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This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili (KdV-KP) equation. This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation. The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric, hyperbolic, and rational solutions. The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations. Furthermore, the obtained solutions have not been reported in the previous literature and might have significant impact on future research.

This article analytically investigates a (2+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama equation. We secure the solutions of this equation via the engagement of Lie symmetry reductions alongside direct integration techniques. We gain non-topological 1-soliton as well as periodic function solutions of the equation. Series solution of the underlying equation is also achieved by exploiting power series technique. Furthermore, the utilization of ( G ′ / G ) -expansion method is done in procuring some closed-form solutions of the equation. Graphical exhibition of the dynamical character of the gained results is given in a bid to have a sound understanding of the physical phenomena of the underlying model. Conclusively, we give the conserved vectors of the aforementioned equation by employing both the standard multiplier approach as well as the Ibragimov’s conservation theorem.

The (3+1)-dimensional Kadomtsev-Petviashvili and the modified KdV-Zakharov-Kuznetsov equations have a significant impact in modern science for their widespread applications in the theory of long-wave propagation, dynamics of shallow water wave, plasma fluid model, chemical kinematics, chemical engineering, geochemistry, and many other topics. In this article, we have assessed the effects of wave speed and physical parameters on the wave contours and confirmed that waveform changes with the variety of the free factors in it. As a result, wave solutions are extensively analyzed by using the balancing condition on the linear and nonlinear terms of the highest order and extracted different standard wave configurations, containing kink, breather soliton, bell-shaped soliton, and periodic waves. To extract the soliton solutions of the high-dimensional nonlinear evolution equations, a recently developed approach of the sine-Gordon expansion method is used to derive the wave solutions directly. The sine-Gordon expansion approach is a potent and strategic mathematical tool for instituting ample of new traveling wave solutions of nonlinear equations. This study established the efficiency of the described method in solving evolution equations which are nonlinear and with higher dimension (HNEEs). Closed-form solutions are carefully illustrated and discussed through diagrams.

Fluids are seen in, e.g., mechanical, chemical and biomedical engineering, astrophysics, biology and geophysics. In this paper, we investigate a ( 3 + 1 ) -dimensional generalized variable-coefficient shallow water wave equation for the flow below a pressure surface in a fluid. Via the Kadomtsev–Petviashvili hierarchy reduction, we derive the rational solutions in terms of the Gramian. The first-order, higher-order and multi-lump waves are obtained. We present the first-order lump waves on the periodic, monotonically increasing, parabolic and Gaussian backgrounds. We observe the second-order lump waves: Two lump waves interact with each other and separate into two new lump waves; After the interaction, velocities and amplitudes of the two lump waves change. Two-lump waves are also shown: Different from the second-order lump waves, after the interaction, the two-lump waves propagate with their original velocities and amplitudes.

Nonlinear evolution equations (NLEEs) are primarily relevant to nonlinear complex physical systems in a wide range of fields, including ocean physics, plasma physics, chemical physics, optical fibers, fluid dynamics, biology physics, solid-state physics, and marine engineering. This paper investigates the Lie symmetry analysis of a generalized (3+1)-dimensional breaking soliton equation depending on five nonzero real parameters. We derive the Lie infinitesimal generators, one-dimensional optimal system, and geometric vector fields via the Lie symmetry technique. First, using the three stages of symmetry reductions, we converted the generalized breaking soliton (GBS) equation into various nonlinear ordinary differential equations (NLODEs), which have the advantage of yielding a large number of exact closed-form solutions. All established closed-form wave solutions include special functional parameter solutions, as well as hyperbolic trigonometric function solutions, trigonometric function solutions, dark-bright solitons, bell-shaped profiles, periodic oscillating wave profiles, combo solitons, singular solitons, wave-wave interaction profiles, and various dynamical wave structures, which we present for the first time in this research. Eventually, the dynamical analysis of some established solutions is revealed through three-dimensional sketches via numerical simulations. Some of the new solutions are often useful and helpful for studying the nonlinear wave propagation and wave-wave interactions of shallow water waves in many new high-dimensional nonlinear evolution equations.

