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Journal articles on the topic 'Ito equation'

Author: Grafiati

Published: 30 May 2022

Last updated: 31 May 2022

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1

SAITO, T., and T. ARIMITSU. "QUANTUM STOCHASTIC LIOUVILLE EQUATION OF ITO TYPE." Modern Physics Letters B 07, no. 29n30 (December 30, 1993): 1951–59. http://dx.doi.org/10.1142/s0217984993001983.

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The quantum stochastic Liouville equation of Ito type is derived, for the first time, within Nonequilibrium Thermo Field Dynamics (NETFD), a unified canonical formalism for dissipative and/or stochastic fields. With the stochastic time-evolution generator of Ito type, the whole framework is inspected. Since most of the mathematical formulations of noncommutative stochastic variables are based on the equations of Ito type, the construction of the quantum stochastic Liouville equation has been highly desired. It is expected that the unified framework may provide us with a deeper insight for the mathematical foundation of quantum stochastic variables as well as for the physical one.

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2

Niu, Xiaoxing, Mengxia Zhang, and Shuqiang Lv. "A Darboux Transformation for Ito Equation." Zeitschrift für Naturforschung A 71, no. 5 (May 1, 2016): 427–31. http://dx.doi.org/10.1515/zna-2016-0004.

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AbstractA system proposed by Ito is reconsidered. The corresponding Darboux transformation is presented explicitly. The resulted Bäcklund transformation is shown to be equivalent to the one found by Hirota. Also, a nonlinear superposition formula, which is of differential-algebraic, is obtained.

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3

Ma, Wen-Xiu, Jie Li, and Chaudry Masood Khalique. "A Study on Lump Solutions to a Generalized Hirota-Satsuma-Ito Equation in (2+1)-Dimensions." Complexity 2018 (December 2, 2018): 1–7. http://dx.doi.org/10.1155/2018/9059858.

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The Hirota-Satsuma-Ito equation in (2+1)-dimensions passes the three-soliton test. This paper aims to generalize this equation to a new one which still has abundant interesting solution structures. Based on the Hirota bilinear formulation, a symbolic computation with a new class of Hirota-Satsuma-Ito type equations involving general second-order derivative terms is conducted to require having lump solutions. Explicit expressions for lump solutions are successfully presented in terms of coefficients in a generalized Hirota-Satsuma-Ito equation. Three-dimensional plots and contour plots of a special presented lump solution are made to shed light on the characteristic of the resulting lump solutions.

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4

Ren, Bo, Ji Lin, and Jun Yu. "Supersymmetric Ito equation: Bosonization and exact solutions." AIP Advances 3, no. 4 (April 2013): 042129. http://dx.doi.org/10.1063/1.4802969.

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5

Yi, Zhang, and Chen Deng-Yuan. "N -Soliton-like Solution of Ito Equation." Communications in Theoretical Physics 42, no. 5 (November 15, 2004): 641–44. http://dx.doi.org/10.1088/0253-6102/42/5/641.

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6

Cen, Feng-Jie, Yan-Dan Zhao, Shuang-Yun Fang, Huan Meng, and Jun Yu. "Painlevé integrability of the supersymmetric Ito equation." Chinese Physics B 28, no. 9 (September 2019): 090201. http://dx.doi.org/10.1088/1674-1056/ab38a7.

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7

Tleubergenov, M. I., G. K. Vassilina, and D. T. Azhymbaev. "Construction of the differential equations system of the program motion in Lagrangian variables in the presence of random perturbations." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 105, no. 1 (March 30, 2022): 118–26. http://dx.doi.org/10.31489/2022m1/118-126.

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The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.

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8

Rezazadeh, Hadi, Sharanjeet Dhawan, Savaïssou Nestor, Ahmet Bekir, and Alper Korkmaz. "Computational solutions of the generalized Ito equation in nonlinear dispersive systems." International Journal of Modern Physics B 35, no. 13 (May 20, 2021): 2150172. http://dx.doi.org/10.1142/s0217979221501721.

