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Discrete & Continuous Dynamical Systems - A
October 2010 , Volume 26 , Issue 4
Special Issue on Recent Progress on Partial Differential Equations Modeling Fluids and Complex Fluids
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2010, 26(4): i-iii
doi: 10.3934/dcds.2010.26.4i
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Abstract:
Fluid phenomena are ubiquitous ranging from small scale blood flows in our body to large scale geophysical flows such as the Gulf Stream. The understanding of these phenomena is crucial to many applied areas such as meteorology, oceanography and aerospace industry. Partial Differential Equations (PDEs) are the most fundamental tools in studying fluid phenomena. This special issue of Discrete and Continuous Dynamical Systems is devoted to the analysis and numerics of partial differential equations (PDEs) modeling fluids and complex fluids. It consists of twenty-one papers from invited speakers of a special sessions in the 7th AIMS Conference on Dynamical Systems and Differential Equations held at the University of Texas at Arlington, Texas, May 18 - 21, 2008.
Topics covered by this collection of papers include the analysis and computations of solutions of PDEs modeling surface water waves, Navier-Stokes Equations and related ones including those modeling complex fluids and large scale geophysical flows. Some other related issues in fluid dynamics are also touched. The papers in this special issue are not grouped according to their topics but instead ordered alphabetically by the names of the first authors.
For more information please click the “Full Text” above.
Fluid phenomena are ubiquitous ranging from small scale blood flows in our body to large scale geophysical flows such as the Gulf Stream. The understanding of these phenomena is crucial to many applied areas such as meteorology, oceanography and aerospace industry. Partial Differential Equations (PDEs) are the most fundamental tools in studying fluid phenomena. This special issue of Discrete and Continuous Dynamical Systems is devoted to the analysis and numerics of partial differential equations (PDEs) modeling fluids and complex fluids. It consists of twenty-one papers from invited speakers of a special sessions in the 7th AIMS Conference on Dynamical Systems and Differential Equations held at the University of Texas at Arlington, Texas, May 18 - 21, 2008.
Topics covered by this collection of papers include the analysis and computations of solutions of PDEs modeling surface water waves, Navier-Stokes Equations and related ones including those modeling complex fluids and large scale geophysical flows. Some other related issues in fluid dynamics are also touched. The papers in this special issue are not grouped according to their topics but instead ordered alphabetically by the names of the first authors.
For more information please click the “Full Text” above.
2010, 26(4): 1121-1139
doi: 10.3934/dcds.2010.26.1121
+[Abstract](2232)
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Abstract:
Considered here is the well-posedness of a KdV-type Boussinesq system modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid when the motion is sensibly two dimensional. Solutions are obtained in a range of Sobolev-type spaces, from the energy level to the analytic Gevrey spaces. In addition, a criterion for detecting the possibility of blow-up in finite time in terms of loss of analyticity is derived.
Considered here is the well-posedness of a KdV-type Boussinesq system modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid when the motion is sensibly two dimensional. Solutions are obtained in a range of Sobolev-type spaces, from the energy level to the analytic Gevrey spaces. In addition, a criterion for detecting the possibility of blow-up in finite time in terms of loss of analyticity is derived.
2010, 26(4): 1141-1151
doi: 10.3934/dcds.2010.26.1141
+[Abstract](2610)
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Abstract:
In this paper we consider the three-dimensional Navier-Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one direction derivative of the velocity field, namely, $u_z$, for the regularity of strong solutions to the three-dimensional Navier-Stokes equations.
In this paper we consider the three-dimensional Navier-Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one direction derivative of the velocity field, namely, $u_z$, for the regularity of strong solutions to the three-dimensional Navier-Stokes equations.
2010, 26(4): 1153-1184
doi: 10.3934/dcds.2010.26.1153
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Abstract:
Considered herein are certain Boussinesq systems with the presence of large surface tension. The existence and stability of solitary waves are established using techniques introduced earlier by Buffoni [7] and Lions [9, 10].
Considered herein are certain Boussinesq systems with the presence of large surface tension. The existence and stability of solitary waves are established using techniques introduced earlier by Buffoni [7] and Lions [9, 10].
2010, 26(4): 1185-1196
doi: 10.3934/dcds.2010.26.1185
+[Abstract](2534)
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Abstract:
We provide a proof of global regularity of solutions of coupled Navier-Stokes equations and Fokker-Planck equations, in two spatial dimensions, in the absence of boundaries. The proof yields a priori estimates for the growth of spatial gradients.
