A geometrically two-dimensional motion down a slope is considered. Let t be time, (x, z) be the coordinates, and (gx, gz) the gravity accelerations along and perpendicular to the slope, respectively. Let h and u be the flow depth and the mean flow velocity along the slope. Similarly, let γ, αs, and μ be the density ratio between the fluid and the particles (γ=ρf/ρs), volume fraction of the solid particles (coarse and fine solid particles), and the basal friction coefficient (μ=tan⁡δ), where δ is the basal friction angle, in the mixture material. Furthermore, K is the Earth pressure coefficient as a function of internal and basal friction angles, and CDV is the viscous drag coefficient.

We start with the multi-phase mass flow model (Pudasaini and Mergili, 2019) and include viscous drag (Pudasaini and Fischer, 2020). For simplicity, we assume that the relative velocity between coarse and fine solid particles (us, ufs) and the fluid phase (uf) in the landslide (debris) material is negligible: that is, us≈ufs≈uf=:u, and so is the viscous deformation of the fluid. This means, for simplicity, we are considering an effectively single-phase mixture flow. Then, by summing up the mass and momentum balance equations (Pudasaini and Mergili, 2019; Pudasaini and Fischer, 2020), we obtain a single mass and momentum balance equation describing the motion of a landslide as 1∂h∂t+∂∂x(hu)=0,2∂∂t(hu)+∂∂xhu2+(1-γ)αsgzKh22=hgx-(1-γ)αsgzμ-gz1-(1-γ)αs∂h∂x-CDVu2, where -(1-αs)gz∂h/∂x emerges from the hydraulic pressure gradient associated with possible interstitial fluids in the landslide. Moreover, the term containing K on the left-hand side and the other terms on the right-hand side in the momentum equation (Eq. ) represent all the involved forces. The first term in the square brackets on the left-hand side of Eq. () describes the advection, while the second term (in the square brackets) describes the extent of the local deformation that stems from the hydraulic pressure gradient of the free surface of the landslide. The first, second, third, and fourth terms on the right-hand side of Eq. () are the gravitational acceleration, effective Coulomb friction (which includes lubrication (1-γ), liquefaction (αs)) (because if there is no solid or a substantially low amount of solid, the mass is fully liquefied, e.g., lahar flows), the term associated with buoyancy and the fluid-related hydraulic pressure gradient, and the viscous drag, respectively. Note that the term with 1-γ or γ originates from the buoyancy effect. By setting γ=0 and αs=1, we obtain a dry landslide, grain flow, or an avalanche motion. For this choice, the third term on the right-hand side vanishes. However, we keep γ and αs to also include possible fluid effects in the landslide (mixture).

The momentum balance equation (Eq. ) can be rewritten as 3h∂u∂t+u∂u∂x+u∂h∂t+∂∂x(hu)=hgx-(1-γ)αsgzμ-gz((1-γ)K+γ)αs+1-αs∂h∂x-CDVu2. Note that for K=1 (which mostly prevails for extensional flows; Pudasaini and Hutter, 2007), the third term on the right-hand side associated with ∂h/∂x simplifies drastically because {((1-γ)K+γ)αs+(1-αs)} becomes unity. So, the isotropic assumption (i.e., K=1) loses some important information about the solid content and the buoyancy effect in the mixture. Employing the mass balance equation (Eq. ), the momentum balance equation (Eq. ) can be rewritten as 4∂u∂t+u∂u∂x=gx-(1-γ)αsgzμ-gz((1-γ)K+γ)αs+1-αs∂h∂x-CDVu2. The gradient ∂h/∂x might be approximated, say as hg, and still include its effect as a parameter that may be estimated. Here, we are mainly interested in developing a simple but more general landslide velocity model than the existing ones that can be solved analytically and highlight its essence to enhance our understanding of landslide dynamics.

Now, with the notation α:=gx-(1-γ)αsgzμ-gz{((1-γ)K+γ)αs+(1-αs)}hg (this includes the forces gravity, friction, lubrication, and liquefaction as well as the surface gradient) and β:=CDV (the viscous drag coefficient), Eq. () becomes a simple model equation: 5∂u∂t+u∂u∂x=α-βu2, where α and β constitute the net driving and the resisting forces in the system. We call Eq. () the landslide velocity equation.