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This papers presents new exact analytical solutions of a generalized Ito equation having three nonlinear terms, third- and fifth-order derivative forms that model the dynamics of traveling waves in nonlinear dispersive systems. With the help of Riccati equation method, we obtain different kinds of exact traveling wave solutions containing dark, singular, trigonometric, rational and other form of waves solutions that are more general than classical ones existing in the literature. Despite the originality of the new results obtained, the method used here is very efficient, powerful and can be extended to other types of nonlinear equations and more. Moreover, the behaviors of traveling waves solutions are portrayed graphically by selecting suitable values for the physical parameters.

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9

Zhou, Yuan, and Solomon Manukure. "Complexiton solutions to the Hirota‐Satsuma‐Ito equation." Mathematical Methods in the Applied Sciences 42, no. 7 (February 3, 2019): 2344–51. http://dx.doi.org/10.1002/mma.5512.

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10

Ma, Hongcai, Xiangmin Meng, Hanfang Wu, and Aiping Deng. "A class of lump solutions for ito equation." Thermal Science 23, no. 4 (2019): 2205–10. http://dx.doi.org/10.2298/tsci1904205m.

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In this paper, we investigate the exact solutions for the (1+1)-D Ito equation. Some lump solutions are obtained by using Hirota?s bilinear method, and the conditions to guarantee analytical and rational localization of the lump solutions are presented. Suitable choices of the involved parameters guaranteeing analyticity of the solution are given. The 3-D plots with particular choices of the involved parameters are illustrated.

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11

V. Krishnan, E. "On the Ito-Type Coupled Nonlinear Wave Equation." Journal of the Physical Society of Japan 55, no. 11 (November 15, 1986): 3753–55. http://dx.doi.org/10.1143/jpsj.55.3753.

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12

Carkovs, Jevgeņijs, and Oksana Pavlenko. "Stochastic Modelling for Dynamics of Interacting Populations." Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences. 73, no. 5 (October 1, 2019): 455–61. http://dx.doi.org/10.2478/prolas-2019-0070.

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Abstract The paper deals with a mathematical model for two interacting populations. Under the assumption of fast stochastic contacting of populations, we derive stochastic Poisson-type differential equations with a small parameter and propose an approximative algorithm for quantitative analysis of population dynamics that consists of two steps. First, we derive an ordinary differential equation for mean value of each population growth and analyse the average asymptotic population behaviour. Then, applying diffusion approximation procedure, we derive a stochastic Ito differential equation for small random deviations on the average motion in a form of a linear non-homogeneous Ito stochastic differential equation and analyse the probabilistic characteristics of the Gaussian process given by this equation.

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13

Inc, Mustafa, E. A. Az-Zo’bi, Adil Jhangeer, Hadi Rezazadeh, Muhammad Nasir Ali, and Mohammed K. A. Kaabar. "New Soliton Solutions for the Higher-Dimensional Non-Local Ito Equation." Nonlinear Engineering 10, no. 1 (January 1, 2021): 374–84. http://dx.doi.org/10.1515/nleng-2021-0029.

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Abstract In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

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14

Fan, Xinghua, and Shasha Li. "Bifurcation of Traveling Wave Solutions of the Dual Ito Equation." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/153139.

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The dual Ito equation can be seen as a two-component generalization of the well-known Camassa-Holm equation. By using the theory of planar dynamical system, we study the existence of its traveling wave solutions. We find that the dual Ito equation has smooth solitary wave solutions, smooth periodic wave solutions, and periodic cusp solutions. Parameter conditions are given to guarantee the existence.

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15

Salas, Alvaro H., Cesar A. Gómez S, and Bernardo Acevedo Frias. "Computing Exact Solutions to a Generalized Lax-Sawada-Kotera-Ito Seventh-Order KdV Equation." Mathematical Problems in Engineering 2010 (2010): 1–7. http://dx.doi.org/10.1155/2010/524567.

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The Cole-Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada-Kotera-Ito KdV equation (Sawada-Kotera-Ito KdV7).

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16

SAITO, T., and T. ARIMITSU. "QUANTUM STOCHASTIC EQUATIONS FOR A NON-LINEAR DAMPED OSCILLATOR." Modern Physics Letters B 07, no. 09 (April 20, 1993): 623–31. http://dx.doi.org/10.1142/s0217984993000606.