In this paper, a generalized (3+1)-dimensional Kadomtsev–Petviashvili Benjamin–Bona–Mahony equation which describes the fluid flow in the case of an offshore structure, is investigated. Here, making use of the bilinear form and symbolic computation, we construct four kinds of rogue wave solutions consisting of independent breathers. Among these solutions, the fourth order rogue wave solution is rarely considered in nonlinear system. Exact locations of the highest and lowest peaks as well as the extreme values of the wave heights are systematically analyzed. The obtained rogue waves observe certain “circularity structure”, the highest or lowest peaks both sit at the same circular. Moreover, we show that the rogue waves are stable during the propagation.

In the present endeavour, the shifted Legendre collocation method is extended to obtain the solution of nonlinear fractional order (2+1)-dimensional advection–reaction–diffusion solute transport equation. The variations of solute concentration of the model for different fractional order space and time derivatives are presented graphically for various particular cases. The main feature of the present contribution is the graphical exhibitions of the effects of advection term, reaction term and fractional-order parameters on the solution profile. To authenticate the effectiveness of the method, a drive has been taken to compare the obtained results with the existing analytical results of the integer-order form of the considered model through error analysis which are displayed in tabular and pictorial forms.

The potential Korteweg-de Vries equation arises in the study of water waves and is reported in the dynamics of tsunami waves. The fractional order potential Korteweg-de Vries equation is more flexible and generalized than its classical form. In this work, the modified auxiliary equation technique and residual power series method are utilized to build new exact and analytical approximate solutions of the time-fractional potential Korteweg-de Vries equation. The dynamics of the solutions obtained are explored by drawing them in two and three dimensions. Comparisons between the new results and the solutions available in literature show that the presented approaches of nonlinear problem resolution are highly effective and reliable. The obtained solutions will be helpful to understand the dynamical framework of many nonlinear physical phenomena.

Different resonance constraints enrich the behavior of soliton solutions. The soliton molecules, which are the bound states of solitons, can be set off by the velocity resonance. The lump waves, which are localized in all directions in space, are theoretically regarded as a limit form of soliton in some ways. In this paper, a (2+1)-dimensional Sharma–Tasso–Olver–Burgers (STOB) equation is investigated. Soliton (kink) molecule, half periodic kink(HPK) and HPK molecule are studied. Then the lump solution is obtained and the interactions between lump and kink molecule are discussed. The kink molecule-lump solutions exhibit a fusion phenomenon and a rogue (instanton) phenomenon, respectively.

In this paper, we implemented the Exp-function method and the extended tanh method to finds the analytical solutions of one of the important nonlinear partial differential equations with variable coefficients called the new (3 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa with time-dependent coefficients. Moreover, we find the numerical solutions to that equation using the finite difference method. The results demonstrate the effectiveness and convenience of the used methods. Comparison between our results through some tables and graphs to illustrate the accuracy of the applied methods. We obtained the different types of solutions-like bright, dark, periodic, rational and singular periodic soliton solutions using the different values of constant parameters. The obtained exact solutions may be useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in (3 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa with time-dependent coefficients model.

Within the Hirota bilinear formulation, we construct N -soliton solutions and analyze the Hirota N -soliton conditions in (2+1)-dimensions. A generalized algorithm to prove the Hirota conditions is presented by comparing degrees of the multivariate polynomials derived from the Hirota function in N wave vectors, and two weight numbers are introduced for transforming the Hirota function to achieve homogeneity of the related polynomials. An application is developed for a general combined nonlinear equation, which provides a proof of existence of its N -soliton solutions. The considered model equation includes three integrable equations in (2+1)-dimensions: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, and the (2+1)-dimensional Hirota–Satsuma–Ito equation, as specific examples.

In this paper, the new travelling wave solutions of the (2+1)-dimensional extended Calogero-Bogoyavlenskii-Schiff (ECBS) equation are investigated. The main aim of this work is to find the new exact solutions with the aid of relatively new ( G ′ G ′ + G + A ) -expansion method. Moreover, the physical interpretation of the nonlinear phenomena is reported through the exact solutions, which indicate the efficacy of the proposed method. Furthermore, the recovered solutions are periodic and solitary wave solutions which are presented graphically.