Equation () constitutes a novel class of nonlinear advective–dissipative system and involves dynamic interactions between the nonlinear advective (or inertial) term u∂u/∂x and the external forcing (source) term α-βu2. However, in contrast to the viscous Burgers' equation wherein the dissipation is associated with the (viscous) diffusion, here, dissipation stems from the viscous drag, -βu2. In form, Eq. () is similar to the classical shallow-water equation. However, from the mechanics and the material composition, it is much wider as such a model does not exist in the literature. From the physical and mathematical point of view, there are two crucial novel aspects associated with the model (Eq. ). First, it explains the dynamics of a deforming landslide and thus extends the classical Voellmy model (Voellmy, 1955; Salm, 1966; McClung, 1983; Pudasaini and Hutter, 2007) due to the broad physics carried by the model parameters α and β as well as the dynamics described by the new term u∂u/∂x. These parameters and the term u∂u/∂x control the landslide deformation and motion. Second, it extends the classical nonlinear inviscid Burgers' equation by including the nonlinear source term, α-βu2, as a quadratic function of u, taking into account many different forces.

From the structure, Eq. () is a fundamental nonlinear partial differential equation, or a nonlinear transport equation with a source, where the source is the external physical forcing. Such an equation explains the nonlinear advection with source term that contains the physics of the underlying problem through the parameters α and β. The form of this equation is very important as it may describe the dynamical state of many extended (compared to the Voellmy and Burgers models) physical and engineering problems appearing in nature, science, and technology, including viscous–fluid flow, traffic flow, shock theory, gas dynamics, landslides, and avalanches (Burgers, 1948; Hopf, 1950; Cole, 1951; Nadjafikhah, 2009; Pudasaini, 2011; Montecinos, 2015).

For a sufficiently long time and sufficiently long slope, the time-independent steady-state motion can be developed. Then, Eq. () reduces to a simplified equation for the landslide velocity down the entire slope: 6∂∂x12u2=α-βu2. Equivalently, this also represents a mass point velocity along the slope. Classically, Eq. () is called the center of mass velocity of a dry avalanche of flow type (Perla et al., 1980) for γ=0, αs=1, K=1, and a negligible free-surface pressure gradient. This will be discussed in detail in Sect. 3.2.

Assume no or negligible local deformation (e.g., ∂u/∂x≈0) or a Lagrangian description; both are equivalent to the mass point motion. In this situation, only the ordinary differentiation with respect to time is involved, and ∂u/∂t can be replaced by du/dt. Then, the model (Eq. ) reduces to 10dudt=α-βu2. Perla et al. (1980) also called Eq. () the governing equation for the center of mass velocity but for a dry avalanche of flow type. This is a simple nonlinear first-order ordinary differential equation. This equation can be solved to obtain an exact analytical solution for the landslide velocity in terms of a tangent hyperbolic function (solution C): 11u(t;α,β)=αβtanh⁡αβt-t0+tanh⁡-1βαu0, where u0=u(t0) is the initial velocity at time t=t0. Equation () provides the time evolution of the velocity of the coherent (without fragmentation and substantial deformation) sliding mass until the time it fragments and/or moves like an avalanche. After that, we must use the full dynamical mass flow model (Pudasaini, 2012; Pudasaini and Mergili, 2019) or Eqs. () and (). For more detail, see Sect. 6.1. For a sufficiently long time, the viscous force brings the motion to a non-accelerating state (steady, uniform). Then, from Eq. () we obtain 12lim⁡t→∞u=αβ=:uTt, where uTt stands for the terminal velocity of the motion of a point mass.

Here, we present the detailed derivation of the analytical solution (Eq. ) to the landslide velocity equation (Eq. ). We derive the functions ϕ, ϕ-1, l, and loϕ that are involved in Theorem 4.1. The first function ϕ is given by 20ϕ(u)=∫1f(u)du=∫1α-βu2du=12αβln⁡α/β+uα/β-u. With the substitution τ=ϕ(u) (which implies u=ϕ-1(τ)), we obtain 21ϕ-1(τ)=αβexp⁡(2αβτ)-1exp⁡2αβτ+1=αβtanh⁡(αβτ). So, now the second function ϕ-1 can be written in terms of u. However, we must be consistent with the physical dimensions of the involved variables and functions. The quantities u, αβ, α/β, and τ have dimensions of m s-1, s-1, m s-1, and s. Thus, for dimensional consistency, the following mapping introduces a new multiplier λ with the dimension of 1 / m s-2. Therefore, we have 22ϕ-1(u)=αβtanh⁡λαβu. With this, the third function l(u) yields 23l(u)=∫gϕ-1(u)du=∫ϕ-1(u)du=αβ∫tanh⁡λαβudu=1λβln⁡cosh⁡λαβu. The fourth function l(ϕ(u))=(loϕ)(u) is instantly achieved: 24l(ϕ(u))=χλ1βln⁡cosh⁡(ξλ)αβϕ(u), where, as before, the multipliers χ and ξ emerge due to the transformation, and for dimensional consistency, they have the dimensions of 1 / (m s-2) and m s-2, respectively. The nice thing about the groupings (χ/λ) and (ξλ) is that they are now dimensionless and equal to unity.