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A unified framework of stochastic differential equations for quantum systems, formulated within Non-Equilibrium Thermo Field Dynamics (NETFD), is applied to a model of non-linear damped oscillator. The quantum stochastic Liouville equation and the quantum Langevin equations (both of Ito and Stratonovich type), which are consistent with the corresponding master equation, are written down explicitly in the case of a non-conventional treatment. This solves Kubo's third problem: how one can obtain the correlations of random force operators for the Langevin equation compatible with the master equation derived by the non-conventional treatment of the damping theory.

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17

Tang, Yaning, Jinli Ma, Wenxian Xie, and Lijun Zhang. "Interaction solutions for the (2+1)-dimensional Ito equation." Modern Physics Letters B 33, no. 13 (May 10, 2019): 1950167. http://dx.doi.org/10.1142/s0217984919501677.

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In this paper, two classes of interaction solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional Ito equation are studied in the case of Hirota bilinear form. As the results, the interaction solutions between the rational function and a periodic function as well as the interaction solution between the hyperbolic function and a periodic function are obtained. Based on the interaction solutions, a new transformation is proposed to analyze and discuss the influence of parameters. Furthermore, two kinds of lump solutions can be obtained via the limit behavior of the interaction solutions and the dynamical properties of these solutions are also illustrated.

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18

Hu, Xiaorui, Shuning Lin, and Shoufeng Shen. "New interaction solutions to (1+1)-dimensional Ito equation." Applied Mathematics Letters 101 (March 2020): 106071. http://dx.doi.org/10.1016/j.aml.2019.106071.

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19

Zhang, Hai-Qiang, Xia Gao, Zhi-jie Pei, and Fa Chen. "Rogue periodic waves in the fifth-order Ito equation." Applied Mathematics Letters 107 (September 2020): 106464. http://dx.doi.org/10.1016/j.aml.2020.106464.

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20

Ding, Liyuan, Wen-Xiu Ma, and Yehui Huang. "Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation." Modern Physics Letters B 35, no. 26 (August 13, 2021): 2150437. http://dx.doi.org/10.1142/s0217984921504376.

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A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

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21

Wang, Jiao, Tian-Zhou Xu, and Gang-Wei Wang. "Numerical algorithm for time-fractional Sawada-Kotera equation and Ito equation with Bernstein polynomials." Applied Mathematics and Computation 338 (December 2018): 1–11. http://dx.doi.org/10.1016/j.amc.2018.06.001.

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22

Zhang, Lijun, and Chaudry Masood Khalique. "Exact Solitary Wave and Periodic Wave Solutions of a Class of Higher-Order Nonlinear Wave Equations." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/548606.

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We study the exact traveling wave solutions of a general fifth-order nonlinear wave equation and a generalized sixth-order KdV equation. We find the solvable lower-order subequations of a general related fourth-order ordinary differential equation involving only even order derivatives and polynomial functions of the dependent variable. It is shown that the exact solitary wave and periodic wave solutions of some high-order nonlinear wave equations can be obtained easily by using this algorithm. As examples, we derive some solitary wave and periodic wave solutions of the Lax equation, the Ito equation, and a general sixth-order KdV equation.

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23

Aibarakati, Wafaa, Aly Seadaw, and Noufe Aljahdaly. "Application of mathematical methods for the non-linear seventh order Sawada-Kotera-Ito dynamical wave equation." Thermal Science 23, Suppl. 6 (2019): 2081–93. http://dx.doi.org/10.2298/tsci190705373a.

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This article deal with finding travelling wave solutions for the 7th order Sawada-Kotera-Ito dynamical wave equation which describes the evolution of steeper waves of shorter wavelength than KdV equations using modified extended direct algebraic method. The new solutions derived have various physical structure, we also give graphic representation of the exact and stable solutions.

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24

Zhang, Yingnan, Xingbiao Hu, and Jianqing Sun. "Numerical calculation of N -periodic wave solutions to coupled KdV–Toda-type equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2245 (January 2021): 20200752. http://dx.doi.org/10.1098/rspa.2020.0752.