We provide a proof of global regularity of solutions of coupled Navier-Stokes equations and Fokker-Planck equations, in two spatial dimensions, in the absence of boundaries. The proof yields a priori estimates for the growth of spatial gradients.
2010, 26(4): 1197-1211
doi: 10.3934/dcds.2010.26.1197
+[Abstract](2755)
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Abstract:
We study the critical and super-critical dissipative quasi-geostrophic equations in $\R^2$ or $\T^2$. An optimal local smoothing effect of solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a main application, we establish the global well-posedness for the critical 2D quasi-geostrophic equations with periodic $H^1$ data. Some decay in time estimates are also provided.
We study the critical and super-critical dissipative quasi-geostrophic equations in $\R^2$ or $\T^2$. An optimal local smoothing effect of solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a main application, we establish the global well-posedness for the critical 2D quasi-geostrophic equations with periodic $H^1$ data. Some decay in time estimates are also provided.
2010, 26(4): 1213-1240
doi: 10.3934/dcds.2010.26.1213
+[Abstract](2471)
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Abstract:
The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in $\mathbb{R}^{d}$, $d \geq 2$, when the diffusivity parameter ε is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to ε, of the heat solution is obtained at all orders. It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of ε and ε1/2 must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.
The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in $\mathbb{R}^{d}$, $d \geq 2$, when the diffusivity parameter ε is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to ε, of the heat solution is obtained at all orders. It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of ε and ε1/2 must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.
2010, 26(4): 1241-1268
doi: 10.3934/dcds.2010.26.1241
+[Abstract](2620)
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Abstract:
This article examines a class of singular perturbation systems in the presence of a small white noise. Modifying the renormalization group procedure developed by Chen, Goldenfeld and Oono [6], we derive an associated reduced system which we use to construct an approximate solution that separates scales. Rigorous results demonstrating that these approximate solutions remain valid with high probability on large time scales are established. As a special case we infer new small noise asymptotic results for a class of processes exhibiting a physically motivated cancellation property in the nonlinear term. These results are applied to some concrete perturbation systems arising in geophysical fluid dynamics and in the study of turbulence. For each system we exhibit the associated renormalization group equation which helps decouple the interactions between the different scales inherent in the original system.
This article examines a class of singular perturbation systems in the presence of a small white noise. Modifying the renormalization group procedure developed by Chen, Goldenfeld and Oono [6], we derive an associated reduced system which we use to construct an approximate solution that separates scales. Rigorous results demonstrating that these approximate solutions remain valid with high probability on large time scales are established. As a special case we infer new small noise asymptotic results for a class of processes exhibiting a physically motivated cancellation property in the nonlinear term. These results are applied to some concrete perturbation systems arising in geophysical fluid dynamics and in the study of turbulence. For each system we exhibit the associated renormalization group equation which helps decouple the interactions between the different scales inherent in the original system.
2010, 26(4): 1269-1290
doi: 10.3934/dcds.2010.26.1269
+[Abstract](2415)
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Abstract:
In this paper, we consider the initial-value problem for the Degasperis-Procesi equation with a linear dispersion, which is an approximation to the incompressible Euler equation for shallow water waves. We establish local well-posedness and some global existence of solutions for certain initial profiles and determine the wave breaking phenomena for the equation. Finally, we verify the occurrence of the breaking waves by numerical simulations.
In this paper, we consider the initial-value problem for the Degasperis-Procesi equation with a linear dispersion, which is an approximation to the incompressible Euler equation for shallow water waves. We establish local well-posedness and some global existence of solutions for certain initial profiles and determine the wave breaking phenomena for the equation. Finally, we verify the occurrence of the breaking waves by numerical simulations.
2010, 26(4): 1291-1304
doi: 10.3934/dcds.2010.26.1291
+[Abstract](3321)
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Abstract:
We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy law. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic systems in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic systems. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous $C^0$ finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.
We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy law. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic systems in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic systems. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous $C^0$ finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.
2010, 26(4): 1305-1318
doi: 10.3934/dcds.2010.26.1305
+[Abstract](2678)
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Least-squares finite element methods for second order elliptic partial differential equations such as Darcy flows are considered. While there has been a significant progress in terms of obtaining error estimates for the methods, the estimates are essentially based on $L_2$-norm of the error. In this paper, we provide maximum norm error estimates for the primary variable using a smoothed Green's function introduced in [33] and maximum norm error for the dual variables by taking advantage of the fact that least-squares solutions are higher-order perturbations of Galerkin solutions [8].