Utilizing these functions in Theorem 4.1, we finally constructed the exact analytical solution (Eq. ).

The amazing fact is that the newly constructed general analytical solution (Eq. ) is strong and includes both the mass point solutions for velocity (Eq. ) and the position (Eq. ). For this, consider a vacuum solution F(0)≡0, which implies t=ϕ(u). Then, with the functional relation of ϕ(u) in Eq. (), we obtain 25u=αβtanh⁡αβt. Up to the constant of integration parameters (with u0=0 at t0=0), Eq. () is Eq. (). So, the first assertion is proved. Second, using F(0)≡0 and ϕ(u)=t in Eq. () immediately yields 26x=1βln⁡cosh⁡αβt. Again, up to the constant of integration parameters (with x0=0 and u0=0 at t0=0), Eq. () is Eq. (). This proves the second assertion.

Moreover, we mention that Eqs. () and () can also be obtained formally. This proves that the conditions used on F are legitimate. To see this, we differentiate Eq. () with respect to t to yield 27u=dxdt=αβtanh⁡αβϕ(u)dϕdt+F′t-ϕ(u)1-dϕdt. But, differentiating ϕ in Eq. () with respect to t and employing Eq. (), we obtain dϕ/dt=1, or ϕ=t. Now, by substituting these in Eqs. () and () we respectively recover Eqs. () and ().

However, we note that F in Eq. () is a general function. So, Eq. () provides a wide spectrum of analytical solutions for the landslide velocity as a function of time and space much wider than Eqs. () and ().

Here, we present some interesting particular exact solutions of Eq. () in the limit as β→0. For this purpose, first we consider Eq. () with β→0 and introduce the new variables t̃=αt and x̃=αx. Then, Eq. () can be written as 28∂u∂t̃+u∂u∂x̃=1. We apply Theorem 4.1 to Eq. (). So, f(u)=1 implies ϕ(u)=u, l(u)=u2/2, and l(ϕ(u))=u2/2. Following the procedure as for Eq. (), we obtain the solution to Eq. () as x̃=u22+F(t̃-u). However, the direct application of ϕ(u)=u in Eq. () leads to the solution (that is more complex in its form) x̃=1βln⁡[cosh⁡(βu)]+F(t̃-u). Then, in the limit, we must have 29lim⁡β→01βln⁡cosh⁡βu=u22. This is an important mathematical identity we obtained as a direct consequence of Theorem 4.1 and Eq. (). Furthermore, the identity (Eq. ) when applied to Eq. () implies 30lim⁡β→0x=lim⁡β→01βln⁡cosh⁡αβt=lim⁡β→01βln⁡cosh⁡βαt=12αt2. Thus, x=12αt2, which is the travel distance in time when the viscous drag is absent.

Moreover, with the definition of x̃, for the particular choice of F≡0, x̃=u22+F(t̃-u) results in u(x;α)=2αx, which is Eq. (). Furthermore, with the choice of x̃=0 and F=t̃-u, we obtain u=1-1-2αt, which for small t can be approximated as u≈αt. But, in the limit as β→0, Eq. () brings about u=αt, which is valid for all t values. Thus, Eq. () generalizes both solutions in Eqs. () and () in numerous ways.

Interestingly, by directly taking the limit as β→0, from Eq. () we obtain 31x=u22α+Ft-uα, which can be written as 32u2+2αFt-uα-2αx=0. Importantly, for any choice of the function F, Eq. () satisfies 33∂u∂t+u∂u∂x=α, which reduces to the classical inviscid Burgers' equation when α→0.

For a properly selected function F, Eq. () can be solved exactly for u. For example, consider a constant F, F=Λ. Then, an explicit exact solution is obtained as 34u=αβtanh⁡12exp⁡2cosh⁡-1exp⁡(β(x-Λ)). Figure  shows the velocity distribution given by Eq. () with u≈28 m s-1 at x=0 and Λ=0, which reaches the steady state at about x=150 m; this is faster than the solution given by Eq. () in Fig. . However, other more general solutions could be found by considering different F functions in Eq. ().

