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We show that the Cauchy problem for the Camassa–Holm equation has a unique, global, weak, and dissipative solution for any initial data u 0 ∈ H 1 (R) , such that u 0 , x is bounded from above almost everywhere. In particular, we establish a one-to-one correspondence between the properties specific to the dissipative solutions and a solution operator associating to each initial data exactly one solution. [ABSTRACT FROM AUTHOR]

In this article, we mainly prove low regularity conservation laws for the Fokas-Lenells equation in Besov spaces with small initial data both on the line and on the circle. We develop a new technique in Fourier analysis and complex analysis to obtain the a priori estimates. It is based on the perturbation determinant associated with the Lax pair introduced by Killip, Vişan, and Zhang for completely integrable dispersive partial differential equations. Additionally, we also utilize the perturbation determinant to derive the global a priori estimates for the Schwartz solutions to the Camassa-Holm (CH) equation in H1{H}^{1}. Even though the energy conservation law of the CH equation is a fact known to all, the perturbation determinant method indicates that we cannot get any conserved quantities for the CH equation in Hk{H}^{k} except k=1k=1.

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa–Holm equation on the real-line case. We show that the data-to-solution map of the Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces B p , r s (R) with s > 1 2 and 1 ≤ p , r < ∞ , which improves the previous works Himonas et al. (2007) [23] , Li et al. (2020) [32] and Li et al. (2021) [31]. Furthermore, we present a strengthening of our previous work in Li et al. (2020) [32] and prove that the data-to-solution map for the Camassa–Holm equation is nowhere uniformly continuous in B p , r s (R) with s > max ⁡ { 1 + 1 / p , 3 / 2 } and (p , r) ∈ [ 1 , ∞ ] × [ 1 , ∞). The method applies also to the b-family of equations which contain the Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue for the Camassa-Holm equation y ″ = 1 4 y + λ m (t) y , with the Neumann-Dirichlet boundary conditions, where potential m admits to change sign. We first study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations. Then based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for this indefinite Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term. [ABSTRACT FROM AUTHOR]

Based on the ∂ ‾ -generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space H 4 , 2 (R). With a new scale (y , t) and a Riemann-Hilbert problem associated with the initial value problem, we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane { (y , t) : − ∞ < y < ∞ , t > 0 } is divided into four asymptotic regions: 1. Fast decay region, y / t ∈ (− ∞ , − 1 / 4) with an error O (t − 1 / 2) ; 2. Modulation-soliton region, y / t ∈ (2 , + ∞) , the result can be characterized with an modulation-solitons with residual error O (t − 1 / 2) ; 3. Zakhrov-Manakov region, y / t ∈ (0 , 2) and y / t ∈ (− 1 / 4 , 0). The asymptotic approximation is characterized by the dispersion term with residual error O (t − 3 / 4) ; 4. Two transition regions, | y / t | ≈ 2 and | y / t | ≈ − 1 / 4 , the asymptotic results are described by the solution of Painlevé II equation with error order O (t − 1 / 2). [ABSTRACT FROM AUTHOR]

The Camassa-Holm equation is a shallow water model possessing solutions breaking in finite time, whereas from a geometric viewpoint, its smooth solutions determine pseudospherical surfaces. Despite having been known for a long time, these two different aspects have never been considered conjunctively. As a result, it is not know if a solution developing breaking wave can also determine a pseudospherical surface nor, that being so, whether or how the singularities of the solutions can be, or even are always, inherited by the corresponding first fundamental form. We prove that certain solutions, with enough regularity and developing breaking wave, determine a pseudospherical surface defined on a strip of finite height whose corresponding metric tensor blows up near the strip's boundary. To this end, we introduce the notion of pseudospherical surface modelled by a function space and reformulate the notion of generic solution of an equation describing pseudospherical surfaces. These two new viewpoints enable us to proceed with a geometric analysis study of the surfaces determined by solutions of Cauchy problems involving the Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

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We study special properties of solutions to the IVP associated to theCamassa-Holm equation on the line related to the regularity and the decay ofsolutions. The first aim is to show how the regularity on the initial data istransferred to the corresponding solution in a class containing the "peakonsolutions". In particular, we shall show that the local regularity is similarto that exhibited by the solution of the inviscid Burger's equation with thesame initial datum. The second goal is to prove that the decay results obtainedin a paper of Himonas, Misio{\l}ek, Ponce, and Zhou extend to the class ofsolutions considered here.