The D’Alembert solution is an important basic formula in linear partial differential theory due to that it can be considered as a general solution of the wave motion equation. However, the study of the D’Alembert wave is few works in nonlinear partial differential systems. In this paper, one construct the D’Alembert solution of a (2+1)-dimensional generalized breaking soliton equation which possesses the nonlinear terms. This D’Alembert wave has one arbitrary function in the traveling wave variable. We investigate the dynamics of the three soliton molecule, the soliton molecule by bound as an asymmetry soliton and one-soliton, the interaction between the half periodic wave and two-kink, and the interaction among the half periodic wave, one-kink and a kink soliton molecule of the (2+1)-dimensional generalized breaking soliton equation by selecting the appropriate parameters.

In this article, the bilinear form, Backlund transformation, Lax pair and infinite conservation laws of the (3+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation with time-dependent coefficients are constructed based on the Bell polynomials approach. N-soliton solutions are studied by means of introducing the complex conjugate condition technique and selecting appropriate test functions and parameters, including the hybrid solution of the a-order kink waves, b-order periodic-kink waves and c-order periodic-breather waves. The homoclinic test method is applied to investigate their dynamical interaction properties between different forms of hybrid-type solutions. Besides, a number of examples are presented by choosing different types of interactions among the hybrid-type solutions. Finally, we analyze the wave propagation direction and velocity to reflect the novel evolutionary behaviors in the three-dimensional profile of the model. These results are helpful to the study of local wave interactions in nonlinear mathematical physics.

The deformed soliton solutions of the (2+1)-dimensional nonlinear Schrodinger Maxwell–Bloch (NLS-MB) equations are investigated by the n-fold Darboux transformation method based on the seed function $$q=p=v=0,\,\eta =1$$ . New soliton solutions, including order-1 and order-2 types, are obtained and analyzed in detail. For the order-1 solutions, polynomial, trigonometric, and hyperbolic solutions are discussed. Especially, the analytical formulas of $$|q^{[1]}|$$ and soliton trajectories are obtained, which are involved by an arbitrary smooth function $$f(y+2\lambda t)$$ . For the order-2 solutions, besides the polynomial, trigonometric, and hyperbolic types, the mixed-type solutions are also derived and exhibited through several typical examples.

Some new solitary solutions of (3+1)-dimensional Jimbo-Miwa equation such as breather solutions, double breather solutions and mixed solutions of different forms are studied via applying Hirota's b...

We study solutions of type IIB string theory dual to $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory on half of ℝ3,1 coupled to holographic three-dimensional superconformal field theories (SCFTs) at the edge of this half-space. The dual geometries are asymptotically AdS5×S5 with boundary geometry ℝ2,1×ℝ+, with a geometrical end-of-the-world (ETW) brane cutting off the other half of the asymptotic region of the would-be Poincaré AdS5×S5. We show that by choosing the 3D SCFT appropriately, this ETW brane can be pushed arbitrarily far towards the missing asymptotic region, recovering the “missing” half of Poincaré AdS5×S5. We also show that there are 3D SCFTs whose dual includes a wedge of Poincaré AdS5×S5 with an angle arbitrarily close to π, with geometrical ETW branes on either side.

Purpose This paper aims to study the Gramian solutions and solitonic interactions of a (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, which models the nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth. Design/methodology/approach Pfaffian technique is used to construct the Gramian solutions of the (2 + 1)-dimensional BKK system. Asymptotic analysis is applied on the two-soliton solutions to study the interaction properties. Findings N-soliton solutions in the Gramian with a real function ζ(y) of the (2 + 1)-dimensional BKK system are constructed and proved, where N is a positive integer and y is the scaled space variable. Conditions of elastic and inelastic interactions between the two solitons are revealed asymptotically. For the three and four solitons, elastic, inelastic interactions and soliton resonances are discussed graphically. Effect of the wave numbers, initial phases and ζ(y) on the solitonic interactions is also studied. Originality/value Shallow water waves are studied for the applications in environmental engineering and hydraulic engineering. This paper studies the shallow water waves through the Gramian solutions of a (2 + 1)-dimensional BKK system and provides some phenomena that have not been studied.