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In this paper, we study the N -periodic wave solutions of coupled Korteweg–de Vries (KdV)–Toda-type equations. We present a numerical process to calculate the N -periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N = 3, we give some detailed examples to show the N -periodic wave solutions to the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

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25

Grigoriu, M. "A Monte Carlo Solution of Heat Conduction and Poisson Equations." Journal of Heat Transfer 122, no. 1 (August 31, 1999): 40–45. http://dx.doi.org/10.1115/1.521435.

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A Monte Carlo method is developed for solving the heat conduction, Poisson, and Laplace equations. The method is based on properties of Brownian motion and Ito^ processes, the Ito^ formula for differentiable functions of these processes, and the similarities between the generator of Ito^ processes and the differential operators of these equations. The proposed method is similar to current Monte Carlo solutions, such as the fixed random walk, exodus, and floating walk methods, in the sense that it is local, that is, it determines the solution at a single point or a small set of points of the domain of definition of the heat conduction equation directly. However, the proposed and the current Monte Carlo solutions are based on different theoretical considerations. The proposed Monte Carlo method has some attractive features. The method does not require to discretize the domain of definition of the differential equation, can be applied to domains of any dimension and geometry, works for both Dirichlet and Neumann boundary conditions, and provides simple solutions for the steady-state and transient heat equations. Several examples are presented to illustrate the application of the proposed method and demonstrate its accuracy. [S0022-1481(00)02201-5]

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26

Zou, Li, Zong-Bing Yu, Shou-Fu Tian, Lian-Li Feng, and Jin Li. "Lump solutions with interaction phenomena in the (2+1)-dimensional Ito equation." Modern Physics Letters B 32, no. 07 (March 5, 2018): 1850104. http://dx.doi.org/10.1142/s021798491850104x.

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In this paper, we consider the (2+1)-dimensional Ito equation, which was introduced by Ito. By considering the Hirota’s bilinear method, and using the positive quadratic function, we obtain some lump solutions of the Ito equation. In order to ensure rational localization and analyticity of these lump solutions, some sufficient and necessary conditions are provided on the parameters that appeared in the solutions. Furthermore, the interaction solutions between lump solutions and the stripe solitons are discussed by combining positive quadratic function with exponential function. Finally, the dynamic properties of these solutions are shown via the way of graphical analysis by selecting appropriate values of the parameters.

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27

Hu, Xing-Biao, and Yong Li. "Nonlinear superposition formulae of the Ito equation and a model equation for shallow water waves." Journal of Physics A: Mathematical and General 24, no. 9 (May 1, 1991): 1979–85. http://dx.doi.org/10.1088/0305-4470/24/9/010.

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28

Seadawy, Aly, Asghar Ali, and Noufe Aljahdaly. "The nonlinear integro-differential Ito dynamical equation via three modified mathematical methods and its analytical solutions." Open Physics 18, no. 1 (March 10, 2020): 24–32. http://dx.doi.org/10.1515/phys-2020-0004.

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AbstractIn this work, we construct traveling wave solutions of (1+1) - dimensional Ito integro-differential equation via three analytical modified mathematical methods. We have also compared our achieved results with other different articles. Portrayed of some 2D and 3D figures via Mathematica software demonstrates to understand the physical phenomena of the nonlinear wave model. These methods are powerful mathematical tools for obtaining exact solutions of nonlinear evolution equations and can be also applied to non-integrable equations as well as integrable ones. Hence worked-out results ascertained suggested that employed techniques best to deal NLEEs.

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29

Meng, Xiangmin, and Hongcai Ma. "The Lump Solutions of the (1 + 1)-Dimensional Ito-Equation." Open Journal of Applied Sciences 09, no. 03 (2019): 121–25. http://dx.doi.org/10.4236/ojapps.2019.93011.

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30

Li, Chunxia, and Yunbo Zeng. "Soliton solutions to a higher order Ito equation: Pfaffian technique." Physics Letters A 363, no. 1-2 (March 2007): 1–4. http://dx.doi.org/10.1016/j.physleta.2006.10.080.

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31

Chen, Yaming, Songhe Song, and Huajun Zhu. "Multi-symplectic methods for the Ito-type coupled KdV equation." Applied Mathematics and Computation 218, no. 9 (January 2012): 5552–61. http://dx.doi.org/10.1016/j.amc.2011.11.045.