For example, with F=1βln⁡[ccosh⁡{αβ(t-ϕ(u))}], where c is a constant, Eq. () can be solved explicitly for u in terms of x and t: 35u=αβtanh⁡12cosh⁡-12cexp⁡(βx)-cosh⁡αβt+αβt. The velocity profile along the slope as given by Eq. () is presented in Fig.  for t=1 m s-1 and c=1. This solution is quite different to that in Fig.  produced by Eq. (). From the dynamical perspective, the solution (Eq. ) is better than the mass point solution (Eq. ). The important observation is that the solution given by Eq. () substantially overestimates the legitimate more general solution (Eq. ) that includes both the local time and space variation of the velocity field. The lower velocity with Eq. () corresponds to the energy consumption due to the deformation associated with the velocity gradient ∂u/∂x in Eq. (). This will be discussed in more detail in Sect. 4.6 and 4.7.

Furthermore, Fig.  presents the time evolution of the velocity field given by Eq. () for x=25 m and c=-2. This corresponds to the decelerating motion down the slope that starts with a very high velocity and finally asymptotically approaches the steady-state velocity.

A crucial aspect of a complex analytical solution is its proper interpretation. The general solution (Eq. ) can be plotted as a function of x and t. For the purpose of comparing the results with those derived previously, we select F as F=[Fk(t-ϕ(u))]pw+Fc with parameter values Fk=5000, Fc=-500, and pw=1/2. Furthermore, x is a parameter while plotting the velocity as a function of t. In these situations, in order to obtain a physically plausible solution, x0=-600 is selected. To match the origin of the mass point solution, in plotting, the time has been shifted by -2. Figure  depicts the two solutions given by Eq. () for the mass point motion and the general solution given by Eq. () that also includes the internal deformation associated with u∂u/∂x in Eq. (). They behave essentially differently right after the mass release. The mass point model substantially overestimates landslide velocity derived by the more realistic general model. However, the reduced dimensional models and solutions considered here may give upper bounds to reality because they do not account for the lateral spreading of the landslide mass. Such problems can only be solved comprehensively by considering the numerical simulations on a full three-dimensional digital terrain model (Mergili et al., 2020a, b; Shugar et al., 2021) and employing the full dynamical mass flow model equations (Pudasaini and Mergili, 2019) without constraining the lateral spreading.

The new general solution (Eq. ) and its plot in Fig.  provide a fundamentally new aspect in our understanding of landslide velocity. The physics behind the substantially, but legitimately, reduced velocity provided by the general velocity (Eq. ) compared to the mass point velocity (Eq. ) are revealed here for the first time. The gap between the two solutions increases steadily until a substantially large time (here about t=20 s); then the gap is reduced slowly. This is so because after t=20 s the mass point velocity is close to its steady value (about 60.1 m s-1). In the meantime, after t=20 s, the general velocity continues to increase but slowly, and after a long time, it also tends to approach the steady state. This substantially lower velocity in the general solution is realistic. Its mechanism can be explained. It becomes clear by analyzing the form of the model equation (Eq. ). For ease of analysis, we assume accelerating flow down the slope. For such a situation, both u and ∂u/∂x are positive, and thus u∂u/∂x>0. The model (Eq. ) can also be written as 36∂u∂t=α-βu2-u∂u∂x. Then, from the perspective of the time evolution of u, the last term on the right-hand side can be interpreted as a negative force additional to the system (Eq. ) describing the mass point motion. This is responsible for the substantially reduced velocity profile given by Eq. () compared to that given by Eq. (). The lower velocity in Eq. () can be perceived as the outcome of the energy consumed in the deformation of the landslide associated with the spatial velocity gradient that can also be inferred by the negative force attached with -u∂u/∂x in Eq. (). Moreover, u∂u/∂x in Eq. () can be viewed as the inertial term of the system (Bertini et al., 1994). However, after a sufficiently long time the drag is dominant, resulting in the decreased value of ∂u/∂x. Then, the effect of this negative force is reduced. Consequently, the difference between the mass point solution and the general solution decreases. However, these statements must be further scrutinized.

The solution method involves some sophisticated mathematical procedures. However, here we present a compact but quick solution description to our problem. The equivalent ordinary differential equation to the partial differential equation system (Eq. ) is 41du^dt=α-βu^2,u^(0)=s(0), which has the solution 42u^(t)=E(t,s(0))=αβtanh⁡αβt+tanh⁡-1βαs(0). Consider a curve x in the x–t plane that satisfies the ordinary differential equation 43dxdt=Et,s0(y)=αβtanh⁡αβt+tanh⁡-1βαs0(y),x(0)=y. Solving the system (Eq. ), we obtain 44x=y+Ft,s0(y)=y+1βln⁡cosh⁡αβt+tanh⁡-1βαs0(y)-1βln⁡cosh⁡tanh⁡-1βαs0(y). So, the exact solution to the problem (Eq. ) is given by 45u(x,t)=Et,s0(y)=αβtanh⁡αβt+tanh⁡-1βαs0(y), where y satisfies Eq. ().