The (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation is a significant physical model. By using the long wave limit method and confining the conjugation conditions on the interrelated solitons, the general M-lump, high-order breather, and localized interaction hybrid solutions are investigated, respectively. Then we implement the numerical simulations to research their dynamical behaviors, which indicate that different parameters have very different dynamic properties and propagation modes of the waves. The method involved can be validly employed to get high-order waves and study their propagation phenomena of many nonlinear equations.

In this paper, we study the $$(2 + 1)$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation, which has certain applications in fluids and plasmas. Via the Kadomtsev–Petviashvili hierarchy reduction, we derive two types of the breather solutions in terms of Gramian. Based on the first type breather solutions, we observe the breathers and periodic waves, while we observe the breathers and solitons according to the second type breather solutions. Taking the long-wave limits technique for the first type breather solutions, we derive semi-rational and rational solutions. The semi-rational solutions describe the interactions between the rogue waves/lumps and breathers, while the rational solutions give birth to the rogue waves and lumps.

In this paper, we consider a generalized $$(2+1)$$ -dimensional nonlinear wave equation. Based on the bilinear method, the N-soliton solutions are obtained. The resonance Y-type soliton, which is similar to the capital letter Y in the spatial structure, and the interaction solutions between different types of resonance solitons are constructed by adding some new constraints to the parameters of the N-soliton solutions. The new type of two-opening resonance Y-type soliton solutions is presented by choosing some appropriate parameters in 3-soliton solutions. The hybrid solutions consisting of resonance Y-type solitons, breathers and lumps are investigated. The trajectories of the lump waves before and after the collision with the resonance Y-type solitons are analyzed from the perspective of mathematical mechanism. Furthermore, the multi-dimensional Riemann-theta function is employed to investigate the quasi-periodic wave solutions. The one-periodic and two-periodic wave solutions are obtained. The asymptotic properties are systematically analyzed, which establish the relations between the quasi-periodic wave solutions and the soliton solutions. The results may be helpful to provide some effective information to analyze the dynamical behaviors of solitons, fluid mechanics, shallow water waves and optical solitons.

In this paper, we investigate a (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation in fluid dynamics. Based on the Hirota method, we give a bilinear auto-Backlund transformation. Via the truncated Painleve expansion, we get a Painleve-type auto-Backlund transformation. With the aid of the symbolic computation, we derive some one- and two-kink soliton solutions. We present the oblique and parallel elastic interactions between the two-kink solitons. Via the extended homoclinic test technique, we construct some breather-wave solutions. Besides, we derive some lump solutions with the periods of the breather-wave solutions to the infinity. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. Based on the polynomial-expansion method, travelling-wave solutions are constructed.

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painleve expansion. Three kinds of non-auto and auto Backlund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the n th Backlund transformation. The consistent Riccati expansion method is employed to derive a Backlund transformation. Furthermore, the soliton solutions, fusion-type N -solitary wave solutions and soliton–cnoidal wave solutions are gained through Backlund transformations.

Multi-loop scattering amplitudes/null polygonal Wilson loops in ${\mathcal N}=4$ super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an $1+1$ dimensional subspace of Minkowski spacetime (or boundary of the $\rm AdS_3$ subspace). Since the edges of a $2n$-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of $G(2,n)/T \sim A_{n{-}3}$. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping $A_2$ functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters $v, 1+v, w, 1+w$ of $A_1 \times A_1$, there are two letters $v-w, 1- v w$ mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping $A_2$ functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to $2n$-gons in terms of $A_2$ functions and beyond., 26 pages, several figures and tables, an ancilary file

In the present paper, new analytical solutions for the conformable space-time fractional (2+1)-dimensional breaking soliton, third-order KdV and Burger's equations are obtained by using the simplified tan(ϕ(ξ)2)tan⁡(ϕ(ξ)2)-expansion method (SITEM). Here, fractional derivatives are described in conformable sense. The obtained traveling wave solutions are expressed by the trigonometric, hyperbolic, exponential and rational functions. Simulation of the obtained solutions are given at the end of the paper.