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32

Zhang, Yi, Yun-Cheng You, Wen-Xiu Ma, and Hai-Qiong Zhao. "Resonance of solitons in a coupled higher-order Ito equation." Journal of Mathematical Analysis and Applications 394, no. 1 (October 2012): 121–28. http://dx.doi.org/10.1016/j.jmaa.2012.03.030.

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33

Yakup; YAŞAR, YILDIRIM. "WRONSKIAN SOLUTIONS OF (2+1) DIMENSIONAL NON-LOCAL ITO EQUATION." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 67, no. 2 (2018): 126–38. http://dx.doi.org/10.1501/commua1_0000000867.

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34

Van Gorder, Robert A. "Solutions to a Novel Casimir Equation for the Ito System." Communications in Theoretical Physics 56, no. 5 (November 2011): 801–4. http://dx.doi.org/10.1088/0253-6102/56/5/02.

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35

Zhao, Hai-qiong. "Soliton solution of a multi-component higher-order Ito equation." Applied Mathematics Letters 26, no. 7 (July 2013): 681–86. http://dx.doi.org/10.1016/j.aml.2013.01.008.

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36

Feng, Binlu, Bo Han, and Huanhe Dong. "Abundant new travelling wave solutions for the coupled Ito equation." Chaos, Solitons & Fractals 39, no. 1 (January 2009): 393–98. http://dx.doi.org/10.1016/j.chaos.2007.04.012.

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37

Tu, Jian Xin, Zhi Ren Wang, Han Zhu, and Ping Wang. "The Nonlinear Random Vibration of a Clamped Rectangular Thin Plate in Magnetic Field." Applied Mechanics and Materials 628 (September 2014): 127–32. http://dx.doi.org/10.4028/www.scientific.net/amm.628.127.

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In this paper, the magneto-elastic nonlinear random vibration of a clamped rectangular thin plate in magnetic field is studied. According to the magneto-elastic theory of plates and shells and the theory of structural random vibration, the magneto-elastic nonlinear random vibration equation of a clamped rectangular thin plate in a magnetic field is derived. Then the nonlinear random vibration equation is transferred into the Ito differential equation, and the Ito differential equation is solved using FPK equation method. Thus the numerical characteristics of displacement response and velocity response of the rectangular thin plate are obtained. Finally, through a numerical example, the influences of magnetic field parameters on the numerical characteristics are discussed, and some methods which can be used to effectively control the random vibration responses of the plate are given.

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38

Yong-Yan, Fan, Jalil Manafian, Syed Maqsood Zia, Dinh Tran Ngoc Huy, and Trung-Hieu Le. "Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave." Advances in Mathematical Physics 2021 (October 12, 2021): 1–26. http://dx.doi.org/10.1155/2021/1164838.

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In the current study, an analytical treatment is studied starting from the 2 + 1 -dimensional generalized Hirota-Satsuma-Ito (HSI) equation. Based on the equation, we first establish the evolution equation and obtain rational function solutions by means of the bilinear form with the help of the Hirota bilinear operator. Then, by the suggested method, the periodic, cross-kink wave solutions are also obtained. Also, the semi-inverse variational principle (SIVP) will be utilized for the generalized HSI equation. Two major cases were investigated from two different techniques. Moreover, the improved tan χ ξ method on the generalized Hirota-Satsuma-Ito equation is probed. The 3D, density, and contour graphs illustrating some instances of got solutions have been demonstrated from a selection of some suitable parameters. The existing conditions are handled to discuss the available got solutions. The current work is extensively utilized to report plenty of attractive physical phenomena in the areas of shallow water waves and so on.

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39

Egorov, A. D. "On composite formulas for mathematical expectation of functionals of solution of the Ito equation in Hilbert space." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 2 (June 28, 2019): 158–68. http://dx.doi.org/10.29235/1561-2430-2019-55-2-158-168.