It is interesting to observe the structure of the solutions given by Eqs. () and (). For a constant initial condition, e.g., s0(x)=λ0, s0(y)=λ0, Eqs. () and () are decoupled. Then, Eq. () reduces to 46u(x,t)=αβtanh⁡αβt+tanh⁡-1βαλ0. For t=0, u(x,0)=u0(x)=λ0, which is the initial condition. Furthermore, Eq. () takes the form 47x=x0+1βln⁡cosh⁡αβt+tanh⁡-1βαλ0-1βln⁡cosh⁡tanh⁡-1βαλ0, from which we see that for t=0, x=y=x0, which is the initial position. With this, we observe that Eqs. () and () are the mass point solutions in Eqs. () and (), respectively.

For the choice of the initial condition s0(x)=αβtanh⁡cosh⁡-1exp⁡(βx), combining Eqs. () and () leads to 48u(x,t)=αβtanh⁡cosh⁡-1exp⁡(βx), which, surprisingly, is the same as the initial condition. However, we can now legitimately compare Eq. () with the previously obtained solution (Eq. ), which is the steady-state motion with viscous drag. These two solutions are presented in Fig. . The very interesting fact is that Eqs. () and () turned out to be the same. For a real-valued parameter β and a real variable x, this reveals an important mathematically identity: 49tanh⁡cosh⁡-1exp⁡(βx)=1-exp⁡(-2βx). This means the very complex function on the left-hand side can be replaced by the much simpler function on the right-hand side. Moreover, taking the limit as β→0 in Eq. () and comparing it with Eq. (), we obtain another functional identity: 50lim⁡β→01βtanh⁡cosh⁡-1exp⁡(βx)=2x. These identities have mathematical significance.

Any initial condition can be applied to the solution system (Eqs.  and ). For the purpose of demonstrating the functionality of this system, here we consider two initial conditions: s0(x)=x0.50 and s0(x)=x0.65. The corresponding results are presented in Fig. . This figure clearly shows time marching of the landslide motion that also stretches as it slides down. Such deformation of the landslide stems from the term u∂u/∂x and the applied forces α-βu2 in our primary model (Eq. ). We will elaborate on this later. This proves our hypothesis on the importance of the nonlinear advection and external forcing for the deformation and motion of the landslide. The mechanism and dynamics of the advection, stretching, and approaching the steady state can be explained with reference to the general solution. For this, consider panel b with initial condition s0(x)=x0.65. At t=0.0 s, Eq. () implies that y=x; then from Eq. (), u(x,t)=s0(x), which is the initial condition. Such a velocity field can take place in a relatively early stage of the developed motion of large natural events (Erismann and Abele, 2001; Huggel et al., 2005; Evans et al., 2009; Mergili et al., 2018). This is represented by the t=0.0 s curve. For the next time, say t=2.0 s, the spatial domain of u expands and shifts to the right as defined by the rule (Eq. ). It has three effects in Eq. (). First, due to the shift of the spatial domain, the velocity field u is relocated to the right (downstream). Second, because of the increased t value and the spatial term associated with tanh⁡-1, the velocity field is elevated. Third, as the tanh⁡ function defines the maximum value of u (about 60.1 m s-1), the velocity field is controlled (somehow appears to be rotated). These dynamics also apply for t>2.0 s. These jointly produce beautiful spatiotemporal patterns in Fig. . Since the maximum of the initial velocity was already close to the steady-state value (the right end of the curve), the front of the velocity field is automatically and strongly controlled, limiting its value to 60.1 m s-1. So, although the rear velocity increases rapidly, the front velocity remains almost unchanged. After a sufficiently long time, t≥15 s, the rear velocity also approaches the steady-steady value. Then, the entire landslide moves downslope with virtually constant steady-state velocity, without any substantial stretching. We can similarly describe the dynamics for Fig. a. However, these two panels reveal an important fact: the initial condition plays an important role in determining and controlling the landslide dynamics.