In this study, we use cutting-edge analytical and numerical approaches to the Gilson–Pickering (GP) problem in order to get precise soliton solutions. This model explains wave propagation in plasma physics and crystal lattice theory. A variety of evolution equations have been developed from the GP model, including the Fornberg–Whitham (FW) equation, the Rosenau–Hyman (RH) equation, and the Fuchssteiner-Fokas-Camassa–Holm (FFCH) equation, to name a few. The GP model has been studied using these evolution equations. To investigate the characterizations of new waves, crystal lattice theory and plasma physics use the Khater II, and He’s variational iteration approaches. Many alternative responses may be achieved by utilizing various formulae; each of these solutions is shown by a distinct graph. The validity of such methods and solutions may be demonstrated by assessing how well the relevant techniques and solutions match up. The results of this study suggest that the technique is preferred for successfully resolving nonlinear equations that emerge in mathematical physics.

In this paper, the extended simple equation method (ESEM) and the generalized Riccati equation mapping (GREM) method are applied to the nonlinear third-order Gilson–Pickering (GP) model to obtain a variety of new exact wave solutions. With the suitable selection of parameters involved in the model, some familiar physical governing models such as the Camassa–Holm (CH) equation, the Fornberg–Whitham (FW) equation, and the Rosenau–Hyman (RH) equation are obtained. The graphical representation of solutions under different constraints shows the dark, bright, combined dark–bright, periodic, singular, and kink soliton. For the graphical representation, 3D plots, contour plots, and 2D plots of some acquired solutions are illustrated. The obtained wave solutions motivate researchers to enhance their theories to the best of their capacities and to utilize the outcomes in other nonlinear cases. The executed methods are shown to be practical and straightforward for approaching the considered equation and may be utilized to study abundant types of NLEEs arising in physics, engineering, and applied sciences. [ABSTRACT FROM AUTHOR]

A solitary wave is the dispersion-less solution of nonlinear evolutionary equations that travels at a constant speed without dissipating its energy. The purpose of this article is to provide insight into the discovery and history of solitons. The different types of the solitons are discussed in brief that is helpful for the researchers. For the discussion of the nature of solitons, the solution behavior of the Korteweg de Vries equation (KdV), the sine-Gordon (SG), the Camassa-Holm (CH) equation, and the nonlinear Schrodinger (NLS) equation are considered. This article deals with the various applications of solitons in different fields such as biophysics, nonlinear optics, Bose-Einstein condensation, plasma physics, Josephson junction, etc. focusing on the properties of solitons based on their classification.

In Zhang et al. (2007) and Zhang (2021) we constructed all single-peak traveling wave solutions of the Camassa–Holm equation including some explicit solutions. In general it is a challenge to construct exact multi-peak traveling wave solutions. As an example a periodic traveling wave (or wavetrain), a special type of spatiotemporal oscillation that is a periodic function of both space and time, plays a fundamental role in many mathematical equations such as shallow water wave equations. In this paper we will construct some new exact periodic traveling wave solutions of the Camassa–Holm equation.

For the famous Camassa-Holm equation, the well-posedness in B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞) and the ill-posedness in B p , r 1 + 1 p (R) with p ∈ [ 1 , ∞ ] , r ∈ (1 , ∞ ] had been studied in [13,14,16,23] , that is to say, it only left an open problem in the critical case B ∞ , 1 1 (R) proposed by Danchin in [13,14]. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in B ∞ , 1 1 (R). Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critical Besov spaces B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞ ] have been completed. Finally, since the norm inflation occurs by choosing an special initial data u 0 ∈ B ∞ , 1 1 (R) but u 0 x 2 ∉ B ∞ , 1 0 (R) (an example implies B ∞ , 1 0 (R) is not a Banach algebra), we then prove that this condition is necessary. That is, if u 0 x 2 ∈ B ∞ , 1 0 (R) holds, then the Camassa-Holm equation has a unique solution u (t , x) ∈ C T (B ∞ , 1 1 (R)) ∩ C T 1 (B ∞ , 1 0 (R)) and the norm inflation will not occur. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