Least-squares finite element methods for second order elliptic partial differential equations such as Darcy flows are considered. While there has been a significant progress in terms of obtaining error estimates for the methods, the estimates are essentially based on $L_2$-norm of the error. In this paper, we provide maximum norm error estimates for the primary variable using a smoothed Green's function introduced in [33] and maximum norm error for the dual variables by taking advantage of the fact that least-squares solutions are higher-order perturbations of Galerkin solutions [8].
2010, 26(4): 1319-1328
doi: 10.3934/dcds.2010.26.1319
+[Abstract](2731)
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Let $u$ be a local weak solution of the Navier-Stokes system in a space-time domain $D\subseteq\mathbb R^{n}\times\mathbb R$. We prove that for every $q>3$ there exists $\epsilon>0$ with the following property: If $(x_0,t_0)\in D$ and if there exists $r_0>0$ such that
Let $u$ be a local weak solution of the Navier-Stokes system in a space-time domain $D\subseteq\mathbb R^{n}\times\mathbb R$. We prove that for every $q>3$ there exists $\epsilon>0$ with the following property: If $(x_0,t_0)\in D$ and if there exists $r_0>0$ such that
sup $|x-x_0|+\sqrt{t-t_0} < r_0$ sup $r\in(0,r_0)$ $ \frac{1}{r^{n+2-q}} \int_{t-r^2}^{t+r^2}\ \ \ \int_{|y-x|\le r} |u(y,s)|^{q}\,dy\,ds \le \epsilon $
then the solution $u$ is regular in a neighborhood of $(x_0,t_0)$. There is no assumption on the integrability of the pressure or the vorticity.
2010, 26(4): 1329-1357
doi: 10.3934/dcds.2010.26.1329
+[Abstract](2562)
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Abstract:
In this work, we show how to construct a pullback exponential attractor associated with an infinite dimensional dynamical system, i.e., a family of time dependent compact sets, with finite fractal dimension, which are positively invariant and exponentially attract in the pullback sense every bounded set of the phase space. Our construction is based on the one in Efendiev et al. [11] in which a uniform forwards (and so also pullback) exponential attractor is constructed. We relax the conditions in [11] in order to obtain an unbounded family of exponential attractors for which the uniform convergence fails so that only the pullback attraction is expected. Thus, by proving that global pullback attractors are included in our family of exponential attractors, we generalize the concept of an exponential attractor to the theory of infinite dimensional non-autonomous dynamical systems. We illustrate our results on a 2D Navier-Stokes system in bounded domains.
In this work, we show how to construct a pullback exponential attractor associated with an infinite dimensional dynamical system, i.e., a family of time dependent compact sets, with finite fractal dimension, which are positively invariant and exponentially attract in the pullback sense every bounded set of the phase space. Our construction is based on the one in Efendiev et al. [11] in which a uniform forwards (and so also pullback) exponential attractor is constructed. We relax the conditions in [11] in order to obtain an unbounded family of exponential attractors for which the uniform convergence fails so that only the pullback attraction is expected. Thus, by proving that global pullback attractors are included in our family of exponential attractors, we generalize the concept of an exponential attractor to the theory of infinite dimensional non-autonomous dynamical systems. We illustrate our results on a 2D Navier-Stokes system in bounded domains.
2010, 26(4): 1359-1382
doi: 10.3934/dcds.2010.26.1359
+[Abstract](1751)
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We study orientational dynamics in sheared polymer-particulate nanocomposites (PNCs) using an approximate tensor model derived from the kinetic theory for flows of PNCs in the weak semiflexible regime. We focus on dynamics induced by shear and enhanced by the interaction between the nanoparticles and the host polymer matrix in a set of selected model parameters, highlighting solution behavior and bifurcations in the semidilute regime of PNCs. We witness the existence of logrolling states, out-of-plane steady states, kayaking and chaotic motions, and flow-aligning steady states of nanoparticle ensembles and corresponding steady states, time-periodic fluttering and chaotic motions in the host polymer matrix in various ranges of shear rates and selected model parameters. A striking feature observed in the polymer matrix is its collective orientational order follows a master curve as the shear rate varies despite that the nanoparticle ensemble experiences a variety of states and motions with widely varying local nematic order.