The stretching (or deformation) of the landslide propagating down the slope depends on the evolution of its front (xf) and rear (xr) positions with maximum and minimum speeds, respectively. This is shown in Fig.  corresponding to the initial condition s0(x)=x0.65 in Fig. . It is observed that the rear position evolves strongly nonlinearly, whereas the front position advances only weakly nonlinearly.

In order to better understand the rate of stretching of the landslide, in Fig.  we also plot the difference between the front and rear positions as a function of time. It shows the stretching (rate) of the rapidly deforming landslide. The stretching dynamic is determined by the front and rear positions of the landslide in time, as shown in Fig. . In the early stages, the stretching increases rapidly. However, in later times (about t≥15 s) it increases only slowly, and after a sufficiently long time, (the rate of) stretching vanishes as the landslide has already been fully stretched. This can be understood because after a sufficiently long time, the motion is in steady state. Nevertheless, the ways the two solutions reach the steady state are different. The two panels in Fig.  also clearly indicate that the stretching (rate) depends on the initial condition.

The dynamics observed in Figs.  and  can be described with respect to the general model in Eq. () or () and its solution given by Eqs. () and (). The nice thing about Eq. () is that it can be analyzed in three different ways: with respect to the first or second or both terms on the right-hand side. If we disregard the first term involving time, then we explicitly see the effect of the second term that is responsible for the spatial variation of u for each time employed in Eq. (). This results in the shift of the solution for u to the right, and in the meantime, the solution stretches but without changing the possible maximum value of u (not shown). Stretching continues for higher times; however, for a sufficiently long time, it remains virtually unchanged. On the other hand, if we consider both the first and second terms on the right-hand side of Eq. () but use the initial velocity distribution only for a very small x domain, say [0, 1], then we effectively obtain the mass point solutions given in Figs.  and  corresponding to Eqs. () and (), respectively, for the spatial and time evolutions of u. This is so because now the very small initial domain for x essentially defines the velocity field as if it was for a center of mass motion. Then, as time elapses, the domain shifts to the right and the velocity increases. Now, plotting the velocity field as a function of space and time recovers the solutions in Figs.  and . In fact, if we collect all the minimum values of u (the left end points) in Fig. b and plot them in space and time, we acquire both the results in Figs.  and . These are effectively the mass point solutions for the spatial and time variation of the velocity field because these results only focus on the left end values of u, akin to the mass point motion. This means Eq. () together with Eq. () is responsible for the dynamics presented in Figs. – corresponding to the term u∂u/∂x and α-βu2 in the general model (Eqs.  or ). So, the dynamics are specially architectured by the advection u∂u/∂x and controlled by the system forcing α-βu2 through the model parameters α and β. This will be discussed in more detail in Sect. 5.7–5.9. This reveals the fact that the shifting, stretching, and lifting of the velocity field stem from the term u∂u/∂x in Eq. (). After a long time, as drag strongly dominates the other system forces, the velocity approaches the steady state, the velocity gradient practically vanishes, and thus the stretching ceases. Then, the landslide just moves down the slope at a constant velocity without any further dynamical complication.

It is compelling to see how the solution system (Eqs.  and ) rolls out an initially constant velocity across specific curves. For this, consider an initial velocity s0(x)=0 in a small domain, say [0, 3], and take a point in it. Then, generate solutions for different times beginning with t=0.0 s, with 2.0 s increments. As shown in Fig. , the space and time evolutions of the velocity fields for a mass point motion given by Eqs. () and () have been exactly rolled up and covered by the system (Eqs.  and ) by transporting the initial velocity along these curves (indicated by the star symbols). As explained earlier, the mechanism is such that, in time, Eq. () shifts the solution point (domain) to the right and Eq. () uplifts the velocity exactly lying on the mass point velocity curves designed by Eqs. () and (). So, the system (Eqs.  and ) generalizes the mass point motion in many different ways.

Next, we show how the new model (Eq. ) and its solution system (Eqs.  and ) can mold the breaking wave in mass transport and describe the folding of a landslide. For this, consider a sufficiently smooth initial velocity distribution given by s0(x)=5exp⁡(-x2/50). Such a distribution can be realized; e.g., as the landslide starts to move, its center might have been moving at the maximum initial velocity due to some localized strength weakening mechanism (examples include liquefaction, frictional strength loss, blasting, seismic shaking), and the strength weakening diminishes quickly away from the center. This later leads to a highly stretchable landslide from the center to the back, while from the center to the front, the landslide contracts strongly. The time evolution of the solution is presented in Fig. . Figure a is for the usual drag as before (β=0.0019), while Fig. b has higher drag (β=0.019). The drag strongly controls the wave breaking and folding, as well as the magnitude of the landslide velocity. Here, we focus on Fig. a, but similar analysis also holds for Fig. b.