Under investigation in this letter is an (3+1)-dimensional Hirota–Satsuma–Ito-like equation, which provide strong support for studying the dynamic behavior of nonlinear waves. Based on a special Cole–Hopf transformation and Hirota bilinear method, the bilinear form of the equation is obtained and this form has never been given. High-order breather solutions, lump solutions and mixed solutions are obtained by using complex conjugate parameters and long-wave limit method. Then, the influence of the coefficient $$g_{t}(t)$$ of the bilinear equation on the interaction of these solutions is analyzed by means of images. It can be found that $$g_{t}(t)$$ changes the interaction of the solutions by influencing the positions and trajectories of higher-order breather solutions, lump solutions and mixed solutions. We find that different values of g(t) make the interaction of solutions different. Finally, the mixed solution of the equation including a breather wave and a line rogue wave is obtained by using the test function, and its dynamic properties are illustrated by means of images.

The lump solutions and interaction solutions are mainly investigated for the (2 + 1)-dimensional KPI equation. According to relations of the undetermined parameters of the test functions, the N-soliton solutions are showed by computations of the Maple using the Hirota bilinear form for (2 + 1)-dimensional KPI equation. One type of the lump solutions for (2 + 1)-dimensional KPI equation has been deduced by the limit method of the N-soliton solutions. In addition, the interaction solutions between the lump and N-soliton solutions of it are studied by the undetermined interaction functions. The sufficient conditions for the existence of the interaction solutions are obtained. Furthermore, the new breather solutions for the (2 + 1)-dimensional KPI equation are considered by the homoclinic test method via new test functions including more parameters than common test functions.

In this paper, a new (3+1)-dimensional integrable Kadomtsev–Petviashvili equation is developed. Its integrability is verified by the Painleve analysis. The bilinear form, multiple-soliton, breather and lump solutions are obtained via using the Hirota bilinear method, a symbolic computation scheme. Furthermore, the abundant dynamical behaviors for these solutions are discovered. It is interesting that there are splitting and fusing phenomena when the lump waves interact. The results can well simulate complex waves and their interaction dynamics in fluids.

In this paper, we investigate a (3+1)-dimensional Hirota–Satsuma–Ito-like system for the shallow water waves. We obtain a Painlev $$\acute{\mathrm{e}}$$ integrable condition of the system. By virtue of the truncated Painlev $$\acute{\mathrm{e}}$$ expansion, we get an auto-Backlund transformation under certain Painlev $$\acute{\mathrm{e}}$$ integrable condition. Based on the bilinear form, we give a bilinear auto-Backlund transformation and a Lax pair under certain Painlev $$\acute{\mathrm{e}}$$ integrable condition. We obtain that a breather and kink waves propagate under certain Painlev $$\acute{\mathrm{e}}$$ integrable condition. The breather has a peak and a trough and the height of the kink wave periodically increases or decreases during the propagation. Furthermore, we get the lump-periodic-wave and solitary-wave solutions and observe that the lump-periodic and solitary waves propagate under certain Painlev $$\acute{\mathrm{e}}$$ integrable conditions. During the propagation, the heights of the lump-periodic waves keep unchanged and height of the solitary wave periodically increases or decreases.

Several $$(2+1)$$ -dimensional integrable coupling systems are derived from two sets of auxiliary linear problems, including the integrable coupling system of a $$(2+1)$$ -dimensional generalization of the dispersive long-wave system, and a $$(2+1)$$ -dimensional generalizations of the Burgers equation and the Davey–Stewartson system. We use the $$\bar\partial$$ -dressing method to investigate these integrable coupling systems and obtain some exact solutions by solving the coupled $$\bar\partial$$ -problem that we introduce. The methods and techniques presented in this paper may be good inspiration for dealing with similar problems and the corresponding integrable coupling systems.