In this paper, we give a construction of u 0 ∈ B p , ∞ σ such that the corresponding solution to the Camassa-Holm equation starting from u 0 is discontinuous at t = 0 in the metric of B p , ∞ σ , which implies the ill-posedness for this equation in B p , ∞ σ. We also apply our method to the b-equation and Novikov equation. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study several problems on the weakly dissipative periodic Camassa-Holm type equation with quadratic and cubic non-linearities. First, in the periodic setting, the local well-posedness of solutions in the Sobolev space is established via Kato's theory. Then, we establish a relation between the behavior of u x and wave breaking phenomena of solutions, and give a sufficient condition on the initial data to guarantee the occurrence of wave breaking. Finally, we obtain the global existence of solutions. [ABSTRACT FROM AUTHOR]

We analyze a singularly Kuramoto-Sivashinsky perturbed Camassa-Holm equation with methods of the geometric singular perturbation theory. Especially, we study the persistence of smooth and peaked solitons. Whether a solitary wave of the original Camassa-Holm equation is smooth or peaked depends on whether there is linear dispersion, i.e. whether 2 k = 0. If 2 k > 0 , then a unique smooth solitary wave persists with selected wave speed under singular Kuramoto-Sivashinsky perturbation just as what happens in the KS-KdV equation. On the other hand, we show that if there is no linear dispersion, i.e. 2 k = 0 , then any observable peaked soliton fails to persist. This case is non-typical since the related slow manifold blows up and the classical geometric singular perturbation theory is not available. [ABSTRACT FROM AUTHOR]

In this paper, the extended simple equation method (ESEM) and the generalized Riccati equation mapping (GREM) method are applied to the nonlinear third-order Gilson–Pickering (GP) model to obtain a variety of new exact wave solutions. With the suitable selection of parameters involved in the model, some familiar physical governing models such as the Camassa–Holm (CH) equation, the Fornberg–Whitham (FW) equation, and the Rosenau–Hyman (RH) equation are obtained. The graphical representation of solutions under different constraints shows the dark, bright, combined dark–bright, periodic, singular, and kink soliton. For the graphical representation, 3D plots, contour plots, and 2D plots of some acquired solutions are illustrated. The obtained wave solutions motivate researchers to enhance their theories to the best of their capacities and to utilize the outcomes in other nonlinear cases. The executed methods are shown to be practical and straightforward for approaching the considered equation and may be utilized to study abundant types of NLEEs arising in physics, engineering, and applied sciences.

For s < 3 / 2 , it is known (see [24]) that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces H s when b > 1. The proof of ill-posedness depends naturally on the value of b , and is based on the construction of peakon–antipeakon solutions with interesting properties which allows to make conclusion on ill-posedness. In this context the construction of such type of the solution for b < 1 is very attractive problem which help shed light on the ill-posedness problem in this case. Thus, in the present paper we consider the ill-posedness of the b-family of equations with additional term for insufficiently investigated case b < 1 on the line. [ABSTRACT FROM AUTHOR]

In the paper, we consider the initial value problem to the Camassa-Holm equation in the real-line case. Based on the local well-posedness result and the lifespan, we proved that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces in the sense of Hadamard. This result improves considerably the previous work given by Himonas–Kenig [23]. [ABSTRACT FROM AUTHOR]

The Degasperis-Procesi equation is an approximating model of shallow-water wave propagating mainly in one direction to the Euler equations. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations with the same asymptotic accuracy, and is completely integrable with the bi-Hamiltonian structure. In the present study, we establish existence and spectral stability results of localized smooth solitons to the Degasperis-Procesi equation on the real line. The stability proof relies essentially on refined spectral analysis of the linear operator corresponding to the second-order variational derivative of the local Hamiltonian of the Degasperis-Procesi equation. [ABSTRACT FROM AUTHOR]