Wave breaking and folding are often-observed important dynamical aspects in mass transport and formation of geological structures. Figure  reveals thrilling dynamics. The most fascinating feature is the velocity wave breaking and how this leads to the emergence of folding of the landslide. This can be explained with respect to the mechanism associated with the solution system (Eqs.  and ). As u∂u/∂x is positive to the left and negative to the right of the maximum initial velocity, the motion to the left of the maximum initial velocity overtakes the velocity to the right of the maximum position. As the position of the maximum velocity accelerates downslope with the fastest speed, after a sufficiently long time, a kink around the front of the velocity wave develops, here after t=2 s. This marks the velocity wave breaking (shock wave formation) and the beginning of the folding. However, the rear stretches continuously. Although mathematically folding may refer to a singularity due to a multi-valued function, here we explain the folding dynamics as a phenomenon that can appear in nature. In time, the folding intensifies, and the folding length increases, but the folding gap decreases. After a long time, the folding gap virtually vanishes and the landslide moves downslope at the steady-state velocity with a perfect fold in the frontal part (not shown), while in the back, it maintains a single large stretched layer. This happened collectively as the system (Eqs.  and ) simultaneously introduced three components of the landslide dynamics: downslope propagation, velocity uplift, and breaking or folding in the frontal part while stretching in the rear. This physically and mathematically proves that nonuniform motion (with its maximum somewhere interior to the landslide) is the basic requirement for the development of the breaking wave and the emergence of landslide folding.

As the external forcing vanishes, i.e., as α→0 and β→0, the landslide velocity equation (Eq. ) reduces to the classical inviscid Burgers' equation. Then, for α→0 and β→0, one would expect that the general solution (Eqs.  and ) should also reduce to the formation of the shock wave and wave breaking generated by the inviscid Burgers' equation. In fact, as shown in Fig. , this has exactly happened. For this, the solution domain remains fixed, and the solutions are not uplifted. This proves that Burgers' equation is a special case of our model (Eq. ).

It is important to understand the dynamic control of the viscous drag on the landslide motion. For this, we set α→0 but increased the value of the viscous drag parameter by 1 and 2 orders of magnitude. The results are shown in Fig. . In connection to Fig. , there are two important observations. First, the translation and stretching of the domain are solely dependent on the net driving force α, and when it is set to zero, the domain remains fixed. Second, the viscous drag parameter β effectively controls the magnitude of the velocity field and the wave breaking. Depending on the magnitude of the viscous drag coefficient, the generation of the shock wave and the wave breaking can be dampened (Fig. a) or fully controlled (Fig. b). Figure b further reveals that with a properly selected viscous drag coefficient, the new model can describe the deposition process of the mass transport and finally brings it to a standstill. In contrast to the classical inviscid Burgers' equation, due to the viscous drag effect, our model (Eq. ) is dissipative and can be recognized as a dissipative inviscid Burgers' equation. However, here the dissipation is not due to the diffusion but due to the viscous drag.

The new model may describe the complex dynamics of many extended physical and engineering problems appearing in nature, science, and technology – connecting different types of complex mass movements and deformations. Specifically, the advantage of the new model equation is that the more general landslide velocity can now be obtained explicitly and analytically, which is very useful in solving relevant engineering and applied problems, and it has enormous application potential.

There are three distinct situations in modeling the landslide motion: (i) the spatial variation of the flow geometry and velocity can be negligible for which the entire landslide effectively moves as a mass point without any local deformation. This refers to the classical Voellmy model. (ii) The geometric deformation of the landslide can be parameterized or neglected; however, the spatial variation of the velocity field may play a crucial role in the landslide motion. In this circumstance, the landslide motion can legitimately be explained by the full form of the new landslide velocity equation (Eq. ). The constructed general solutions in Eqs. () and ()–() of this model have revealed many important features of dynamically deforming and advecting landslide motions. (iii) Both the landslide geometry and velocity may substantially change locally. Then, no assumptions on the spatial gradient of the geometry and velocity can be made. For this, only the full set of the basic model equations (Eqs.  and ) can explain the landslide motion. While models and simulation techniques for situations (i) and (iii) are available in the literature, situation (ii) is entirely new both physically and mathematically. It is evident that dynamically situation (ii) plays an important role, first in making the bridge between the two limiting solutions and, second, by providing the most efficient solution. The solutions in Eqs. () and ()–() include the local deformation associated with the velocity gradient. However, except for parameterization, Eqs. () and ()–() do not explicitly include the geometrical deformation. As long as the spatial change in the landslide geometry is insignificant, we can use Eqs. () or () and () to describe the landslide motion. These solutions also include mass point motions and are valid before fragmentation and/or significant to large geometric deformations. However, when the geometric deformations are significant, we must use Eqs. () and () and solve them numerically with some high-resolution numerical methods (Tai et al., 2002; Mergili et al., 2017; 2020a, b).