In the present paper, new analytical solutions for the space-time fractional Boussinesq and (2 + 1)-dimensional breaking soliton equations are obtained by using the simplified tan (phi(xi)/2)-expansion method. Here, fractional derivatives are defined in the conformable sense. To show the correctness of the obtained traveling wave solutions, residual error function is defined. It is observed that the new solutions are very close to the exact solutions. The solutions obtained by the presented method have not been reported in former literature. (C) 2020 Shanghai Jiaotong University. Published by Elsevier B.V.

The non-autonomous discrete bright–dark soliton solutions(NDBDSSs) of the 2 + 1-dimensional Ablowitz–Ladik (AL) equation are derived. We analyze the dynamic behaviors and interactions of the obtained 2 + 1-dimensional NDBDSSs. In this paper, we present two kinds of different methods to control the 2 + 1-dimensional NDBDSSs. In first method, we can only control the wave propagations through the spatial part, since the time function has not effect in the phase part. In second method, we can control the wave propagations through both the spatial and temporal parts. The different propagation phenomena can also be produced through two kinds of managements. We obtain the novel “ $$\pi $$ ”-shape non-autonomous discrete bright soliton solution(NDBSS), the novel “ $$\curlywedge $$ ”-shape non-autonomous discrete dark soliton solution(NDDSS) and their interaction behaviors. The novel behaviors are considered analytically, which can be applied to the electrical and optical fields.

This article focuses on the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. By using bilinear neural network method, the new test functions are constructed to find the analytical solution of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Using symbolic computing technology, periodic-soliton and periodic-type solutions of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation are obtained. These new test functions can also be used to obtain the exact analytical solutions of other nonlinear partial differential equations.

In this paper, based on the bilinear form, we give multi-solitary wave solutions of the $$(4+1)$$ -dimensional Fokas equation. From the obtained multi-solitary wave solutions, with special parameters, we derive resonant solutions of N-solitary waves. For complex conjugate parameters, we analyze interactions of two periodic waves, interactions of a solitary wave and a periodic wave by making use of their phase shifts. Particularly, the intermediate processes of elastic interactions are analyzed in detail, and interesting fusional and fissionable phenomena are found. The asymptotic interactional behaviors for these solutions are analyzed theoretically and illustrated graphically.

In this paper, the generalized exponential rational function method is applied to obtain analytical solutions for the conformable (2+1)-dimensional chiral nonlinear Schrodinger equation. We obtain novel soliton, traveling waves and kink-type solutions with complex structures. We also present the two- and three-dimensional shapes for the real and imaginary part of the obtained solutions. It is illustrated that generalized exponential rational function method is simple and efficient method to reach the various types of the soliton solutions.

Purpose This paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both right- and left-going waves in a like manner to the Boussinesq equation. Design/methodology/approach This formally uses the simplified Hirota’s method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space. Findings This paper confirms the complete integrability of the newly developed (3 + 1)-dimensional model in the Painevé sense. Research limitations/implications This paper addresses the integrability features of this model via using the Painlevé analysis. Practical implications This paper presents a variety of lump solutions via using a variety of numerical values of the included parameters. Social implications This work formally furnishes useful algorithms for extending integrable equations and for the determination of lump solutions. Originality/value To the best of the author’s knowledge, this paper introduces an original work with newly developed integrable equation and shows useful findings of solitons and lump solutions.

The Cauchy problem for the well-known Benjamin–Bona–Mahoney–Burgers equation in the class of Holder initial functions from $${{\mathbb{C}}^{{2 + \alpha }}}({{\mathbb{R}}^{3}})$$ with $$\alpha \in (0,1]$$ is considered. For such initial functions, it is proved that the Cauchy problem has a unique time-unextendable classical solution in the class $${{\mathbb{C}}^{{(1)}}}([0,T];{{\mathbb{C}}^{{2 + \lambda }}}({{\mathbb{R}}^{3}}))$$ for any $$T \in (0,{{T}_{0}});$$ moreover, either $${{T}_{0}} = + \infty $$ or $${{T}_{0}} < + \infty $$ and, in the latter case, $${{T}_{0}}$$ is the solution blow-up time. To prove the solvability of the Cauchy problem, we examine volume and surface potentials associated with the Cauchy problem in Holder spaces. Finally, a Schauder estimate is obtained.