We study orientational dynamics in sheared polymer-particulate nanocomposites (PNCs) using an approximate tensor model derived from the kinetic theory for flows of PNCs in the weak semiflexible regime. We focus on dynamics induced by shear and enhanced by the interaction between the nanoparticles and the host polymer matrix in a set of selected model parameters, highlighting solution behavior and bifurcations in the semidilute regime of PNCs. We witness the existence of logrolling states, out-of-plane steady states, kayaking and chaotic motions, and flow-aligning steady states of nanoparticle ensembles and corresponding steady states, time-periodic fluttering and chaotic motions in the host polymer matrix in various ranges of shear rates and selected model parameters. A striking feature observed in the polymer matrix is its collective orientational order follows a master curve as the shear rate varies despite that the nanoparticle ensemble experiences a variety of states and motions with widely varying local nematic order.
2010, 26(4): 1383-1398
doi: 10.3934/dcds.2010.26.1383
+[Abstract](1806)
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I shall briefly survey the current status on more rigorous studies of chaos in fluids by focusing along the line of chaos phenotypes: sensitive dependence on initial data, and recurrence.
I shall briefly survey the current status on more rigorous studies of chaos in fluids by focusing along the line of chaos phenotypes: sensitive dependence on initial data, and recurrence.
2010, 26(4): 1399-1417
doi: 10.3934/dcds.2010.26.1399
+[Abstract](2150)
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In this article, we present a mathematical theory of the Walker circulation of the large-scale atmosphere over the tropics. This study leads to a new metastable state oscillation theory for the El Niño Southern Oscillation (ENSO), a typical inter-annual climate low frequency oscillation. The mathematical analysis is based on 1) the dynamic transition theory, 2) the geometric theory of incompressible flows, and 3) the scaling law for proper effect of the turbulent friction terms, developed recently by the authors.
In this article, we present a mathematical theory of the Walker circulation of the large-scale atmosphere over the tropics. This study leads to a new metastable state oscillation theory for the El Niño Southern Oscillation (ENSO), a typical inter-annual climate low frequency oscillation. The mathematical analysis is based on 1) the dynamic transition theory, 2) the geometric theory of incompressible flows, and 3) the scaling law for proper effect of the turbulent friction terms, developed recently by the authors.
2010, 26(4): 1419-1440
doi: 10.3934/dcds.2010.26.1419
+[Abstract](2331)
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Mathematical modeling and numerical simulation of smectic C liquid crystals which possess the spontaneous polarization are considered in this paper. In particular, the model allows for a system with a zero net polarization which is one of the ubiquitous systems of the polarized liquid crystals. Theoretical and numerical investigations are conducted to study effects of the energy associated with the polarization, switching patterns between two uniform states by an externally applied field and random noise, as well as a relation between polarization and applied field near the phase transition from the smectic A and smectic C.
Mathematical modeling and numerical simulation of smectic C liquid crystals which possess the spontaneous polarization are considered in this paper. In particular, the model allows for a system with a zero net polarization which is one of the ubiquitous systems of the polarized liquid crystals. Theoretical and numerical investigations are conducted to study effects of the energy associated with the polarization, switching patterns between two uniform states by an externally applied field and random noise, as well as a relation between polarization and applied field near the phase transition from the smectic A and smectic C.
2010, 26(4): 1441-1469
doi: 10.3934/dcds.2010.26.1441
+[Abstract](2105)
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We study asymptotic behavior of the Darcy-Boussinesq system at large Darcy-Prandtl number. We prove that the global attractors for this system converge to that of the infinite Darcy-Prandtl number model. We also show the convergence of statistical properties including invariant measures.
We study asymptotic behavior of the Darcy-Boussinesq system at large Darcy-Prandtl number. We prove that the global attractors for this system converge to that of the infinite Darcy-Prandtl number model. We also show the convergence of statistical properties including invariant measures.
2010, 26(4): 1471-1490
doi: 10.3934/dcds.2010.26.1471
+[Abstract](1970)
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We consider the Navier-Stokes system on R2. It is well-known that solutions with $L^2$ data become instantly smooth and persist globally. In this note, we show that the solution map is Lipschitz, when acting in $L^\infty $Hσ (R2) and $L^2_t$Hσ+1 (R2), when $0\leq $ σ<1. This generalizes an earlier result of Gallagher and Planchon [7], who showed the Lipschitzness in $L^2$(R2). The question for the Lipschitzness of the map in Hσ (R2), σ$\geq 1$ remains an interesting open problem, which hinges upon the validity of an endpoint estimate for the trilinear form $(\phi, v, w)\to \int$R2(∂Φ/∂x ∂v/∂y - ∂Φ/∂y ∂v/∂x)wdx.