It is well-known that peakons in the Camassa–Holm equation are H 1 -orbitally stable thanks to conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise C 1 perturbations to peakons grow in time in spite of their stability in the H 1 -norm. We also show that the linearized stability analysis near peakons contradicts the H 1 -orbital stability result, hence passage from linear to nonlinear theory is false in H 1. [ABSTRACT FROM AUTHOR]

Here we shall discuss analyticity results for several important partial differential equations. This includes the analytic regularity of sub-Laplacians under the finite type condition; the analyticity of the solution in both variables to the Cauchy problem for the Camassa–Holm equation with analytic initial data by using the Ovsyannikov theorem, which is a Cauchy–Kowalevski type theorem for nonlocal equations; the Cauchy problem for BBM with analytic initial data; the Cauchy problem for KdV with analytic initial data examining the evolution of uniform radius of spatial analyticity; and finally the time regularity of KdV solutions, which is Gevrey 3. [ABSTRACT FROM AUTHOR]

The fractional Camassa-Holm equation is generally used as a powerful tool in computer simulations of water waves in shallow water, coastal and harbor models. In this paper, new wave solutions of this equation are obtained by using a new extended direct algebraic method. Thirty-six completely new solutions are obtained and are graphically represented. These solutions may motivate future research on the topic. [ABSTRACT FROM AUTHOR]

In this article, we sutdy a multi-component Camassa-Holm-type system. By employing the characteristic method, we obtain a class of perturbational self-similar solutions with elliptical symmetry, whose velocity components are governed by the generalized Emden equations. In particular, when n=1, these solutions constitute a generalization of that obtained by Yuen in [38]. Interestingly, numerical simulations show that the analytical solutions obtained can be used to describe the drifting phenomena of shallow water flows. In addition, the method proposed can be extended to other mathematical physics models such as higher-dimensional Hunter-Saxton equations and Degasperis-Procesi equations.

The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation \[u_t-\alpha^2 u_{xxt}+2\omega u_x +3uu_x+\gamma u_{xxx}=\alpha^2 (2u_x u_{xx} + uu_{xxx})\] with zero boundary conditions (as \(|x|\to\infty\)) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented.

The Camassa–Holm equation can be used in fluids and other fields. Under investigation in this paper, the bilinear form, implicit soliton solution and multi-peakon solution of the generalized nonautonomous Camassa–Holm equation under constraints are derived. Based on these, time varying influence factors of solution amplitude, velocity and background are discussed, which are caused by inhomogeneity of boundaries and media. Furthermore, the phenomena of nonlinear tunnelling, soliton collision and split are constructed to show the characteristic of nonautonomous solitons and peakons in the propagation. [ABSTRACT FROM AUTHOR]

In this paper we construct global weak conservative solutions of the Camassa–Holm (CH) equation on the periodic domain. We first express the equation in Lagrangian flow variable η and then transform it using the change of variables ρ = η x . The new variable removes the singularity of the CH equation, and we obtain both global weak conservative solutions and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter–Saxton equation using the geometric interpretation. [ABSTRACT FROM AUTHOR]

Abstract We study herein the Camassa–Holm-type equation, which can be considered as a model in the shallow water for the long-crested waves propagating near the equator with effect of the Coriolis force due to the Earth's rotation. This quasi-linear equation is nonlocal with higher-order nonlinearities compared to the classical Camassa–Holm equation. We establish the global existence and uniqueness of the energy conservative weak solutions in the energy space H 1 to this model equation. [ABSTRACT FROM AUTHOR]

In this research, our purpose is to investigate some types of solutions to a simpli- fied modified form of the Camassa-Holm equation. The Jacobi elliptic function expansion method is applied to this equation. Then, a lot of travelling wave solutions are obtained. The derived solutions are in the form of Jacobi elliptic functions, hyperbolic functions, and trigonometric functions. Graphics of solutions are drawn in order to determine the types of the solutions. Furthermore, different kinds of solutions such as the singular kink wave solution, the kink wave solution, and the periodic solution are achieved. [ABSTRACT FROM AUTHOR]

