New Analytical Solutions for Coupled Stochastic Korteweg-de Vries Equations via Generalized Derivatives

In this paper, the coupled nonlinear KdV (CNKdV) equations are solved in a stochastic environment. Hermite transforms, generalized conformable derivative, and an algorithm that merges the white noise instruments and the (G′/G2)-expansion technique are utilized to obtain white noise functional conformable solutions for these equations. New stochastic kinds of periodic and soliton solutions for these equations under conformable generalized derivatives are produced. Moreover, three-dimensional (3D) depictions are shown to illustrate that the monotonicity and symmetry of the obtained solutions can be controlled by giving a value of the conformable parameter. Furthermore, some remarks are presented to give a comparison between the obtained wave solutions and the wave solutions constructed under the conformable derivatives and Newton’s derivatives.

[1]  E. M. Özkan,et al.  On exact solutions of some important nonlinear conformable time-fractional differential equations , 2022, SeMA Journal.

[2]  N. Aljahdaly,et al.  On the modified (G′G2)-expansion method for finding some analytical solutions of the traveling waves. , 2021, Journal of Ocean Engineering and Science.

[3]  Ahmed H. Soliman,et al.  An extended Kudryashov technique for solving stochastic nonlinear models with generalized conformable derivatives , 2021, Commun. Nonlinear Sci. Numer. Simul..

[4]  Abd-Allah Hyder,et al.  A new generalized θ-conformable calculus and its applications in mathematical physics , 2020, Physica Scripta.

[5]  M. M. Bhatti,et al.  Editorial: Recent Trends in Computational Fluid Dynamics , 2020, Frontiers in Physics.

[6]  Prakash Kumar Das.  New multi-hump exact solitons of a coupled Korteweg-de-Vries system with conformable derivative describing shallow water waves via RCAM , 2020, Physica Scripta.

[7]  A. Abouelregal,et al.  The Size-Dependent Thermoelastic Vibrations of Nanobeams Subjected to Harmonic Excitation and Rectified Sine Wave Heating , 2020 .

[8]  A. Yulin,et al.  Multi-frequency radiation of dissipative solitons in optical fiber cavities , 2020, Scientific Reports.

[9]  Abd-Allah Hyder.  White noise theory and general improved Kudryashov method for stochastic nonlinear evolution equations with conformable derivatives , 2020 .

[10]  A. Kara,et al.  A (2+1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions , 2020, Nuclear Physics B.

[11]  Mustafa Bayram,et al.  Theory and application for the system of fractional Burger equations with Mittag leffler kernel , 2020, Appl. Math. Comput..

[12]  M. Barakat,et al.  General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics , 2020, Physica Scripta.

[13]  Alexander N. Churilov.  Orbital stability of periodic solutions of an impulsive system with a linear continuous-time part , 2020 .

[14]  M. A. Sohaly,et al.  The new exact solutions for the deterministic and stochastic (2+1)-dimensional equations in natural sciences , 2019, Journal of Taibah University for Science.

[15]  N. Aljahdaly.  Some applications of the modified (G′/G2)-expansion method in mathematical physics , 2019, Results in Physics.

[16]  M. Belić,et al.  Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized exp(−Φ(ξ))-expansion , 2019, Optik.

[17]  Dazhi Zhao,et al.  A new framework for multivariate general conformable fractional calculus and potential applications , 2018, Physica A: Statistical Mechanics and its Applications.

[18]  M. Hafez,et al.  Nonlinear time fractional Korteweg-de Vries equations for the interaction of wave phenomena in fluid-filled elastic tubes , 2018, The European Physical Journal Plus.

[19]  M. Hafez,et al.  Oblique closed form solutions of some important fractional evolution equations via the modified Kudryashov method arising in physical problems , 2018, Journal of Ocean Engineering and Science.

[20]  Abdul-Majid Wazwaz,et al.  A two-mode modified KdV equation with multiple soliton solutions , 2017, Appl. Math. Lett..

[21]  E. Bodenschatz,et al.  Convective instability and boundary driven oscillations in a reaction-diffusion-advection model. , 2017, Chaos.

[22]  Abdullah Akkurt,et al.  A new Generalized fractional derivative and integral , 2017, 1704.03299.

[23]  M. Luo,et al.  General conformable fractional derivative and its physical interpretation , 2017, Calcolo.

[24]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

[25]  R. Grimshaw.  Coupled Korteweg–de Vries Equations , 2013 .

[26]  Nikolai A. Kudryashov,et al.  One method for finding exact solutions of nonlinear differential equations , 2011, 1108.3288.

[27]  Willy Hereman,et al.  Shallow Water Waves and Solitary Waves , 2013, Encyclopedia of Complexity and Systems Science.

[28]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[29]  Chao-Qing Dai,et al.  Jacobian elliptic function method for nonlinear differential-difference equations , 2006 .

[30]  Hong-qing Zhang,et al.  A generalized F-expansion method to find abundant families of Jacobi Elliptic Function solutions of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation , 2006 .

[31]  Yingchao Xie,et al.  Exact solutions for Wick-type stochastic coupled KdV equations☆ , 2004 .

[32]  I. Akhatov,et al.  Sound-ultrasound interaction in bubbly fluids: Theory and possible applications , 2001 .

[33]  A. D. Bouard,et al.  On the Stochastic Korteweg–de Vries Equation , 1998 .

[34]  D. Crighton.  Applications of KdV , 1995 .

[35]  S. Korsunsky.  Soliton solutions for a second-order KdV equation , 1994 .

[36]  R. Hirota,et al.  Soliton solutions of a coupled Korteweg-de Vries equation , 1981 .