the aid of Painlevé analysis. To get the analytic solutions of the considered equation, the auto-Bäcklund transformation method is employed. Three new analytical solution families including complex solution have been derived successfully via the auto-Bäcklund transformation method. All the results have been plotted in three-dimension by taking different parameter values and functions. These solutions depict the kink-antikink wave, periodic wave, complex periodic wave and complex kink-antikink wave solutions for the considered equation.
•(3 + 1)-dimensional variable coefficient Date–Jimbo–Kashiwara-Miwa equation investigated.•Integrability of this equation has been evaluated by mean of Painlevé property.•Auto-Bäcklund transformation (ABT) of the considered equation is being employed.•Three new analytical solution families including complex solution have been derived.•These solutions have been obtained successfully via the ABT method.
In this paper, the Lie group analysis is used to carry out the similarity reduction and exact solutions of the
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3
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1
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-dimensional modified KdV–Zakharov–Kuznetsov equation. This research deals ...with the similarity solutions of mKdV–ZK equation. The mKdV–ZK equation has been reduced into a new partial differential equation with less number of independent variables, and again using Lie group symmetry method, the new partial differential equation is reduced into an ordinary differential equation. We have obtained the infinitesimal generators, commutator table of Lie algebra, symmetry group, and similarity reduction for the mKdV–ZK equation. In addition to that, solitary wave solutions of the mKdV–ZK equation are derived from the reduction equation. Thus, we obtain some new exact explicit solutions of the
(
3
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1
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-dimensional mKdV–ZK equation which describes the dynamics of nonlinear waves in plasmas.
In this paper, the Lie symmetry analysis method has been proposed for finding similarity reduction and exact solutions of nonlinear evolution equation. Here for illustrating the effectiveness and ...accuracy of proposed method, we have taken (3+1) dimensional Yu–Toda–Sasa–Fukuyama (YTFS) equation. Also by using symmetry reduction method, we have reduced nonlinear partial differential equation (NPDE) to nonlinear ordinary differential equation, which has advantage to provide semi analytical solution. We have obtained infinitesimal generators, the entire geometric vector field, commutator table of Lie algebra and symmetry group by using Lie symmetry analysis method. Then, we have used tanh method for finding new analytical exact solutions of some reduced transform equations.
The time-fractional resonant nonlinear Schrödinger equation is studied in this paper using the modified auxiliary equation approach. This effort yields several innovative optical soliton solutions to ...the investigated problem. An equivalent nonlinear ordinary differential equation with integer-order has been obtained from the time-fractional RNLSE using the modified Riemann–Liouville derivative along with fractional complex transform, and then the emerged equations are solved using the most impressive direct method, the modified auxiliary equation method. As a consequence, novel optical soliton solutions have been successfully developed, with several 3-D graphs demonstrating their behaviour.
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