We consider the Navier-Stokes system on R2. It is well-known that solutions with $L^2$ data become instantly smooth and persist globally. In this note, we show that the solution map is Lipschitz, when acting in $L^\infty $Hσ (R2) and $L^2_t$Hσ+1 (R2), when $0\leq $ σ<1. This generalizes an earlier result of Gallagher and Planchon [7], who showed the Lipschitzness in $L^2$(R2). The question for the Lipschitzness of the map in Hσ (R2), σ$\geq 1$ remains an interesting open problem, which hinges upon the validity of an endpoint estimate for the trilinear form $(\phi, v, w)\to \int$R2(∂Φ/∂x ∂v/∂y - ∂Φ/∂y ∂v/∂x)wdx.
2010, 26(4): 1491-1508
doi: 10.3934/dcds.2010.26.1491
+[Abstract](2236)
+[PDF](222.0KB)
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In this article, we study the existence and uniqueness of the global and periodic strong solutions of the primitive equations of the ocean. We prove that there exists a unique periodic strong solution provided that the heat source is regular and small enough. The proof of the existence is based on approximate solutions and a fixed point argument. We also derive some a priori estimates on the strong solutions.
In this article, we study the existence and uniqueness of the global and periodic strong solutions of the primitive equations of the ocean. We prove that there exists a unique periodic strong solution provided that the heat source is regular and small enough. The proof of the existence is based on approximate solutions and a fixed point argument. We also derive some a priori estimates on the strong solutions.
2010, 26(4): 1509-1523
doi: 10.3934/dcds.2010.26.1509
+[Abstract](3713)
+[PDF](216.9KB)
Abstract:
It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Kortewg-de Vries (KdV) equation and the damped Benjamin-Bona-Mahony (BBM) equation. In this paper we intend to show the same results hold for the pure KdV equation (without the damping terms) posed on a bounded domain. Consideration is given to the initial-boundary-value problem
It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Kortewg-de Vries (KdV) equation and the damped Benjamin-Bona-Mahony (BBM) equation. In this paper we intend to show the same results hold for the pure KdV equation (without the damping terms) posed on a bounded domain. Consideration is given to the initial-boundary-value problem
$ u_t +u_x +$uux$ +$uxxx$=0, \quad u(x,0)
=\phi (x), \qquad $ 0 < x < 1, t > 0, (*)
$ u(0,t) =h (t), \qquad
u(1,t)=0, \qquad u_x (1,t) =0, \quad t>0. $
It is shown that if the boundary forcing $h$ is periodic with small amplitude, then the small amplitude solution $u$ of (*) becomes eventually time-periodic. Viewing (*) (without the initial condition ) as an infinite-dimensional dynamical system in the Hilbert space $L^2(0,1)$, we also demonstrate that for a given periodic boundary forcing with small amplitude, the system (*) admits a (locally) unique limit cycle, or forced oscillation, which is locally exponentially stable. A list of open problems are included for the interested readers to conduct further investigations.
2010, 26(4): 1525-1536
doi: 10.3934/dcds.2010.26.1525
+[Abstract](2206)
+[PDF](201.0KB)
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This paper extends the dual-Petrov-Galerkin method proposed by Shen [21], further developed by Yuan, Shen and Wu [27] to general fifth-order KdV type equations with various nonlinear terms. These fifth-order equations arise in modeling different wave phenomena. The method is implemented to compute the multi-soliton solutions of two representative fifth-order KdV equations: the Kaup-Kupershmidt equation and the Caudry-Dodd-Gibbon equation. The numerical results imply that this scheme is capable of capturing, with very high accuracy, the details of these solutions such as the nonlinear interactions of multi-solitons.
This paper extends the dual-Petrov-Galerkin method proposed by Shen [21], further developed by Yuan, Shen and Wu [27] to general fifth-order KdV type equations with various nonlinear terms. These fifth-order equations arise in modeling different wave phenomena. The method is implemented to compute the multi-soliton solutions of two representative fifth-order KdV equations: the Kaup-Kupershmidt equation and the Caudry-Dodd-Gibbon equation. The numerical results imply that this scheme is capable of capturing, with very high accuracy, the details of these solutions such as the nonlinear interactions of multi-solitons.
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