Abstract We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space H 3 / 2 and even in the Besov space B p , r 1 + 1 / p for p ∈ [ 1 , ∞ ] , r ∈ (1 , ∞ ]. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5,14,16]). [ABSTRACT FROM AUTHOR]

The aim of this paper is to answer the question left in Gui and Liu (2015). We prove that given initial data u 0 ∈ H s (R) with s > 3 2 and for some T > 0 , the solution of the Camassa–Holm equation converges strongly in L ∞ (0 , T ; H s (R)) to the inviscid Burgers equation as the filter parameter α tends to zero. [ABSTRACT FROM AUTHOR]

An important point in looking for period solutions of the Camassa–Holm equation is to understand the associated spectral problem y ″ = 1 4 y + λ m ( t ) y . The first aim of this paper is to study the dependence of eigenvalues for the periodic Camassa–Holm Equation on potentials as an infinitely dimensional parameter. To be precise, we prove that as nonlinear functionals of potentials, eigenvalues for the periodic Camassa–Holm Equation are continuous in potentials with respect to the weak topologies in the L p Lebesgue spaces. The second aim of this paper is to find the optimal lower bound of the lowest eigenvalue for the periodic Camassa–Holm Equation when the L 1 norm of potentials are given. In order to make our results more applicable, we will find the optimal lower bound for the lowest eigenvalue in the more general setting of measure differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the infinite propagation speed and the asymptotic behavior for a generalized Camassa–Holm equation with cubic nonlinearity. First, we get the infinite propagation speed in the sense that the corresponding solution with compactly supported initial data does not have compact support any longer in its lifespan. Then, the asymptotic behavior of the solution at infinity is investigated. Especially, we prove that the solution decays algebraically with the same exponent as that of the initial data. [ABSTRACT FROM AUTHOR]

The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa−Holm (CH), modified Camassa−Holm (mCH), and Degasperis−Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering.

In this study, we consider the Cauchy problem for the second-order Camassa–Holm equation with periodic initial data u 0 . Using the vanishing viscosity method, the local weak solution of the equation is obtained in the finite energy space. A continuous semigroup of weak conservative solutions in Lagrangian coordinates is constructed. In particular, for any two solutions u ( t ) and v ( t ) of the equation, a Lipschitz metric d D is constructed with the property that d D ( u ( t ) , v ( t ) ) ≤ e C t d D ( u 0 , v 0 ) . [ABSTRACT FROM AUTHOR]

Global analytic solution in both the time and the space variables is proved for the Cauchy problem of a generalized CH equation, which contains as its members two integrable equations, namely the Camassa–Holm and the Novikov equations. The main assumptions are that the initial datum u 0 ( x ) is analytic on the line, it has uniform radius of analyticity r ( 0 ) > 0 , and is such that the McKean quantity m 0 ( x ) ≐ ( 1 − ∂ x 2 ) u 0 ( x ) does not change sign. Furthermore, an explicit lower bound on the radius of space analyticity at later times is obtained, which is of the form L 3 exp ⁡ ( − L 1 exp ⁡ ( L 2 t ) ) , where L 1 , L 2 and L 3 are appropriate positive constants. [ABSTRACT FROM AUTHOR]

We present a multi-component modified Yajima- Oikawa hierarchy and a multi-component Novikov hierarchy. We get generalized Hamiltonian structures for the two hierarchies with the aid of trace identity. [ABSTRACT FROM AUTHOR]

We study the blow-up phenomena of the strong solutions for the Fokas–Qiao–Xia–Li (FQXL) equation with quadratic and cubic nonlinearities, which include the celebrated Camassa–Holm equation and the Fokas–Olver–Rosenau–Qiao (FORQ) equation as its special case. The FQXL equation models the unidirectional propagation of the shallow water waves over a flat bottom and describes the physics more accurately than its celebrated counterpart, the Gardner equation. Two kinds of conditions on the initial data, including cases where the initial momentum density is not of one sign, are formulated to guarantee finite time blow-up for the strong solutions for this equation. The first condition relies on the sign persistence property of the momentum density m . The second condition is based on the full use of the conservation laws of the equation, especially the conservation quantity H 2 [ u 0 ] . [ABSTRACT FROM AUTHOR]