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This article is devoted to constructing composite approximate formulas for calculation of mathematical expectation of nonlinear functionals of solution of the linear Ito equation in Hilbert space with additive noise. As the leading process, the Wiener process taking values in Hilbert space is examined. The formulas are a sum of the approximations of the nonlinear functionals obtained by expanding the leading random process into a series of independent Gaussian random variables and correcting approximating functional quadrature formulas that ensure an approximate accuracy of compound formulas for third-order polynomials. As a test example, the application of the obtained formulas to the case of a one-dimensional wave equation with a leading Wiener process indexed by spatial and temporal variables is considered. This article continues the research begun in [1].The problem is motivated by the necessity to calculate the nonlinear functionals of solution of stochastic partial differential equations. Approximate evaluation of mathematical expectation of stochastic equations with a leading random process indexed only by the time variable is considered in [2–11]. Stochastic partial equations in various interpretations are considered [12–16]. The present article uses the approach given in [12].

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40

Chebotarev, Alexander M. "Quantum Stochastic Differential Equation is Unitarily Equivalent to a Symmetric Boundary Value Problem in Fock Space." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 02 (April 1998): 175–99. http://dx.doi.org/10.1142/s0219025798000120.

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We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.

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41

Czechowski, Zbigniew. "On reconstruction of the Ito-like equation from persistent time series." Acta Geophysica 61, no. 6 (April 4, 2013): 1504–21. http://dx.doi.org/10.2478/s11600-013-0117-1.

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42

Chen, Jun-Chao, Yong Chen, Bao-Feng Feng, and Han-Min Zhu. "Pfaffian-Type Soliton Solution to a Multi-Component Coupled Ito Equation." Chinese Physics Letters 31, no. 11 (November 2014): 110502. http://dx.doi.org/10.1088/0256-307x/31/11/110502.

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43

Liu, Jian-Guo, Guo-Ping Ai, and Wen-Hui Zhu. "Mixed type exact solutions to the (2+1)-dimensional Ito equation." Modern Physics Letters B 32, no. 28 (October 4, 2018): 1850343. http://dx.doi.org/10.1142/s0217984918503438.

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Using a direct test function based on the Hirota’s bilinear form, two classes of mixed type exact solutions to the (2[Formula: see text]+[Formula: see text]1)-dimensional Ito equation are found through symbolic computations with Mathematica. These mixed type exact solutions contain exponential function, trigonometric function and hyperbolic function. The physical structures and characteristics for these resulting mixed type exact solutions are illustrated by some three-dimensional plots and contour plots.

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44

Zhao, Zhanhui, Zhengde Dai, and Chuanjian Wang. "Extend three-wave method for the (1+2)-dimensional Ito equation." Applied Mathematics and Computation 217, no. 5 (November 2010): 2295–300. http://dx.doi.org/10.1016/j.amc.2010.06.059.

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45

Ma, Wen-Xiu, Xuelin Yong, and Hai-Qiang Zhang. "Diversity of interaction solutions to the (2+1)-dimensional Ito equation." Computers & Mathematics with Applications 75, no. 1 (January 2018): 289–95. http://dx.doi.org/10.1016/j.camwa.2017.09.013.

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Hu, Xiaorui, Shoufeng Shen, and Yongyang Jin. "Rogue wave and interaction phenomenon to (1+1)-dimensional Ito equation." Applied Mathematics Letters 90 (April 2019): 99–103. http://dx.doi.org/10.1016/j.aml.2018.10.018.

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Li, J. B., and F. J. Chen. "Exact traveling wave solutions and bifurcations of the dual Ito equation." Nonlinear Dynamics 82, no. 3 (July 14, 2015): 1537–50. http://dx.doi.org/10.1007/s11071-015-2259-y.

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Ma, Wen-Xiu. "Interaction solutions to Hirota-Satsuma-Ito equation in (2 + 1)-dimensions." Frontiers of Mathematics in China 14, no. 3 (May 30, 2019): 619–29. http://dx.doi.org/10.1007/s11464-019-0771-y.

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Zhou, Yuan, Solomon Manukure, and Wen-Xiu Ma. "Lump and lump-soliton solutions to the Hirota–Satsuma–Ito equation." Communications in Nonlinear Science and Numerical Simulation 68 (March 2019): 56–62. http://dx.doi.org/10.1016/j.cnsns.2018.07.038.

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Peng, Li-Juan. "Different wave structures for the completely generalized Hirota–Satsuma–Ito equation." Nonlinear Dynamics 105, no. 1 (June 15, 2021): 707–16. http://dx.doi.org/10.1007/s11071-021-06602-0.

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