The models in Eq. () or Eqs. ()–() and Eqs. ()–() are amicable and can be directly coupled. Such a coupling between the geometrically negligibly or slowly deforming landslide motion described by Eq. () or Eqs. ()–() and the full dynamical solution with any large to catastrophic deformations described by Eqs. () and () is novel. First, this allows us to consistently couple the negligible or slowly deformable landslide with a fast (or rapidly) deformable flow-type landslide (or debris flow). Second, our method provides a very efficient simulation due to the instant exact solution given by Eq. () or Eqs. () and () prior to the large external geometric deformation that is then linked to the full model equations (Eqs.  and ). Computational software such as r.avaflow (Mergili et al., 2017, 2020a, b; Pudasaini and Mergili, 2019) can substantially benefit from such a coupled solution method. Third, importantly, this coupling is valid for single-phase or multi-phase flows because the corresponding model (Eq. ) is derived by reducing the multi-phase mass flow model (Pudasaini and Mergili, 2019).

Burgers' equation has no external forcing term. The solution domain remains fixed and does not stretch and propagate downslope. So, the initial velocity profile deforms and the wave breaks within the fixed domain. In contrast, our model (Eq. ) is fundamentally characterized and explained simultaneously by the nonlinear advection u∂u/∂x and external forcing, α-βu2. The first designs the main dynamic feature of the wave, while the latter induces rapid downslope propagation, stretching of the wave domain, and quantification of the wave form and magnitude. These special features of our model are often-observed phenomena in mass transport and are freshly revealed here.

Within their scopes and structures, many of the analytical solutions constructed in Sects. 3–5 are similar. This effectively implies the physical aspects of our general landslide velocity model (Eq. ) and also the compatibility and reliability of all the solutions. The solutions in Eqs. () and ()–() recover all the mass point motions given by Eqs. () and (). From the physical and dynamical point of view, the velocity profiles given by Eqs. () and ()–() as solutions of the general model for the landslide velocity (Eq. ) are much wider and better than those given by Eqs. () and () as solutions of the mass point model (Eq. ).

Structurally, the solutions presented in Sect. 3 are only partly new, yet they are physically substantially advanced. However, in Sects. 4 and 5 we have presented entirely novel solutions both physically and structurally. From a physical and mathematically point of view, particularly important is the form of the general velocity model (Eq. ). First, it extends the classical Voellmy mass point model (Voellmy, 1955) by including (i) much wider physical aspects of landslide types and motions as well as (ii) the landslide dynamics associated with the internal deformation as described by the spatial velocity gradient associated with the advection. Second, the model (Eq. ) is the direct extension of the inviscid Burgers' equation by including a (quadratic) nonlinear source as a function of the state variable. This source term contains all the applied forces appearing from the physics and mechanics of the landslide motion.

Moreover, as viewed from the general structure of the model (Eq. ), all the solutions constructed here can be utilized for any physical problems that can be cast and represented in form (Eq. ), but independent of the definition of the model parameters α and β, as well as the state variable u (Faraoni, 2022).

The new model (Eq. ) and its solutions have broad implications mathematically, physically, and technically. By deriving a general landslide velocity model and its various analytical exact solutions, we made a breakthrough in correctly determining the velocity of a deformable landslide that is controlled by several applied forces as it propagates down the slope. We achieve a novel understanding that the inertia and the forcing terms ultimately regulate the landslide motion and provide a physically more appropriate analytical description of landslide velocity, dynamic impact, and inundation. This addresses the long-standing scientific question of explicit and full analytical representation of the velocity of deformable landslides. Such a description of the state of landslide velocity is innovative.

As the analytically obtained values represent the velocity of natural landslides well, technically, this provides a very important tool for landslide engineers and practitioners in quickly, efficiently, and accurately determining landslide velocity. The general solutions presented here reveal an important fact: accurate information about the mechanical parameters, state of the motion, and initial condition is very important for the proper description of the landslide motion. We have extracted some interesting particular exact solutions from the general solutions. As direct consequences of the new general solutions, some important and nontrivial mathematical identities have been established that replace very complex expressions by straightforward functions.